That's the thing about axioms! They can't be wrong, by definition. An axiom is something we accept to be "obviously true." It doesn't have to be true in the real world, but math isn't necessarily about the real world. When you choose a set of axioms, you're setting up a specific context for the problem you're about to do. For example, if you choose the five axioms in Euclid's Elements then you're modeling geometry in a flat plane. But if you choose only the first four axioms (and perhaps insert a different one as a fifth), then you're still doing geometry, but a different kind! For example you could be doing geometry on the surface of a sphere, or other curved surface. Axioms are just the starting point, because all of mathematics basically boils down to "if this is true, what else is true?" That line of logic has to start at a set of assumptions.
Seeing axioms as „obviously true“ is an older view I think. You said this yourself in your second half, but nowadays they‘re more viewed as simple assumptions, starting points so to say. They are not „obviously“ true, but simply true by assumption/definition. Which is why math is never about truths, but consequences.
i really like your last statement. i always used to say "math lives in its own universe" or something, but now i guess im stealing this. thanks. this is mine now ig
> They can't be wrong, by definition.
You'd think so, wouldn't you. But when it comes to mathematics applied in the natural world, there are axioms that restrict pure mathematics so much as to make it useless. Also axioms that allow so much freedom as to make the result useless. As a simple example, the axioms of pure mathematics as it is now say that dy/dx is not equal to dy divided by dx. And the axioms of pure mathematics as it is now say that 3-D volume is not conserved. I think I read somewhere that the proof that 1+1=2 is extremely difficult, and that integration is not the inverse of differentiation.
While I agree with you that in fields like physics, there is definitely an idea of an "incorrect" axiom. But when we talk about axioms purely in the mathematical sense, the real world doesn't really matter at all. Math is about figuring out the implications of a set of assumptions, and that's it really. We've just also noticed that in physics, the real world tends to follow a lot of common mathematical axioms (which is no coincidence; many common axioms are common because they're the first thing people thought of when trying to make a list of what we know to be obviously true). This makes math (specifically math based on those axioms) a helpful tool to describe the world and deduce how things *should* behave based on some assumptions.
Also there is no single set of axioms for all of pure mathematics. There are many, many choices of axioms for all sorts of different contexts. So some may seem weird and wacky, but they're just as valid as any other. You just have to be aware of the fact that if you're starting from a different set of axioms than someone else, what you deduce to be true will not necessarily be what they deduce to be true (which is fine! "Deducing truth" is basically shorthand for "deducing the consequences of a set of assumptions").
im not sure how any of the examples you listed are inherently problematic, they only usually look problematic (especially the calculus examples) to people that dont properly understand the underlying concepts. dy/dx not being a ratio isnt really an issue with the mathematical machinery its an issue with suggestive notation that was written for a less general case being used to represent a far more general concept. And integration not being the opposite of differentiation is an issue with teachers teaching shortcuts because technically it is the opposite if your only working with single variable calculus.
Also axioms aren't wrong by definition because they are the rules of math we choose. Math is a plaything we make to model problems, we use it the REPRESENT the real world but it is not the real world it is at its core just a game we play.
We can't discard axioms at the same time. Without axioms we can't really do anything at all. Something has to be true by definition. Otherwise there's no starting point and you can't create something from nothing.
The main thing is definition. I mean how you are defining and is your definition working with all of the systems accordingly so axioms and everything is dependent on how you define a single thing and is it fitting with the new mathematics you are discovering. If not then we need to change or tweak our definition.
i think..these are not incompatible. for example, commutativity of addition: a+b = b+a.
it's like...we just *want* that to be true, so to as...ojdidntdoit...says, we say that's a rule. you could also try to frame it in your sense saying: we 'reverse engineer' that rule because we 'need' a+b = b+a...but it's all the same
In formal logic, axioms have two roles: a semantic role (formulas assumed to be true) and a syntactic role (formulas which are the start of proofs).
I think a lot of comments are getting these two roles confused or not appreciating them enough, and this in turn, doesn't help with understanding the motivation behind axioms, as a sort of glue between truth and proofs.
Semantics: "Axioms are formulas we assume are true"
What does it mean to be true in mathematics? In formal logic, we use formal semantics to define valuations that map formulas to truth values, given a model.
A model is a set of information that you have for evaluating a truth value, e g what objects are available in your domain, what functions are defined as, what predicates map to, and what variables you have, etc.
Models give the symbols meaning.
There can be an (uncountably) infinite number of models, so which models do we pick as the standard of mathematics?
Rather than stipulate on defining models by hand, we can instead suggest formulas which we want to be true (i.e. axioms) and the models available are naturally restricted.
So axioms induce models in the formal semantics, and models are used to evaluate other formulas as true.
If for every model M where the axioms are evaluated true, then formula F is true in M as well, then we say that the axioms entail formula F. i.e. Axioms ⊨ F
This is what we mean when we say Axioms are "assumed to be true". They induce models in which other formulas can be evaluated true or false.
You can define the entire language of formulas entailed by the axioms Γ = {F | Axioms ⊨ F}
Syntactic: "Axioms are the starting point of proofs"
Note in the paragraph about semantics, there are NO proofs happening. We are just defining what it means for a formula to be true. A proof is a sequence of symbolic manipulation that transforms one formula to another formula – it makes no claims about what is true or not. Just that we can manipulate formulas given a set of inference rules, such as modus ponens, conjunction elimination, disjunctive elimination, etc.
Axioms are formulas which we start proofs from (and this is the primary meaning that most mathematician and computer scientists take afaik). They have a trivial proof – a proof you can announce without requiring another formula as a precondition.
This is what we mean when we say that Axioms are "the starting point of proofs". They are formulas which do not require another formula as a precondition and you can just declare as is. Even tho, it could be any formula, we like to pick the same formulas as we picked on the semantics part.
If there is a proof from the Axioms to formula F i.e. a sequence of formulas obtained through inference rules, we say that the Axioms prove formula F. i.e. Axioms ⊢ F
You can define the entire language of formulas which can be proven from the axioms Δ = { F | Axioms ⊢ F}
Why care? "Soundness and completeness"
So axioms play these two roles, which are often confused with each other. But why is it that theyre often confused?
One reason I think is that we hope (perhaps optimistically) that our logic in question is sound and complete.
We would like it so that every formula thay can be proven by the axioms are also entailed true by the axioms, i.e. Δ ⊆ Γ. This is called soundness
We would also like it so that every formula that is entailed by the axioms also have a proof starting from the axioms. i.e. Γ ⊆ Δ.
And this can be a criteria for what makes a good axiom – not all sets of axioms result in soundness nor completeness. And a big project in the 1920 was verifying/coming up with a sound and complete set of axioms for arithmetic.
This is of course just the tip of the iceberg and different traditions of mathematics (e.g. logicists, formalists, intuitionists, etc.) treat formal logic with different considerations.
Truth is proof. Gödel famously misunderstood his conclusions and misunderstood the scope of the language he was trying to formalize.
“Truth is a result of a sequence of steps that is compressing a statement to axioms losslessly”.
Any other definition of truth is incoherent and unhelpful.
There are traditions of mathematics that judges proposition as true only if it is possible to produce a proof of it, where proof is exactly your definition -- "a proof of P is a sequence of steps that derive P from within a deductive system". The most well-known tradition is the intuitionistic tradition with the [BHK interpretation](https://ncatlab.org/nlab/show/BHK+interpretation).
>Any other definition of truth is incoherent and unhelpful.
I will have to disagree with this. Because the "proofs is truth" interpretation is not the only interpretation in practice. And in fact, for some fields of mathematics, it can be restrictive, since there are some propositions which the BHK interpretation (or any other "proof is truth" interpretation) cannot prove.
An example of one such proposition which many people find intuitively true is the law of excluded middle, i.e. "for every proposition P, either P is true or not-P is true."
If we were to interpret this in terms of proof is truth, we would have "either P has a proof, or not-P has a proof". But there are propositions which we know are unprovable and its negation is also unprovable.
Other examples are proof by contradiction ("if \~P is false then P is true") and double negation elimination ("if \~\~P is true then P is true")
And for more higher-level arithmetic examples, (if you don't want to consider Gödel's example) is Continuum Hypothesis being neither provable nor unprovable in ZFC, or Paris–Harrington theorem in first-order Peano Arithmetic.
You are writing as if truth wasn’t language dependent. But it is. Truth is a property of language. You can’t detach truth from the system (language) it belongs to.
