Does set theory serve as the foundation of ALL math?

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Set theory can serve as a foundational framework for all mathematics, but it is not the only option available. As long as a starting point satisfies the Peano Axioms, various methods can construct the same mathematical structures. The choice of foundational framework is less significant than the axioms it adheres to, as identical axioms yield similar mathematical outcomes. While sets provide an effective organizing principle for defining numbers and functions, they are not strictly necessary for mathematical applications. Ultimately, the relationship between set theory and specific mathematical operations, like integration and symmetry, highlights the interconnectedness of different mathematical concepts.
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What, if anything, does set theory have to do with integrating x^2 or finding the center of symmetry for a polygon?
 
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Set theory can serve as the foundation of all math. That does not mean it HAS TO.

As long as you start with a group of entities that satisfy the Peano Axioms, you are set to construct the whole mathematics as we know it. Sets are one way to construct this group of entities. There are, indeed, other methods possible.

No matter what foundation you take as a starting point, if it satisfies the Peano Axioms, the mathematics you will arrive at will be pretty much like the one we use now. This means the foundation of mathematics is not that important compared to the axioms it satisfies. Same axioms mean same mathematics, even if the foundations are different.
 
Before you can have functions on numbers, like f(x)= x^2, you first must have numbers. And while, as millenial said, it is not NECESSARY to use sets, a good definition of "real numbers" is as certain sets of rational numbers ("Dedekind cuts"), a good definition of rational numbers is as equivalence classes of integers, under a certain equivalence relation, a good definition of integers is as equivalence classes of whole numbers, under a certain equivalence relation, and a good definition of whole numbers is as sets: 0 is the empty set, {}, 1 is the set whose only member is the empty set, {{}}, 2 is the set whose only members are 0 and 1, {0, 1}= {{},{{}}}, etc.

It is not necessary to use sets, but they are a good "organizing principle".
In any case, one seldom refers to this when 'applying' mathematics. It is rather a matter of the logical foundation of mathematics.
 
HallsofIvy said:
a good definition of whole numbers is as sets: 0 is the empty set, {}, 1 is the set whose only member is the empty set, {{}}, 2 is the set whose only members are 0 and 1, {0, 1}= {{},{{}}}, etc.

I read that in Devlin's The Langauge of Mathematics. He said that was "very clever." Being young in mathematics, I thought that was beyond clever. It blew my mind a little bit.

I'd suggest that book to anyone. I never thought I would read a mathematics book.
 
thetaobums said:
What, if anything, does set theory have to do with integrating x^2 or finding the center of symmetry for a polygon?

Indefinite integration as an operation is a mapping between sets of functions (definite integration maps from sets of functions to the set of real numbers). The symmetries of a polygon are deeply related to group theory, which studies certain kinds of operations defined on sets.
 
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