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Having known about what axioms are, I want to know whether there are some "fundamental axioms of mathematics" on which every branch of mathematics depends. If yes, what are they ?

Or Do we have different axioms in each branch of mathematics??

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  • $\begingroup$ The rule of detachment is crucial almost everywhere. It says that if $A$ is true, and if $A$ implies $B$, then you can conclude $B$. $\endgroup$
    – MJD
    Commented Oct 10, 2014 at 17:17

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Whatever axioms that you might be working with, someone will ask themselves what happens if we remove them, and a new branch will be formed.

If we confine ourselves to mainstream mathematics, then I suppose that induction axioms, modus ponens, and existential instantiation (along with the Leibniz laws about equality) would make the fundamental axioms.

But axioms describe objects. They tell us what are the formal properties of objects are, so we can all work with them, even though we might not be able to comprehend these objects as physical manifestations (e.g. numbers which are very very large, or very very small); or infinite sets or so on.

Since different fields of mathematics deal with different objects, they will care about the axioms relevant to those objects. In a field where the research focuses on categories, the axioms of a category will be fundamental; in a field where sets are the basis, the axioms of set theory will be fundamental. Sometimes we can study one field using a different field (e.g. study set theory via category theory, or vice versa) in which case the importance of some axioms will change from here to there and from one study to another.

But again, if pressed to the wall, I'd say that a few logical axioms, inference rules and induction are the most fundamental. Although ultrafinitists might reject those as well.

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    $\begingroup$ Can i know what is modus ponens and existential instantiation. I am assuming induction axioms are nothing but the principle of mathematical inductions. $\endgroup$
    – Jasser
    Commented Oct 10, 2014 at 16:52
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    $\begingroup$ Well, that is getting of the head, it will be very nice of you to tell me any resources to start these things off since it will be better to know from an expert the proper way to study. $\endgroup$
    – Jasser
    Commented Oct 10, 2014 at 17:01
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    $\begingroup$ I see. I'd study some logic, maybe some naive set theory wouldn't hurt too. $\endgroup$
    – Asaf Karagila
    Commented Oct 10, 2014 at 17:55
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    $\begingroup$ @user291957: That's not foundations, that's probability. $\endgroup$
    – Asaf Karagila
    Commented Oct 13, 2014 at 14:16
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    $\begingroup$ @user291957 foundations are about what the most basic level of mathematical reasoning is. It is about what logic should be picked (e.g. First Order Logic or Second Order Logic) so it can be used to build all contemporary mathematics or whether such logic exists. What is the strength of particular logic and what justify its use. What is intuitive enough to be taken as axiom/reasoning an why. Those are often philosophical questions. Anyway if you want to start you should read some book on First Order Logic. "Mathematical Logic" by Ian Chiswell and Wilfrid Hodges is good for beginners. $\endgroup$ Commented Oct 14, 2014 at 10:17
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As others have pointed out, there are certain axioms that are fundamental, in the sense they are assumed to be true in any other field of maths, but I think it is worth adding that axioms are in many ways very pragmatic things: they are there because they serve a practical purpose, not because they are connected to some sort of "Deep Truth". As an example take the group axioms:

A group is a pair, $(G,\cdot)$, where $G$ is a set, and $\cdot$ is a function $\cdot \colon G \times G \rightarrow G$, so that

  • $\forall a, b, c \in G \colon (a \cdot b) \cdot c = a \cdot (b \cdot c)$
  • $\exists \ e \in G$ so that $\forall a \in G \colon e \cdot a = a \cdot e = a$
  • $\forall a \in G \ \exists \ a^{-1} \in G \colon a \cdot a^{-1} = a^{-1} \cdot a = e$

These axioms are not really what one would call obviously true - they are simply what you need to get a working theory with useful properties.

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  • $\begingroup$ I thought that was the definition of a group, not an axiom? ie. If pair has these properties, then we give them a special name: group. $\endgroup$
    – Elliott
    Commented Sep 2, 2021 at 5:24
  • $\begingroup$ @Elliott Well, the point is that axioms are nothing more than the fundamental definitions, that you need for some concept to work. Some axioms are universally useful and seem 'obviously true', but one still has to make the decision to accept them as true. Consider the axiom choice - it is very useful and in some forms it seems to be obviously true, but in others less so; but it is very useful, so most mathematicians choose to accept it, although probably with some misgivings. $\endgroup$
    – j4nd3r53n
    Commented Sep 2, 2021 at 7:27
  • $\begingroup$ axioms are nothing more than the fundamental definitions this isn't true. Consider the reflective axiom: Things equal to the same thing are equal to each other; this is not just a definition. In theory we could choose a naming as an axiom: All cats with fur are called Fluffy Cats, but the axiom literally adds no additional provability to the axiom set, and in practice this is never done (we aim for minimal sets). It's just true by its own definition. It's simply a naming. This isn't the same as it being "obviously true", like the reflective axiom or the axiom of choice. $\endgroup$
    – Elliott
    Commented Sep 2, 2021 at 8:38
  • $\begingroup$ @Elliott I think we probably have to simply politely disagree; to me nothing is absolutely true - we always have to decide that we believe in any axiom. It may be very counter-intuitive to reject some axioms, like the one you quote, but it is not, IMO, true in and of itself - I decide to call it true and it makes good sense to most, but I could decide that it isn't. And be called crazy, but that's an entirely different matter :-) $\endgroup$
    – j4nd3r53n
    Commented Sep 2, 2021 at 10:03
  • $\begingroup$ Sorry if I'm coming across rude - I'm just passionate. I actually completely agree with you that nothing is truly knowable. For instance I assume that I exist, I don't believe that I can prove that I exist. Dealing with axioms we do get close to these sorts of non-mathematical, philosophical arguments, but what we're talking about really isn't at that level. An axiom isn't just a naming. $\endgroup$
    – Elliott
    Commented Sep 2, 2021 at 10:20

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