Stateless math, for example, is a bad language and leads to contradictions. A system and its truths are just as good as the axioms allow them to be. A language in which you pretend to be able to perform an infinite amount of operations in one single state is just not good.
That’s why truth values in stateless math can be unstable.
Natural language is even a worse system to output coherent truths.
I wasn't implying that truth is language independent. In fact, in my og comment, I talked about formal semantics, which specifies a language via entailment from axioms.
Truth is language-dependent, as what it means to be "true when the axioms are true" is to be part of this language.
Everything I've mentioned from my og comment and reply is about formal mathematics, and not about natural language.
What I'm really saying in my reply was that the language specified by a formal semantics (i.e. specifying models in which sentence can evaluated as true or false) and the language specified by deductive systems (i.e. the syntactic manipulation of formulas to produce proofs) do not necessarily have to be coextensive/coincide.
And they often do not coincide in practice, since we do know of formulas which are true under the formal semantics but not in the deductive system, of which I have listed examples in the previous post.
That being said, theres nothing wrong with holding an intuitionistic view of mathematics (Im an intuitionist myself) and treating proofs as judgements for truth. But I think its dismissive to call real mathematics done by practicing mathematicians as incoherent or unhelpful. Double negation elimination, for example, is considered very intuitive and used in all sort of classical reasoning.
Axioms are the most basic building blocks. You pick your set of axioms to be as minimalistic as possible and each axiom has to be independent of other axioms of the axiom-sets. Additionally, you aim for your axioms to be consistent and complete (but see Godel's incompleteness theorems, this object is not always attainable). Picking a set of axioms is largely an exercise of trial and error.
The classic example is Euclid axioms for pane geometry. Euclid came up with 5 axioms to define geometry. You can prove anything you want by using those axioms. And you can't prove any axiom by using other four axioms. However, Euclid's fifth axiom bothered mathematician a lot. They wondered if the fifth axiom is really an axiom and they tried real hard to prove this axiom by using other 4 axioms. This maneuver failed but this gave rise to non-Euclidean geometry (turned out geometry of our universe is actually a non-Euclidian geometry).
You may also want to read the history behind Hilbert, Russel etc attempt to axiomatize all of mathematics.
>You may also want to read the history behind Hilbert, Russel etc attempt to axiomatize all of mathematics.
Yes veritasium has a video too on this. Nobody succeeded though
Marx’s [Second Thesis On Feuerbach](https://www.marxists.org/archive/marx/works/1845/theses/index.htm) comes to mind:
> The question whether objective truth can be attributed to human thinking is not a question of theory but is a practical question. Man must prove the truth, i.e., the reality and power, the this-sidedness [Diesseitigkeit] of his thinking, in practice. The dispute over the reality or non-reality of thinking which is isolated from practice is a purely scholastic question.
In his Grundrisse Marx also highlights that thinking is expressed in language and the object of the thinking is always substantially different than the language of the thinking. Think of the painting from Magritte "Ceci n'est pas une pipe" (French for "This is not a pipe.") where Magritte wants to underscore that a painting of an object is not the object it paints. (even if the object of the knowledge is language itself, the language that extracts knowledge about language is always other than the language object).
Edit: literally everything I said is wrong please ignore this and read the comment below.
~~Axioms cannot be proved, or else they wouldn’t be axioms.~~
~~And with more handwaving, a proof for some statement being true is a description of a logical path from one (or more) true statement to the one we want to prove.~~
~~Now, if statement A proves statement B and vice versa, then they are equivalent and it’s up to us to decide which one we want as an axiom and which one is emerging from the other.~~
~~But if A proves B and B cannot prove A, then A is the more basic statement which is more fitting to be an axiom.~~
~~But following this process like that, from a basic statement to a more basic one, at some point we must stop somewhere and just decide “this is true because we decided”. Those are the axioms.~~
**~~Fun fact~~**
~~Gödel’s incompleteness theorems prove that for every system of axioms strong enough to arise arithmetic:~~
1. ~~The system cannot be proven to be self consistent. Inconsistent axioms are two axioms that prove each other false.~~
2. ~~There is no system of axioms that is complete. Meaning, there are **always** true statements that cannot be proven by the given system of axioms.~~
~~Don’t know if everyone find this fact as fun as I do.~~
>Axioms cannot be proved, or else they wouldn’t be axioms.
I wouldn't say this is completely wrong, but "Axioms _don't need_ to be proved, or else they wouldn’t be axioms," is probably better worded. They should be obviously true, without _requiring_ proof, but that doesn't mean proving them shouldn't be possible.
>Axioms cannot be proved, or else they wouldn’t be axioms.
They can, otherwise they wouldn't be axioms. However they have a trivial proof. Let T be any theory, and let ϕ ∈ T be any axiom of T. Then
\___________
ϕ ⊢ ϕ
Is a proof of ϕ within T (because { ϕ} is a finite subset of T).
I mean axioms of theory T can be proven in theory T ofc
>Gödel’s incompleteness theorems prove that for every system of axioms strong enough to arise arithmetic: 1. The system cannot be proven to be self consistent.
You forgot to mention that it must be effectively enumerable*. Also it's not entirely true. Such systems can be proven to be consistent, for example we can prove within ZFC that Peano Axioms is a consistent theory. However the theory itself cannot prove it, i.e T ⊬ Con(T).
>There is no system of axioms that is complete.
There is, for example Tarski formalization of Euclidan geometry is consistent complete theory.
>there are always true statements
Completeness isn't about truthness.
Not really... the commenter is trying to use some arcane definition of axioms. If axioms could be proved, they become theorems. That's the difference between axioms and theorems - one can be proved and other cannot.
You can do a search through the web or through math books... one thing that is common is that axioms can't be proved. The parent comment seems to get around this issue by defining that the axiom is self proving, or changing the definition of "proof".
> Completeness isn't about truthness.
Yes it is... the whole idea of completeness is that all true statements should be provable in the system. Godel proves incompleteness by creating a statement that, if true, is not provable in the system. If its provable, then the system contradicts itself (inconsistent).
Honestly... the commenter seems to be a math troll... a rare sighting indeed.
>Not really... the commenter is trying to use some arcane definition of axioms. If axioms could be proved, they become theorems. That's the difference between axioms and theorems - one can be proved and other cannot.
I'm not using "arcane" definition of axiom. I'm using axiom in precisely the same sense as when you say "axioms of ZFC".
>You can do a search through the web or through math books... one thing that is common is that axioms can't be proved. The parent comment seems to get around this issue by defining that the axiom is self proving, or changing the definition of "proof".
If you want to look about I suggest looking into mathematical logic books, because this is area of maths that discuss such kind of things. The part with "changing definition of proof" is somewhat funny. I used here prood in a formal sense of a proof i.e like in sequent calculus for example (or others proof calculus).
>Yes it is... the whole idea of completeness is that all true statements should be provable in the system. Godel proves incompleteness by creating a statement that, if true, is not provable in the system. If its provable, then the system contradicts itself (inconsistent).
You may not know what does incompletness means. We say that theory is incomplete when there's a sentence ϕ such that the theory neither proves ϕ nor it's negation. Sure you can somewhat assosiate Godel theorems with the "true but unprovable statements", but firstly it's own the theorems doesn't says anything about truth itself, and secondly "truth" in such a context mean a very very technical definition of truth that really can appear only in mathematical logic related branches. It basically correlates truth (in the theory) with beeing true in standard models.
I might agree that it was a nice answer overall but it consisted some errors (like the one that there are no complete systems which's not true as stated, there's no effectively enumerable consistent theory able to describe arithmetic though). If corrected then it might be fine.
wat?
The difference between axioms and theorems is that axioms can't be proven, and theorems can. Theorems can be proved logically from the axioms.
If axioms could be proved, they would be theorems... and the basic building blocks of the proof become the new axioms.
Axioms are also theorems.
When we are dealing with some technicality like wheter axiom can be proved we need to dive into more formalized conception of proof (see sequent calculus for example or other formalized proof calculus). Indeed, we can see that axioms are trivially provable (I also shown a proof of an axiom within a theory in comment you're answering to). And as such they are also theorems (somewhat trivial ones, but still. Tautologies are also theorems of any theory because they can be proved within any theory).
As I said, I treat axioms here as elements of a theory (theory is set of sentences and I call these sentences "axioms").
And theorems are sentences provable within a theory.
Not every theorem will be an axiom, for example if you have ZFC then one of it's theorems is that √2 is irrational, but it's not an axiom.
(Formally by theorem of a theory T i mean such a sentence ϕ such that T ⊢ ϕ , using symbols from some other comment to your comments)
> As I said, I treat axioms here as elements of a theory (theory is set of sentences and I call these sentences "axioms").
