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comment Why is the Axiom of Infinity necessary?
@NoahSchweber So, if you do not assume a non-empty universe in your rules of logic you could not infer the existence of any sets in ZFC without AOI. Then FOL gives your something for nothing. Not a bad deal!
3h
comment Why is the Axiom of Infinity necessary?
The Axiom of Infinity (AOI) is the only ZFC axiom the postulates the existence of any set or object. Except for the Axiom of Extensionality that defines set equality, all other axioms tell us only how to construct a set from other set(s) that are presumed to exist. Therefore, to prove the existence of any set in ZFC, you must start with the infinite set postulated to existent by AOI. See my revised answer.
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revised Why is the Axiom of Infinity necessary?
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answered Why is the Axiom of Infinity necessary?
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comment Can I do universal instantiation on this predicate?
It is certainly possible in "standard" FOL, but I don't think you will ever see it applied in analysis, geometry, or any kind of algebra.
1d
comment Which is the most powerful language, set theory or category theory?
I have attempted to formalize the set theory and rules of logic that it seems the average mathematician actually uses. They are the basis for software I have developed to teach undergrads the basic methods of proof (visit my website at dcproof.com ). There is only a passing resemblance to ZFC and "standard" FOL, but I think my system is much more intuitive and user-friendly. Anything you can prove using my system should be provable in ZFC and FOL, however.
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revised Use logic quantifiers to write…
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