Comment by wisnesky
2 years ago
A foundation for mathematics is any formalism sufficient to prove the results typically taken as axioms in practical mathematics. For example, in ZFC you can define numbers as sets in many different ways and prove that 1+1=2 for each of them - other foundations include higher order logic, topos theory, other set theories, etc.
You're getting at the problem with saying "foundation". If you can reach the same conclusions from a large number of starting points, there isn't any special set of ideas on which everything rests.
Why does the foundation need to be unique? It seems intuitive to me that they must not be unique.
Is it also possible to define set theory in terms of arithmetic?
Yes, at least with a strong enough arithmetic (such as Peano's), but that is usually more complicated; for example, you might have to create a Godel numbering or some other "deep embedding" to represent each set as a number. There's also so-called "reverse mathematics", which tries to determine the weakest axiom system capable of establishing a particular result.
Interesting, thanks!
You can certainly define some set theories using arithmetic but arithmetic as formalized by first-order Peano arithmetic can not define a set theory powerful enough to prove results about uncountable sets, such as large cardinal axioms or even properties of real numbers.
Any links/examples of why this isn't possible? My thinking is - if all existing set theory and set theoretic proofs are expressed in finite set-theoretic statements/expressions - can't we 'encode' these expressions within the system of arithmetic?
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