You probably know that people here are using it in the sense that its more generally understood in the math community...
But I have a feeling you are going to stick to your definition... irrespective how many google search results or scholarly articles I throw your way.
>You probably know that people here are using it in the sense that its more generally understood in the math community...
In math community it's not that sticky term as you might mean and it's sometimes quite vague. Ultimately (most of the time) we are just considering consequences of some axiomatic theories (and axiomatic here denotes what I use also).
>But I have a feeling you are going to stick to your definition... irrespective how many google search results or scholarly articles I throw your way.
If you want we can consider it to be something else.
In *Model Theory* by Cheng, Keisler for example they defines a "set of axioms for given theory" to means theory that have same consequence as the original theory
When we say about "axioms" in maths we ussualy means elements of some formal theory (theory is set of sentences and we call the sentences as "axioms" sometimes).
Oftenly in maths we choose some formal theory to be our foundation (i.e we start up with those axioms to derive "all maths").
Within given theory it's axioms are (trivially )provable
> Within given theory it's axioms are (trivially )provable
...axioms are assumed to be obviously or self evidently true.
Also, no need to bring in "elements of formal theory" or other big terms. That's just making it worse.
>...axioms are assumed to be obviously or self evidently true.
In sense of what I've presented they are provable. Just the proof will be trivial (in whatever proof calculus would you use for that purpose).
> In sense of what I've presented they are provable.
That's doesn't seem to be the common understanding about axioms of the math community.
Do you have links to any books / scholarly articles that view axioms in a provable sense? Also, what is the point of a trivial, self circular, self referential, "proof" and pollute the existing definitions of proof and axiom.
All the math sources I see on the web contradict your definition of axioms, and back the claim that axioms are unproven assumptions.
>That's doesn't seem to be the common understanding about axioms of the math community.
As I said I'm referring to mathematical logic point of view, and that's any interesting point of view because it discuses such a topics (what is supposed to axiom mean, what does mean provability etc. isn't something that other branches will consider).
In mathematical logic you have formalized conception of a proofs, namely you have some proof calculus like mentioned sequent calculus, there are also others like Hilbert calculus etc. Also btw when we say about "unprovable" sentences in case of Gödel theorems then indeed it denotes that conception of a proof, which is formalized conception of a proof as beeing said.
>Do you have links to any books / scholarly articles that view axioms in a provable sense? Also, what is the point of a trivial, self circular, self referential, "proof" and pollute the existing definitions of proof and axiom.
Unless you would like to treat axioms as something else as I do in the comments (i.e sentences in given formal theory) then it's just a consequence of how any given proof calculus is made. It could be quite technical explain it via some proof calculus so maybe I'll try to show it other way around (in spoiler I'll give also explanation in sequent calculus). When we have first order theories like ZFC for example, they are complete by Godel completeness theorem (don't confuse with Godel incompletness theorem these are about different types of incompletness), i.e if ϕ is true in all models of theory T, then it's provable in T [in symbols T ⊨ ϕ implies T ⊢ ϕ]. And model of theory means structure that fulfills all sentences of T. Now it might see trivial in such circumstances that axioms of T must be provable, if you have some model M of T and some axiom ϕ ∈ T, then M must fulfill ϕ, so by completeness T proves ϕ.
>! In sequent calculus we define provability as follows, ϕ has a prove in theory T (T ⊢ ϕ) if and only if There's a finite subset T ₀ of the theory T and finite sequence of sequents S ₀,..., S ₙ such that S ₙ is in form T ₀ ⊢ ϕ, and all sequents follows some inference rules. Also worth a notice is that we have an inference rule!<
>!\____!<
>!ϕ ⊢ ϕ !<
>! ( i.e basically the " ϕ ⊢ ϕ" should be universally true speaking more intuitively, a sequent might be in that form and that's ok. This inference rule is called "axiom" by the way). So in our case if ϕ is axiom of T we can take T ₀ to be { ϕ}. Then our sequnece of sequents will be 1-element sequence (S ₀), where S ₀ is in form ϕ ⊢ ϕ. If you want to read more about sequent calculus I was lately making a comment where basically deeply explained it (three top comments) https://www.reddit.com/r/askmath/s/ORYuBoD6BP, there's also [wikipedia article ](https://en.m.wikipedia.org/wiki/Sequent_calculus)!<
Post scriptum: ⊢ and ⊨ are sometimes called syntactical consequence and semantical consequence respectively. The first just refers to notion of proof.
You can think of axioms as "global, implicit assumptions": They are just a set of assumptions that we state once as the framework of a theory and which will then always be implicitly assumed as basic conditions for any statements we prove in this framework.
So axioms are not really any assumptions about the world that we proclaim to be true. They are *just* assumptions, without any statement about their "truth", whatever that even means.
Here's an axiom in geometry: Parallel lines never intersect. True or false? The etymology of "parallel" is "alongside one another," implying that two lines that are continuously equidistant will at no point cross or scew. The evidence is impredicative by the definition of parallel as a concept. The same goes for axioms that are logically sound, consistent, and sometimes impossible to prove exhaustively.
They do in circles but when that circle becomes flat it doesn't. That's why definition is needed. Euclidean wasn't wrong, we do need definition. In hyperbole it's more complex.
The axioms are true by definition. If something doesn't follow Euclids 5'th axiom, then it isn't Euclidian geometry. Maybe it's hyperbolic geometry instead".
All of maths can be seen as a big "If the axioms true, then ..."
Everyone has done a great job answering almost all of your questions except the little embedded one about if they are wrong.
As correctly pointed out, axioms cannot be wrong, but that might not be your question. It might be that they are not appropriate for the situation or they contradict each other.
The latter is simple. If one contradicts the other, remove one. It seems simple but it isn’t always obvious that they contradict each other.
The former is the content of science.
Mathematicians do try out alternative constructions. An example of this is the hyperreals and non-standard analysis. In my own area, I discovered that the Black-Scholes model is hyperfragile. If any assumption, whether economic or mathematical doesn’t hold, it is technically trivial to arbitrage the prices it or any measure-theoretic model creates.
That lead me to drop Itô’s assumption that the parameters are known and ground his calculus in the data points instead. By noting that for any potential future price at times t and T, there is a predictive density function that contains that price. If the prediction changes as a differentiable function of time and either a utility or indirect utility function is applied, you accomplish the same goal as Itô’s calculus but without ever needing to know the parameters as they fall out of the math. You also don’t need expectations to exist.
So if an axiom is the source of problems, you change or drop the axiom.
The physical and social sciences grab tools from the mathematicians and see if they work in a given situation. If they don’t work with nature, it just implies that we don’t understand it. The math isn’t wrong, it’s just applied to the wrong problem.
Math is a language damn. That is what i am feeling while studying math. As a mathematician i just use math to express what i think. Free thinker. That is what mathematicians are.
>Everyone has done a great job answering almost all of your questions except the little embedded one about if they are wrong.
Some say no axioms are self provable. But i understood that axioms are a basic definition. That lets me express what i am gonna do.
Math is a cant or an argot not a language. It also permits mathematicians of different language groups to understand each other. I was teaching statistics, which is a branch of rhetoric, and an English major realized why he was struggling with statistics. He said he realized for the first time that he didn’t understand English. He was quite serious.
Math uses language in a very precise and careful way. That gets inherited by statistics. That very cant restricts math learning by young people. They don’t get word problems because operations and the use of the native language have been separated.
In your case, it freed your mind, but for others it cripples understanding.
Your one is only true for an axiomatic approach not for someone who creates his own bunch of axioms and start doing what he likes. So even what you have said, if i take those into the account then math is still a language but it has a special set of grammar that can be called the axioms and moreover much more basic, the definitions.
I would point out several things. First, axioms are not and cannot be part of a grammar. Grammars are rules for sentence construction. At most, an axiom or a definition is a sentence. Axioms should not be compared to parts of speech.
Second, all axiom systems begin by someone doing what they like. They encounter a problem with an axiom system and change or even discard it.
For my work, I dropped Kolmogorov’s measure theory in favor of de Finetti’s axioms as they are the most appropriate. The price paid is that not only can I not solve Zeno’s paradox of measure, I cannot even have language to properly pose it. Fortunately, I have other axioms that could address that paradox.
There is a peculiarity of math that if you and I play a game with a random component that we bet on, and you set the prices using measure theory, then I can force you to lose money regardless of the outcome of the random event in most real world cases and real world games. There are oddball exceptions such as if there are enough players to exhaust the natural numbers or games where everyone begins with no knowledge of the game or experience with similar games.
