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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$\begingroup$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.
$\endgroup$ 14 $\begingroup$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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