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Does set theory serve as the foundation of ALL math?

  1. Jul 28, 2012 #1
    What, if anything, does set theory have to do with integrating x^2 or finding the center of symmetry for a polygon?
     
  2. jcsd
  3. Jul 28, 2012 #2
    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.
     
  4. Jul 28, 2012 #3

    HallsofIvy

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    Before you can have functions on numbers, like [itex]f(x)= x^2[/itex], 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.
     
  5. Jul 28, 2012 #4
    I read that in Devlin's The Language 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.
     
  6. Jul 28, 2012 #5
    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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