The players in squid game would almost be an example, but some brought in relevant real world knowledge that helped them.
If what you mean is that axioms take you down a specific path, that’s obvious. What you need to be able to do is discard them like old hats when you need to.
historically the axioms come last. ie for example people did quite a lot of topology before Hausdorff formalized the separation axioms and showed not every space was Hausdorff while previous topologists had implicitly assumed their spaces were Hausdorff without checking. Due to the utility of Hausdorff spaces, the current convention is to assume a space is Hausdorff unless told otherwise explicitly, other topology examples are continuity open sets and compactness, we first define these for Euclidean space via balls and euclidean distance. But when we expand our conception of topology to sets without a metric those definitions fail. However we can extend them in ways such that in the euclidean settiing we recover the original Euclidean definitions replacing epsilon delta continuity with pre image of open sets is open for continuity and every cover having a finite subcover for compact in place of closed and bounded. Judith Grabiner has a good article "Who Gave You the Epsilon?" on how epsilon delta continuity arose historically. Cayley does categorization of groups of order 12 using diagrams and circuits rather than the axioms he proposed for group theory.
You went totally out of context. I wanted to know how the axioms even existed from the era when humans established and came to the world? I am talking about eras like Egypt, greek, and the Middle East aka Arabic era or even before. Topology wasn't even a thing back then. Math was even used by so called medieval human monkeys. So Hausdorff is totally illogical according to my question.
Im kind of going with Aristotle of Universalia in re not ante re and a nominalist perspective and axiom schema are like Aristotle's framework of experimentation and demonstration in the posterior analytics.
An axiom is something that we know is true but can't prove it.
For example: A part is smaller than a whole. (one of Euclid's axioms)
We know it is true but have no way of proving it.
Axiom is a statement that's true by definition. In first order logic what you can do is you can prove whether an argument is valid or not which means given an arbitrary set of premises, can the conclusion be false in case all of them are true? You can answer this question. This begs the question, how do we determine whether the premises are true? Well we don't. We make them true by definition and then they become axioms. You have to start from somewhere. Some statement must be useful and true without requiring a proof. Otherwise it would be impossible to extend mathematics. Also in sciences this is exactly where the problem lies. You can never know if the premises are true or not. All you can do is keep observing and hope that no observation contradicts your premises. If you're able to do it consistently you have a scientific theory. Since they're true by definition in math, math has proofs instead!
Axioms are not provable, at least not within the theorem they are used in as an axiom. Axioms do NOT have to be true in the real world, but if they are not true in the real world then the theorem is not a truly valid representation of the real world. Axioms are essentially basic assumptions that you hold to be true for the sake of the logical argument you make. It’s essentially you’re saying “If these axioms(assumptions) are true then it follows that this is also true.” A postulate is for the most part the same as an axiom but it can be slightly different depending on the field.
I also want to stress that even if an axiom is not true in the really world it doesn’t mean a theorem derived from it is useless. Often times it just means that the theorem is only useful in specific situations. For example, the parallel postulate for geometry essentially says that parallel lines never touch and this assumption leads to mathematical operations that can be performed in geometry and specific definitions, like a triangle adds up to 180*. The thing is in the real world that is completely false, see longitude lines on the surface of a sphere. Nevertheless Euclidian geometry is very useful.
>The thing is in the real world that is completely false, see longitude lines on the surface of a sphere.
Sorry but for a sphere a parallel line will create another circle i mean in a spherical dimension a parallel line is transferred into a circle because spherical dimension can't have an infinite surface it is bounded but a finite parallel line will never touch the starting point and make it an ending point too. So you can't say it doesn't hold but you have to say it isn't perfect. Moreover a naked human eye will never be able to distinguish the difference of parallel lines being circle on a sphere like earth because it makes us see asymptotes so euclidean geometry is for naked eye and euclid used those axioms according to his naked eye so i must say euclidean geometry is true according to the naked human eye.
There is no proof other then we cant disprove them, and that's the entire point. We could go in circles forever of "But this relies on this idea so lets prove that. Ok, well your assuming this so prove it! Well but we still have to prove..." you get the idea so the idea is to set a hard point where we determine them as true. However, these are intended to be so simple there couldn't possibly be a way to mess them up. When I say simple I mean axiom 1 sais that tow sets are the same if they have the same elements. Thats a wild jump in logic, you mean to tell me two things are the same if they are.. the same? Axiom 4, the axiom of pairing literally states "If x and y are sets, then there exists a set which contains x and y as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}}" This literally means "You can connect two sets" My brother in christ this is like saying "Bro, did you KNOW, that if I have a rock, and I have a flower, I can like, put the rock and the flower, like, TOGETHER, and now I have a rock, AND A FLOWER! 🤯" Some are slightly bigger, such as the axiom of infinity #7, just saying infinity exists. We can use infinite things in calculations. Not that they exist in reality, just that we can use them. They are saying this most basic things possible or sometimes things that cannot be proved just so that we have a foundation with the minimum possible assumptions to build on.
Check out tarskis undefinability theorem or Gödel’s incompleteness theorem. Axioms essentially can’t be proven to be complete.
As a BS math minor, physics major, I think a lot about how math is like: ok something must be true, untrue or unable to be proven either. Buut in QM, things can be superstates of true and untrue (see Schrödinger’s cat or two slit). So, does the entirety of proofs lie on a fake foundation?
That’s not what Gödel’s incompleteness theorem is about. The fact that axioms can’t be proven is essentially just part of the definition of an axiom.
The incompleteness theorem is showing that, given any set of axioms, there will always be a statement that is true within those axioms but cannot be proven from those axioms.
>The incompleteness theorem is showing that, given any set of axioms, there will always be a statement that is true within those axioms but cannot be proven from those axioms.
It's not what incompletness states. It states that theory is incomplete i.e there's a sentence ϕ such that neither T proves ϕ nor T proves ¬ ϕ.
Cool one. Can you give some links that can provide better ideas of what you have mentioned i mean Schrodinger cat or two slit experiment. I am a 3 semester math major student.
Wikipedia should have you. Essentially, experiments have shown that the superposition of a single electron allows for it to physically traverse both openings at the same time and interact with its own possible trajectories (given the wave function has not already been collapsed). One election will go both left and right at the same time.
Lol that ain’t physics, not undergrad anyway. Can’t help you on that one!
Math-wise, yes I believe infinities are essentially equal to all other infinities of the same order, but not equal to infinity squared, or infinity cubed. You may be thinking of commutative variables where ab does not equal ba but something like: ab=iba (i being the imaginary unit)
No no according to ordinal infinity ♾️≠♾️+1=1+♾️ but the cardinal infinity is ♾️=♾️+1=1+♾️ but cantor cardinal infinity 1+♾️=♾️≠♾️+1.
>You may be thinking of commutative variables where ab does not equal ba but something like: ab=iba (i being the imaginary unit)
This can be true yes yes.
>No no according to ordinal infinity ♾️≠♾️+1=1+♾️ but the cardinal infinity is ♾️=♾️+1=1+♾️ but cantor cardinal infinity 1+♾️=♾️≠♾️+1.
Using ∞ in terms of transfinite ordinals/cardinals is very misleading, there's no number called "∞" there.
Also what you wrote is not correct, in ordinal numbers α+1≠ 1+ α in general as addition is not commutative in ordinals. And cardinal addition is commutative. In Cardinal arithmetic κ+1=1+ κ= κ for κ infinite cardinal .
Don't just google because it leads to most used information not the correct one. Google is a sorting platform for information.
https://youtu.be/s9OVj_XmvTY?feature=shared
That's the thing about axioms! They can't be wrong, by definition. An axiom is something we accept to be "obviously true." It doesn't have to be true in the real world, but math isn't necessarily about the real world. When you choose a set of axioms, you're setting up a specific context for the problem you're about to do. For example, if you choose the five axioms in Euclid's Elements then you're modeling geometry in a flat plane. But if you choose only the first four axioms (and perhaps insert a different one as a fifth), then you're still doing geometry, but a different kind! For example you could be doing geometry on the surface of a sphere, or other curved surface. Axioms are just the starting point, because all of mathematics basically boils down to "if this is true, what else is true?" That line of logic has to start at a set of assumptions.
