Foundations of Mathematics
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[(Up)](../../README.md#topics) | _See also: [Mathematics](../Mathematics/README.md#mathematics), [Logic](../Logic/README.md#logic), [Incompleteness](../Incompleteness/README.md#incompleteness)_

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### Web resources

[What is a natural number?](https://philosophy.stackexchange.com/questions/106389/what-is-a-natural-number) ★

[Most \'unintuitive\' application of the Axiom of Choice?](https://mathoverflow.net/questions/20882/most-unintuitive-application-of-the-axiom-of-choice) ★

[Why worry about the axiom of choice?](https://mathoverflow.net/questions/22927/why-worry-about-the-axiom-of-choice) ★

[How much of the axiom of choice do you need in mathematics?](https://mathoverflow.net/questions/452509/how-much-of-the-axiom-of-choice-do-you-need-in-mathematics) ★

[What can be preserved in mathematics if all constructions are carried out in ZF?](https://mathoverflow.net/questions/430838/what-can-be-preserved-in-mathematics-if-all-constructions-are-carried-out-in-zf) ★

[Lists as a foundation of mathematics](https://mathoverflow.net/questions/456649/lists-as-a-foundation-of-mathematics) ★

[Is there any physical or computational justification for non-constructive axioms such as AC or excluded middle?](https://mathoverflow.net/questions/323616/is-there-any-physical-or-computational-justification-for-non-constructive-axioms/323754#323754) ★

[Does changing the universe of set theory change the definition of truth?](https://mathoverflow.net/questions/464234/does-changing-the-universe-of-set-theory-change-the-definition-of-truth) ★

[Bourbaki\'s definition of the number 1](https://mathoverflow.net/questions/357498/bourbakis-definition-of-the-number-1) ★

[The Origin of the Number Zero \| History \| Smithsonian](https://www.smithsonianmag.com/history/origin-number-zero-180953392/)

[How to rewrite mathematics constructively?](https://mathoverflow.net/questions/381908/how-to-rewrite-mathematics-constructively) ★

[How should a \"working mathematician\" think about sets? (ZFC, category theory, urelements)](http://mathoverflow.net/questions/255820/how-should-a-working-mathematician-think-about-sets-zfc-category-theory-ur) ★

[Set theories without \"junk\" theorems?](http://mathoverflow.net/questions/90820/set-theories-without-junk-theorems/90945#90945) ★

[Why hasn\'t mereology succeeded as an alternative to set theory?](http://mathoverflow.net/questions/58495/why-hasnt-mereology-succeeded-as-an-alternative-to-set-theory) ★

[Which is the most powerful language, set theory or category theory?](https://math.stackexchange.com/questions/1639982/which-is-the-most-powerful-language-set-theory-or-category-theory) ★

[Defining the standard model of PA so that a space alien could understand](https://mathoverflow.net/questions/332247/defining-the-standard-model-of-pa-so-that-a-space-alien-could-understand) ★

[ULTRAINFINITISM, or a step beyond the transfinite](http://mathoverflow.net/questions/100981/ultrainfinitism-or-a-step-beyond-the-transfinite) ★

[Set-theoretical multiverse and foundations](http://mathoverflow.net/questions/78018/set-theoretical-multiverse-and-foundations) ★

[New Foundations and weaker forms of choice](http://mathoverflow.net/questions/81577/new-foundations-and-weaker-forms-of-choice) ★

[getting rid of existential quantifiers](http://mathoverflow.net/questions/90324/getting-rid-of-existential-quantifiers) ★

[Paris--Harrington theorem - Wikipedia, the free encyclopedia](https://en.wikipedia.org/wiki/Paris%E2%80%93Harrington_theorem)

[What sorts of extra axioms might we add to ZFC to compute higher Busy Beaver numbers?](https://mathoverflow.net/questions/380971/what-sorts-of-extra-axioms-might-we-add-to-zfc-to-compute-higher-busy-beaver-num) ★

[Are there first-order statements that second order PA proves that first order PA does not?](https://mathoverflow.net/questions/366580/are-there-first-order-statements-that-second-order-pa-proves-that-first-order-pa) ★

_(in [Type Theory](../Type%20Theory/README.md#type-theory))_ [Type Theory and Mathematical Logic \| artagnon.com](https://artagnon.com/logic) ★


### Books

_(in [Logic](../Logic/README.md#logic))_ Mathematical Logic (Kleene) (borrow @ [archive.org](https://archive.org/details/mathematicallogi0000klee)) 🏛️ [💭](commentary/cpressey.md#mathematical-logic-kleene)

_(in [Mathematics](../Mathematics/README.md#mathematics))_ Compactness and Contradiction (online @ [terrytao.wordpress.com](https://terrytao.wordpress.com/wp-content/uploads/2011/06/blog-book.pdf))