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Results tagged with foundations
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user 22131
Mathematical logic, Set theory, Peano arithmetic, Model theory, Proof theory, Recursion theory, Computability theory, Univalent foundations, Reverse mathematics, Frege foundation of arithmetic, Goedel's incompleteness and Mathematics, Structural set theory, Category theory, Type theory.
21
votes
Accepted
Are there substantive differences between the different approaches to "size issues" in categ...
From the point of view of a category theorist, I would say there are many fundamentally non-equivalent way of handling size questions - but they are in a completely different direction than what is me …
- 42.4k
26
votes
Accepted
Why do we care about small sets?
The problem isn't that you are not allowed to make some constructions - there are foundations that let you do pretty much everything you want with as many different sizes as you want - but only that you … As pointed out by Andrej Brauer, in some foundations this might cause problems, but there definitely are foundations that can handle that sort of thing fine - for example the one you are talking about …
- 42.4k
11
votes
Is material set theory conservative over structural set theory?
This will obviously be highly dependent on the concrete theory you are considering. But overall the answer is yes.
The most general version of these results I'm aware of are in Mike Shulman's Comparin …
- 42.4k
38
votes
Accepted
Could groups be used instead of sets as a foundation of mathematics?
The answer is yes, in fact one has a lot better than bi-interpretability, as shown by the corollary at the end. It follows by mixing the comments by Martin Brandenburg and mine (and a few additional d …
- 42.4k