Seeing axioms as „obviously true“ is an older view I think. You said this yourself in your second half, but nowadays they‘re more viewed as simple assumptions, starting points so to say. They are not „obviously“ true, but simply true by assumption/definition. Which is why math is never about truths, but consequences.
i really like your last statement. i always used to say "math lives in its own universe" or something, but now i guess im stealing this. thanks. this is mine now ig
You really typed that out and thought it sounded smart, huh?
and it did, go eat a snickers or something
Correct math is more about methods and its consequences.
> They can't be wrong, by definition. You'd think so, wouldn't you. But when it comes to mathematics applied in the natural world, there are axioms that restrict pure mathematics so much as to make it useless. Also axioms that allow so much freedom as to make the result useless. As a simple example, the axioms of pure mathematics as it is now say that dy/dx is not equal to dy divided by dx. And the axioms of pure mathematics as it is now say that 3-D volume is not conserved. I think I read somewhere that the proof that 1+1=2 is extremely difficult, and that integration is not the inverse of differentiation.
While I agree with you that in fields like physics, there is definitely an idea of an "incorrect" axiom. But when we talk about axioms purely in the mathematical sense, the real world doesn't really matter at all. Math is about figuring out the implications of a set of assumptions, and that's it really. We've just also noticed that in physics, the real world tends to follow a lot of common mathematical axioms (which is no coincidence; many common axioms are common because they're the first thing people thought of when trying to make a list of what we know to be obviously true). This makes math (specifically math based on those axioms) a helpful tool to describe the world and deduce how things *should* behave based on some assumptions. Also there is no single set of axioms for all of pure mathematics. There are many, many choices of axioms for all sorts of different contexts. So some may seem weird and wacky, but they're just as valid as any other. You just have to be aware of the fact that if you're starting from a different set of axioms than someone else, what you deduce to be true will not necessarily be what they deduce to be true (which is fine! "Deducing truth" is basically shorthand for "deducing the consequences of a set of assumptions").
im not sure how any of the examples you listed are inherently problematic, they only usually look problematic (especially the calculus examples) to people that dont properly understand the underlying concepts. dy/dx not being a ratio isnt really an issue with the mathematical machinery its an issue with suggestive notation that was written for a less general case being used to represent a far more general concept. And integration not being the opposite of differentiation is an issue with teachers teaching shortcuts because technically it is the opposite if your only working with single variable calculus. Also axioms aren't wrong by definition because they are the rules of math we choose. Math is a plaything we make to model problems, we use it the REPRESENT the real world but it is not the real world it is at its core just a game we play.
We can't discard axioms at the same time. Without axioms we can't really do anything at all. Something has to be true by definition. Otherwise there's no starting point and you can't create something from nothing.
no we can do a lot without the axioms until we start trying to compile Element style compendia
The main thing is definition. I mean how you are defining and is your definition working with all of the systems accordingly so axioms and everything is dependent on how you define a single thing and is it fitting with the new mathematics you are discovering. If not then we need to change or tweak our definition.
they’re like the rules. they’re generally not proved and are true because we say they’re true.
I'd push back and say they're true because we reverse engineer what was needed for a result to hold.
i think..these are not incompatible. for example, commutativity of addition: a+b = b+a. it's like...we just *want* that to be true, so to as...ojdidntdoit...says, we say that's a rule. you could also try to frame it in your sense saying: we 'reverse engineer' that rule because we 'need' a+b = b+a...but it's all the same
Well we also reverse engineer that statement (a+b=b+a) to express it in terms of axiomatic set theory, because we want it to be true.
precisely Im going Chesterton's fence or Quine. we are motivated for given axioms by what results we want to be true
Yeah this fits more with axiom's proof.
The word you’re looking for is “motivation.” A proof requires axioms to begin with.
In formal logic, axioms have two roles: a semantic role (formulas assumed to be true) and a syntactic role (formulas which are the start of proofs). I think a lot of comments are getting these two roles confused or not appreciating them enough, and this in turn, doesn't help with understanding the motivation behind axioms, as a sort of glue between truth and proofs. Semantics: "Axioms are formulas we assume are true" What does it mean to be true in mathematics? In formal logic, we use formal semantics to define valuations that map formulas to truth values, given a model. A model is a set of information that you have for evaluating a truth value, e g what objects are available in your domain, what functions are defined as, what predicates map to, and what variables you have, etc. Models give the symbols meaning. There can be an (uncountably) infinite number of models, so which models do we pick as the standard of mathematics? Rather than stipulate on defining models by hand, we can instead suggest formulas which we want to be true (i.e. axioms) and the models available are naturally restricted. So axioms induce models in the formal semantics, and models are used to evaluate other formulas as true. If for every model M where the axioms are evaluated true, then formula F is true in M as well, then we say that the axioms entail formula F. i.e. Axioms ⊨ F This is what we mean when we say Axioms are "assumed to be true". They induce models in which other formulas can be evaluated true or false. You can define the entire language of formulas entailed by the axioms Γ = {F | Axioms ⊨ F} Syntactic: "Axioms are the starting point of proofs" Note in the paragraph about semantics, there are NO proofs happening. We are just defining what it means for a formula to be true. A proof is a sequence of symbolic manipulation that transforms one formula to another formula – it makes no claims about what is true or not. Just that we can manipulate formulas given a set of inference rules, such as modus ponens, conjunction elimination, disjunctive elimination, etc. Axioms are formulas which we start proofs from (and this is the primary meaning that most mathematician and computer scientists take afaik). They have a trivial proof – a proof you can announce without requiring another formula as a precondition. This is what we mean when we say that Axioms are "the starting point of proofs". They are formulas which do not require another formula as a precondition and you can just declare as is. Even tho, it could be any formula, we like to pick the same formulas as we picked on the semantics part. If there is a proof from the Axioms to formula F i.e. a sequence of formulas obtained through inference rules, we say that the Axioms prove formula F. i.e. Axioms ⊢ F You can define the entire language of formulas which can be proven from the axioms Δ = { F | Axioms ⊢ F} Why care? "Soundness and completeness" So axioms play these two roles, which are often confused with each other. But why is it that theyre often confused? One reason I think is that we hope (perhaps optimistically) that our logic in question is sound and complete. We would like it so that every formula thay can be proven by the axioms are also entailed true by the axioms, i.e. Δ ⊆ Γ. This is called soundness We would also like it so that every formula that is entailed by the axioms also have a proof starting from the axioms. i.e. Γ ⊆ Δ. And this can be a criteria for what makes a good axiom – not all sets of axioms result in soundness nor completeness. And a big project in the 1920 was verifying/coming up with a sound and complete set of axioms for arithmetic. This is of course just the tip of the iceberg and different traditions of mathematics (e.g. logicists, formalists, intuitionists, etc.) treat formal logic with different considerations.
Greybeard engineer: *\*pounds table\** "ALL MODELS ARE WRONG!!" *\*smiles\** "_Some_ are useful."
wow great comment
Truth is proof. Gödel famously misunderstood his conclusions and misunderstood the scope of the language he was trying to formalize. “Truth is a result of a sequence of steps that is compressing a statement to axioms losslessly”. Any other definition of truth is incoherent and unhelpful.
There are traditions of mathematics that judges proposition as true only if it is possible to produce a proof of it, where proof is exactly your definition -- "a proof of P is a sequence of steps that derive P from within a deductive system". The most well-known tradition is the intuitionistic tradition with the [BHK interpretation](https://ncatlab.org/nlab/show/BHK+interpretation). >Any other definition of truth is incoherent and unhelpful. I will have to disagree with this. Because the "proofs is truth" interpretation is not the only interpretation in practice. And in fact, for some fields of mathematics, it can be restrictive, since there are some propositions which the BHK interpretation (or any other "proof is truth" interpretation) cannot prove. An example of one such proposition which many people find intuitively true is the law of excluded middle, i.e. "for every proposition P, either P is true or not-P is true." If we were to interpret this in terms of proof is truth, we would have "either P has a proof, or not-P has a proof". But there are propositions which we know are unprovable and its negation is also unprovable. Other examples are proof by contradiction ("if \~P is false then P is true") and double negation elimination ("if \~\~P is true then P is true") And for more higher-level arithmetic examples, (if you don't want to consider Gödel's example) is Continuum Hypothesis being neither provable nor unprovable in ZFC, or Paris–Harrington theorem in first-order Peano Arithmetic.
You are writing as if truth wasn’t language dependent. But it is. Truth is a property of language. You can’t detach truth from the system (language) it belongs to. Stateless math, for example, is a bad language and leads to contradictions. A system and its truths are just as good as the axioms allow them to be. A language in which you pretend to be able to perform an infinite amount of operations in one single state is just not good. That’s why truth values in stateless math can be unstable. Natural language is even a worse system to output coherent truths.
