One of the things that we always say is that we can recreate all of mathematics using set theory as a basis. What does that mean? Basically, it means that given some other branch of math, which works with some class of objects O using some set of axioms A, we can define a set-based construction of the objects of S(O), and them prove the axioms A about S(O) using the axioms of ZFC.<\/p>\n
\nLet’s take a look at what that means, by showing how all of the proofs of number theory using natural numbers can be wrapped up in set theory.
\nFirst, we need to define the class of objects that we’re going to talk about. That’s the set of natural numbers. (That switch from class to set *was* deliberate; we know that the natural numbers are a set.) We’ve already seen the basic construction: it’s the axiom of infinity. The set of natural numbers starts with 0 – which we represent as ∅. For each additional natural number N, it’s represented by the set of all numbers smaller than N:
\n1. 1 = { 0 } = {∅}
\n2. 2 = { 0, 1 } = { ∅, {&empty}}
\n3. 3 = {0, 1, 2} = { ∅, {&empty}, {∅, {∅}}}
\n4. …
\nNow, we need to show that the axioms that define the meaning of the natural numbers are true when applied to this construction. For natural numbers, that means we need to show that the [Peano axioms][peano] are true.
\n[peano]: http:\/\/scienceblogs.com\/goodmath\/2007\/01\/basics_natural_numbers_and_int_1.php
\nThe first Peano axiom is that 0 is a natural number. We’ve got that one.
\nThe second Peano axiom is that every number is equal to itself. No problem, we’ve already got set equality defined in terms of subsets, so we’ve got that.
\nThe third and fourth axioms are the symmetry and transitivity of equality. Set equality is transitive and symmetric, so no problem there.
\nThe fifth Peano axiom is that anything equal to a number is a number. There is no way to create a set that will be equal to any of the set-based numbers, but that aren’t exactly identical to one of the set-based numbers.
\nThe sixth Peano axiom is that every number has a successor. In the set number construction, given a number, N, the successor N+1 is N∪{N}. So we’ve got a set based definition of successor; and further, it obviously satisfies the 7th axiom – that a=b if and only if a+1=b+1.
\nThe 8th Peano axiom is an easy one: there is no natural number N such that N+1=0. Since 0=∅, in set terms, that means that there in no N such that N∪{N}=∅. N∪{N} must have *at least* one element: N. So it can’t ever be ∅ and voila, axiom 7 is satisfied.
\nThe ninth Peono axiom is induction: and the axiom of infinity is specifically designed to permit inductive proofs. So we’ve got that.
\nSo – the we’ve got a construction of the natural numbers using sets; and we can easily prove that the Peano axioms are true and valid for that construction. So using that, any proof about the natural numbers can be reduced to a proof in terms of the axioms of ZFC.<\/p>\n","protected":false},"excerpt":{"rendered":"
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