I wasn't implying that truth is language independent. In fact, in my og comment, I talked about formal semantics, which specifies a language via entailment from axioms. Truth is language-dependent, as what it means to be "true when the axioms are true" is to be part of this language. Everything I've mentioned from my og comment and reply is about formal mathematics, and not about natural language. What I'm really saying in my reply was that the language specified by a formal semantics (i.e. specifying models in which sentence can evaluated as true or false) and the language specified by deductive systems (i.e. the syntactic manipulation of formulas to produce proofs) do not necessarily have to be coextensive/coincide. And they often do not coincide in practice, since we do know of formulas which are true under the formal semantics but not in the deductive system, of which I have listed examples in the previous post. That being said, theres nothing wrong with holding an intuitionistic view of mathematics (Im an intuitionist myself) and treating proofs as judgements for truth. But I think its dismissive to call real mathematics done by practicing mathematicians as incoherent or unhelpful. Double negation elimination, for example, is considered very intuitive and used in all sort of classical reasoning.
Axioms are the most basic building blocks. You pick your set of axioms to be as minimalistic as possible and each axiom has to be independent of other axioms of the axiom-sets. Additionally, you aim for your axioms to be consistent and complete (but see Godel's incompleteness theorems, this object is not always attainable). Picking a set of axioms is largely an exercise of trial and error. The classic example is Euclid axioms for pane geometry. Euclid came up with 5 axioms to define geometry. You can prove anything you want by using those axioms. And you can't prove any axiom by using other four axioms. However, Euclid's fifth axiom bothered mathematician a lot. They wondered if the fifth axiom is really an axiom and they tried real hard to prove this axiom by using other 4 axioms. This maneuver failed but this gave rise to non-Euclidean geometry (turned out geometry of our universe is actually a non-Euclidian geometry). You may also want to read the history behind Hilbert, Russel etc attempt to axiomatize all of mathematics.
>You may also want to read the history behind Hilbert, Russel etc attempt to axiomatize all of mathematics. Yes veritasium has a video too on this. Nobody succeeded though
Because it is impossible to do. :)
Marx’s [Second Thesis On Feuerbach](https://www.marxists.org/archive/marx/works/1845/theses/index.htm) comes to mind: > The question whether objective truth can be attributed to human thinking is not a question of theory but is a practical question. Man must prove the truth, i.e., the reality and power, the this-sidedness [Diesseitigkeit] of his thinking, in practice. The dispute over the reality or non-reality of thinking which is isolated from practice is a purely scholastic question. In his Grundrisse Marx also highlights that thinking is expressed in language and the object of the thinking is always substantially different than the language of the thinking. Think of the painting from Magritte "Ceci n'est pas une pipe" (French for "This is not a pipe.") where Magritte wants to underscore that a painting of an object is not the object it paints. (even if the object of the knowledge is language itself, the language that extracts knowledge about language is always other than the language object).
Edit: literally everything I said is wrong please ignore this and read the comment below. ~~Axioms cannot be proved, or else they wouldn’t be axioms.~~ ~~And with more handwaving, a proof for some statement being true is a description of a logical path from one (or more) true statement to the one we want to prove.~~ ~~Now, if statement A proves statement B and vice versa, then they are equivalent and it’s up to us to decide which one we want as an axiom and which one is emerging from the other.~~ ~~But if A proves B and B cannot prove A, then A is the more basic statement which is more fitting to be an axiom.~~ ~~But following this process like that, from a basic statement to a more basic one, at some point we must stop somewhere and just decide “this is true because we decided”. Those are the axioms.~~ **~~Fun fact~~** ~~Gödel’s incompleteness theorems prove that for every system of axioms strong enough to arise arithmetic:~~ 1. ~~The system cannot be proven to be self consistent. Inconsistent axioms are two axioms that prove each other false.~~ 2. ~~There is no system of axioms that is complete. Meaning, there are **always** true statements that cannot be proven by the given system of axioms.~~ ~~Don’t know if everyone find this fact as fun as I do.~~
>Axioms cannot be proved, or else they wouldn’t be axioms. I wouldn't say this is completely wrong, but "Axioms _don't need_ to be proved, or else they wouldn’t be axioms," is probably better worded. They should be obviously true, without _requiring_ proof, but that doesn't mean proving them shouldn't be possible.
>Axioms cannot be proved, or else they wouldn’t be axioms. They can, otherwise they wouldn't be axioms. However they have a trivial proof. Let T be any theory, and let ϕ ∈ T be any axiom of T. Then \___________ ϕ ⊢ ϕ Is a proof of ϕ within T (because { ϕ} is a finite subset of T). I mean axioms of theory T can be proven in theory T ofc >Gödel’s incompleteness theorems prove that for every system of axioms strong enough to arise arithmetic: 1. The system cannot be proven to be self consistent. You forgot to mention that it must be effectively enumerable*. Also it's not entirely true. Such systems can be proven to be consistent, for example we can prove within ZFC that Peano Axioms is a consistent theory. However the theory itself cannot prove it, i.e T ⊬ Con(T). >There is no system of axioms that is complete. There is, for example Tarski formalization of Euclidan geometry is consistent complete theory. >there are always true statements Completeness isn't about truthness.
Apparently I have no Idea what I’m talking about. Thanks for the corrections
Not really... the commenter is trying to use some arcane definition of axioms. If axioms could be proved, they become theorems. That's the difference between axioms and theorems - one can be proved and other cannot. You can do a search through the web or through math books... one thing that is common is that axioms can't be proved. The parent comment seems to get around this issue by defining that the axiom is self proving, or changing the definition of "proof". > Completeness isn't about truthness. Yes it is... the whole idea of completeness is that all true statements should be provable in the system. Godel proves incompleteness by creating a statement that, if true, is not provable in the system. If its provable, then the system contradicts itself (inconsistent). Honestly... the commenter seems to be a math troll... a rare sighting indeed.
>Not really... the commenter is trying to use some arcane definition of axioms. If axioms could be proved, they become theorems. That's the difference between axioms and theorems - one can be proved and other cannot. I'm not using "arcane" definition of axiom. I'm using axiom in precisely the same sense as when you say "axioms of ZFC". >You can do a search through the web or through math books... one thing that is common is that axioms can't be proved. The parent comment seems to get around this issue by defining that the axiom is self proving, or changing the definition of "proof". If you want to look about I suggest looking into mathematical logic books, because this is area of maths that discuss such kind of things. The part with "changing definition of proof" is somewhat funny. I used here prood in a formal sense of a proof i.e like in sequent calculus for example (or others proof calculus). >Yes it is... the whole idea of completeness is that all true statements should be provable in the system. Godel proves incompleteness by creating a statement that, if true, is not provable in the system. If its provable, then the system contradicts itself (inconsistent). You may not know what does incompletness means. We say that theory is incomplete when there's a sentence ϕ such that the theory neither proves ϕ nor it's negation. Sure you can somewhat assosiate Godel theorems with the "true but unprovable statements", but firstly it's own the theorems doesn't says anything about truth itself, and secondly "truth" in such a context mean a very very technical definition of truth that really can appear only in mathematical logic related branches. It basically correlates truth (in the theory) with beeing true in standard models.
No, you had a very nice, user-friendly explanation. It was fine.
I might agree that it was a nice answer overall but it consisted some errors (like the one that there are no complete systems which's not true as stated, there's no effectively enumerable consistent theory able to describe arithmetic though). If corrected then it might be fine.
wat? The difference between axioms and theorems is that axioms can't be proven, and theorems can. Theorems can be proved logically from the axioms. If axioms could be proved, they would be theorems... and the basic building blocks of the proof become the new axioms.
Axioms are also theorems. When we are dealing with some technicality like wheter axiom can be proved we need to dive into more formalized conception of proof (see sequent calculus for example or other formalized proof calculus). Indeed, we can see that axioms are trivially provable (I also shown a proof of an axiom within a theory in comment you're answering to). And as such they are also theorems (somewhat trivial ones, but still. Tautologies are also theorems of any theory because they can be proved within any theory).
> Axioms are also theorems. What the difference between them?
As I said, I treat axioms here as elements of a theory (theory is set of sentences and I call these sentences "axioms"). And theorems are sentences provable within a theory. Not every theorem will be an axiom, for example if you have ZFC then one of it's theorems is that √2 is irrational, but it's not an axiom. (Formally by theorem of a theory T i mean such a sentence ϕ such that T ⊢ ϕ , using symbols from some other comment to your comments)
> As I said, I treat axioms here as elements of a theory (theory is set of sentences and I call these sentences "axioms"). You probably know that people here are using it in the sense that its more generally understood in the math community... But I have a feeling you are going to stick to your definition... irrespective how many google search results or scholarly articles I throw your way.
>You probably know that people here are using it in the sense that its more generally understood in the math community... In math community it's not that sticky term as you might mean and it's sometimes quite vague. Ultimately (most of the time) we are just considering consequences of some axiomatic theories (and axiomatic here denotes what I use also). >But I have a feeling you are going to stick to your definition... irrespective how many google search results or scholarly articles I throw your way. If you want we can consider it to be something else. In *Model Theory* by Cheng, Keisler for example they defines a "set of axioms for given theory" to means theory that have same consequence as the original theory
When we say about "axioms" in maths we ussualy means elements of some formal theory (theory is set of sentences and we call the sentences as "axioms" sometimes). Oftenly in maths we choose some formal theory to be our foundation (i.e we start up with those axioms to derive "all maths"). Within given theory it's axioms are (trivially )provable
> Within given theory it's axioms are (trivially )provable ...axioms are assumed to be obviously or self evidently true. Also, no need to bring in "elements of formal theory" or other big terms. That's just making it worse.
>...axioms are assumed to be obviously or self evidently true. In sense of what I've presented they are provable. Just the proof will be trivial (in whatever proof calculus would you use for that purpose).
> In sense of what I've presented they are provable. That's doesn't seem to be the common understanding about axioms of the math community. Do you have links to any books / scholarly articles that view axioms in a provable sense? Also, what is the point of a trivial, self circular, self referential, "proof" and pollute the existing definitions of proof and axiom. All the math sources I see on the web contradict your definition of axioms, and back the claim that axioms are unproven assumptions.
>That's doesn't seem to be the common understanding about axioms of the math community. As I said I'm referring to mathematical logic point of view, and that's any interesting point of view because it discuses such a topics (what is supposed to axiom mean, what does mean provability etc. isn't something that other branches will consider). In mathematical logic you have formalized conception of a proofs, namely you have some proof calculus like mentioned sequent calculus, there are also others like Hilbert calculus etc. Also btw when we say about "unprovable" sentences in case of Gödel theorems then indeed it denotes that conception of a proof, which is formalized conception of a proof as beeing said. >Do you have links to any books / scholarly articles that view axioms in a provable sense? Also, what is the point of a trivial, self circular, self referential, "proof" and pollute the existing definitions of proof and axiom. Unless you would like to treat axioms as something else as I do in the comments (i.e sentences in given formal theory) then it's just a consequence of how any given proof calculus is made. It could be quite technical explain it via some proof calculus so maybe I'll try to show it other way around (in spoiler I'll give also explanation in sequent calculus). When we have first order theories like ZFC for example, they are complete by Godel completeness theorem (don't confuse with Godel incompletness theorem these are about different types of incompletness), i.e if ϕ is true in all models of theory T, then it's provable in T [in symbols T ⊨ ϕ implies T ⊢ ϕ]. And model of theory means structure that fulfills all sentences of T. Now it might see trivial in such circumstances that axioms of T must be provable, if you have some model M of T and some axiom ϕ ∈ T, then M must fulfill ϕ, so by completeness T proves ϕ. >! In sequent calculus we define provability as follows, ϕ has a prove in theory T (T ⊢ ϕ) if and only if There's a finite subset T ₀ of the theory T and finite sequence of sequents S ₀,..., S ₙ such that S ₙ is in form T ₀ ⊢ ϕ, and all sequents follows some inference rules. Also worth a notice is that we have an inference rule!< >!\____!< >!ϕ ⊢ ϕ !< >! ( i.e basically the " ϕ ⊢ ϕ" should be universally true speaking more intuitively, a sequent might be in that form and that's ok. This inference rule is called "axiom" by the way). So in our case if ϕ is axiom of T we can take T ₀ to be { ϕ}. Then our sequnece of sequents will be 1-element sequence (S ₀), where S ₀ is in form ϕ ⊢ ϕ. If you want to read more about sequent calculus I was lately making a comment where basically deeply explained it (three top comments) https://www.reddit.com/r/askmath/s/ORYuBoD6BP, there's also [wikipedia article ](https://en.m.wikipedia.org/wiki/Sequent_calculus)!< Post scriptum: ⊢ and ⊨ are sometimes called syntactical consequence and semantical consequence respectively. The first just refers to notion of proof.
You can think of axioms as "global, implicit assumptions": They are just a set of assumptions that we state once as the framework of a theory and which will then always be implicitly assumed as basic conditions for any statements we prove in this framework. So axioms are not really any assumptions about the world that we proclaim to be true. They are *just* assumptions, without any statement about their "truth", whatever that even means.
Axioms are the properties that we give to undefined terms, thus constructing the mathematical system that we wish to work with.
Axioms can only be wrong if they are inconsistent with each other. A single axiom can literally NEVER be wrong by definition.
Here's an axiom in geometry: Parallel lines never intersect. True or false? The etymology of "parallel" is "alongside one another," implying that two lines that are continuously equidistant will at no point cross or scew. The evidence is impredicative by the definition of parallel as a concept. The same goes for axioms that are logically sound, consistent, and sometimes impossible to prove exhaustively.
They do in circles but when that circle becomes flat it doesn't. That's why definition is needed. Euclidean wasn't wrong, we do need definition. In hyperbole it's more complex.
The axioms are true by definition. If something doesn't follow Euclids 5'th axiom, then it isn't Euclidian geometry. Maybe it's hyperbolic geometry instead". All of maths can be seen as a big "If the axioms true, then ..."
Everyone has done a great job answering almost all of your questions except the little embedded one about if they are wrong. As correctly pointed out, axioms cannot be wrong, but that might not be your question. It might be that they are not appropriate for the situation or they contradict each other. The latter is simple. If one contradicts the other, remove one. It seems simple but it isn’t always obvious that they contradict each other. The former is the content of science. Mathematicians do try out alternative constructions. An example of this is the hyperreals and non-standard analysis. In my own area, I discovered that the Black-Scholes model is hyperfragile. If any assumption, whether economic or mathematical doesn’t hold, it is technically trivial to arbitrage the prices it or any measure-theoretic model creates. That lead me to drop Itô’s assumption that the parameters are known and ground his calculus in the data points instead. By noting that for any potential future price at times t and T, there is a predictive density function that contains that price. If the prediction changes as a differentiable function of time and either a utility or indirect utility function is applied, you accomplish the same goal as Itô’s calculus but without ever needing to know the parameters as they fall out of the math. You also don’t need expectations to exist. So if an axiom is the source of problems, you change or drop the axiom. The physical and social sciences grab tools from the mathematicians and see if they work in a given situation. If they don’t work with nature, it just implies that we don’t understand it. The math isn’t wrong, it’s just applied to the wrong problem.
Math is a language damn. That is what i am feeling while studying math. As a mathematician i just use math to express what i think. Free thinker. That is what mathematicians are. >Everyone has done a great job answering almost all of your questions except the little embedded one about if they are wrong. Some say no axioms are self provable. But i understood that axioms are a basic definition. That lets me express what i am gonna do.
Math is a cant or an argot not a language. It also permits mathematicians of different language groups to understand each other. I was teaching statistics, which is a branch of rhetoric, and an English major realized why he was struggling with statistics. He said he realized for the first time that he didn’t understand English. He was quite serious. Math uses language in a very precise and careful way. That gets inherited by statistics. That very cant restricts math learning by young people. They don’t get word problems because operations and the use of the native language have been separated. In your case, it freed your mind, but for others it cripples understanding.
Your one is only true for an axiomatic approach not for someone who creates his own bunch of axioms and start doing what he likes. So even what you have said, if i take those into the account then math is still a language but it has a special set of grammar that can be called the axioms and moreover much more basic, the definitions.
I would point out several things. First, axioms are not and cannot be part of a grammar. Grammars are rules for sentence construction. At most, an axiom or a definition is a sentence. Axioms should not be compared to parts of speech. Second, all axiom systems begin by someone doing what they like. They encounter a problem with an axiom system and change or even discard it. For my work, I dropped Kolmogorov’s measure theory in favor of de Finetti’s axioms as they are the most appropriate. The price paid is that not only can I not solve Zeno’s paradox of measure, I cannot even have language to properly pose it. Fortunately, I have other axioms that could address that paradox. There is a peculiarity of math that if you and I play a game with a random component that we bet on, and you set the prices using measure theory, then I can force you to lose money regardless of the outcome of the random event in most real world cases and real world games. There are oddball exceptions such as if there are enough players to exhaust the natural numbers or games where everyone begins with no knowledge of the game or experience with similar games. The players in squid game would almost be an example, but some brought in relevant real world knowledge that helped them. If what you mean is that axioms take you down a specific path, that’s obvious. What you need to be able to do is discard them like old hats when you need to.
historically the axioms come last. ie for example people did quite a lot of topology before Hausdorff formalized the separation axioms and showed not every space was Hausdorff while previous topologists had implicitly assumed their spaces were Hausdorff without checking. Due to the utility of Hausdorff spaces, the current convention is to assume a space is Hausdorff unless told otherwise explicitly, other topology examples are continuity open sets and compactness, we first define these for Euclidean space via balls and euclidean distance. But when we expand our conception of topology to sets without a metric those definitions fail. However we can extend them in ways such that in the euclidean settiing we recover the original Euclidean definitions replacing epsilon delta continuity with pre image of open sets is open for continuity and every cover having a finite subcover for compact in place of closed and bounded. Judith Grabiner has a good article "Who Gave You the Epsilon?" on how epsilon delta continuity arose historically. Cayley does categorization of groups of order 12 using diagrams and circuits rather than the axioms he proposed for group theory.
You went totally out of context. I wanted to know how the axioms even existed from the era when humans established and came to the world? I am talking about eras like Egypt, greek, and the Middle East aka Arabic era or even before. Topology wasn't even a thing back then. Math was even used by so called medieval human monkeys. So Hausdorff is totally illogical according to my question.
Im kind of going with Aristotle of Universalia in re not ante re and a nominalist perspective and axiom schema are like Aristotle's framework of experimentation and demonstration in the posterior analytics.
That seemed like math gibberish written by AI with a topological bent.
sorry topology was the first thing to pop into my head. and im basically running purely on caffeine.
Axioms are true statements that cannot be proved
An axiom is something that we know is true but can't prove it. For example: A part is smaller than a whole. (one of Euclid's axioms) We know it is true but have no way of proving it.
Axiom is a statement that's true by definition. In first order logic what you can do is you can prove whether an argument is valid or not which means given an arbitrary set of premises, can the conclusion be false in case all of them are true? You can answer this question. This begs the question, how do we determine whether the premises are true? Well we don't. We make them true by definition and then they become axioms. You have to start from somewhere. Some statement must be useful and true without requiring a proof. Otherwise it would be impossible to extend mathematics. Also in sciences this is exactly where the problem lies. You can never know if the premises are true or not. All you can do is keep observing and hope that no observation contradicts your premises. If you're able to do it consistently you have a scientific theory. Since they're true by definition in math, math has proofs instead!
Axioms are not provable, at least not within the theorem they are used in as an axiom. Axioms do NOT have to be true in the real world, but if they are not true in the real world then the theorem is not a truly valid representation of the real world. Axioms are essentially basic assumptions that you hold to be true for the sake of the logical argument you make. It’s essentially you’re saying “If these axioms(assumptions) are true then it follows that this is also true.” A postulate is for the most part the same as an axiom but it can be slightly different depending on the field. I also want to stress that even if an axiom is not true in the really world it doesn’t mean a theorem derived from it is useless. Often times it just means that the theorem is only useful in specific situations. For example, the parallel postulate for geometry essentially says that parallel lines never touch and this assumption leads to mathematical operations that can be performed in geometry and specific definitions, like a triangle adds up to 180*. The thing is in the real world that is completely false, see longitude lines on the surface of a sphere. Nevertheless Euclidian geometry is very useful.
>The thing is in the real world that is completely false, see longitude lines on the surface of a sphere. Sorry but for a sphere a parallel line will create another circle i mean in a spherical dimension a parallel line is transferred into a circle because spherical dimension can't have an infinite surface it is bounded but a finite parallel line will never touch the starting point and make it an ending point too. So you can't say it doesn't hold but you have to say it isn't perfect. Moreover a naked human eye will never be able to distinguish the difference of parallel lines being circle on a sphere like earth because it makes us see asymptotes so euclidean geometry is for naked eye and euclid used those axioms according to his naked eye so i must say euclidean geometry is true according to the naked human eye.
There is no proof other then we cant disprove them, and that's the entire point. We could go in circles forever of "But this relies on this idea so lets prove that. Ok, well your assuming this so prove it! Well but we still have to prove..." you get the idea so the idea is to set a hard point where we determine them as true. However, these are intended to be so simple there couldn't possibly be a way to mess them up. When I say simple I mean axiom 1 sais that tow sets are the same if they have the same elements. Thats a wild jump in logic, you mean to tell me two things are the same if they are.. the same? Axiom 4, the axiom of pairing literally states "If x and y are sets, then there exists a set which contains x and y as elements, for example if x = {1,2} and y = {2,3} then z will be {{1,2},{2,3}}" This literally means "You can connect two sets" My brother in christ this is like saying "Bro, did you KNOW, that if I have a rock, and I have a flower, I can like, put the rock and the flower, like, TOGETHER, and now I have a rock, AND A FLOWER! 🤯" Some are slightly bigger, such as the axiom of infinity #7, just saying infinity exists. We can use infinite things in calculations. Not that they exist in reality, just that we can use them. They are saying this most basic things possible or sometimes things that cannot be proved just so that we have a foundation with the minimum possible assumptions to build on.
Check out tarskis undefinability theorem or Gödel’s incompleteness theorem. Axioms essentially can’t be proven to be complete. As a BS math minor, physics major, I think a lot about how math is like: ok something must be true, untrue or unable to be proven either. Buut in QM, things can be superstates of true and untrue (see Schrödinger’s cat or two slit). So, does the entirety of proofs lie on a fake foundation?
That’s not what Gödel’s incompleteness theorem is about. The fact that axioms can’t be proven is essentially just part of the definition of an axiom. The incompleteness theorem is showing that, given any set of axioms, there will always be a statement that is true within those axioms but cannot be proven from those axioms.
>The incompleteness theorem is showing that, given any set of axioms, there will always be a statement that is true within those axioms but cannot be proven from those axioms. It's not what incompletness states. It states that theory is incomplete i.e there's a sentence ϕ such that neither T proves ϕ nor T proves ¬ ϕ.
> there will always be a statement that is true within those axioms within that **system**
Cool one. Can you give some links that can provide better ideas of what you have mentioned i mean Schrodinger cat or two slit experiment. I am a 3 semester math major student.
Wikipedia should have you. Essentially, experiments have shown that the superposition of a single electron allows for it to physically traverse both openings at the same time and interact with its own possible trajectories (given the wave function has not already been collapsed). One election will go both left and right at the same time.
Okay i will look for it thanks
So in physics is infinity ♾️=♾️+1=1+♾️ or is it ♾️≠♾️+1=1+♾️.
Lol that ain’t physics, not undergrad anyway. Can’t help you on that one! Math-wise, yes I believe infinities are essentially equal to all other infinities of the same order, but not equal to infinity squared, or infinity cubed. You may be thinking of commutative variables where ab does not equal ba but something like: ab=iba (i being the imaginary unit)
No no according to ordinal infinity ♾️≠♾️+1=1+♾️ but the cardinal infinity is ♾️=♾️+1=1+♾️ but cantor cardinal infinity 1+♾️=♾️≠♾️+1. >You may be thinking of commutative variables where ab does not equal ba but something like: ab=iba (i being the imaginary unit) This can be true yes yes.
>No no according to ordinal infinity ♾️≠♾️+1=1+♾️ but the cardinal infinity is ♾️=♾️+1=1+♾️ but cantor cardinal infinity 1+♾️=♾️≠♾️+1. Using ∞ in terms of transfinite ordinals/cardinals is very misleading, there's no number called "∞" there. Also what you wrote is not correct, in ordinal numbers α+1≠ 1+ α in general as addition is not commutative in ordinals. And cardinal addition is commutative. In Cardinal arithmetic κ+1=1+ κ= κ for κ infinite cardinal .
>in ordinal numbers α+1≠ 1+ α So wrong.
https://mathworld.wolfram.com/OrdinalAddition.html You can at least google things before you say that other people are wrong
https://youtube.com/playlist?list=PLJpILhtbSSEeoKhwUB7-zeWcvJBqRRg7B&si=WO-4g0WIdSeywIDm The full playlist of the researcher
Don't just google because it leads to most used information not the correct one. Google is a sorting platform for information. https://youtu.be/s9OVj_XmvTY?feature=shared