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Results tagged with lo.logic
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user 49
first-order and higher-order logic, model theory, set theory, proof theory, computability theory, formal languages, definability, interplay of syntax and semantics, constructive logic, intuitionism, philosophical logic, modal logic, completeness, Gödel incompleteness, decidability, undecidability, theories of truth, truth revision, consistency.
61
votes
Accepted
How to rewrite mathematics constructively?
If you want a "general method" that "always works" to turn a classical theorem into a constructive one, there are double-negation translations: if you add enough $\neg\neg$s to a classical theorem, yo …
- 66.8k
39
votes
A better way to explain forcing?
This is an expansion of David Roberts's comment. It may not be the sort of answer you thought you were looking for, but I think it is appropriate, among other reasons because it directly addresses yo …
- 66.8k
29
votes
Accepted
Formalizations of the idea that something is a function of something else?
First of all, it seems to me as though the real question here is "what is a variable quantity?" Most of the definitions you quote from pre-20th century mathematicians assume that the notion of "varia …
- 66.8k
29
votes
Defining $SU(n)$ in HoTT
I think Noah's answer is mostly right, but partly misleading, and explaining why will take too much space for a comment, so I'm posting a separate answer.
As Noah says, the main conceptual point is t …
- 66.8k
28
votes
Accepted
Category of categories as a foundation of mathematics
My personal opinion is that one should consider the 2-category of categories, rather than the 1-category of categories. I think the axioms one wants for such an "ET2CC" will be something like:
Firs …
- 66.8k
24
votes
What is so special about set theory anyway?
It's not clear to me whether your question is more about the role of sets versus other foundational objects, or about how set theory can be extended with large cardinal axioms to discuss models of str …
- 66.8k
21
votes
Accepted
Precise relationship between elementary and Grothendieck toposes?
There are known statements that are true in any Grothendieck topos, but not in every elementary topos with NNO. For instance:
Freyd's theorem that a complete small category is a preorder is not con …
- 66.8k
20
votes
Accepted
Coinduction for all?
This is a question that I've puzzled about myself, and I don't pretend to have The Answer. But here's one thought that I've found illuminating. Let's start by comparing the behavior of induction and …
- 66.8k
19
votes
Accepted
Which kind of foundation are mathematicians using when proving metatheorems?
Your question is much more specific than your title suggests. As to the question itself, my answer is that it doesn't matter. The proof is given in mathematics, not in any formal system. A foundati …
- 66.8k
18
votes
Accepted
What kind of category is generated by Cubical type theory?
There are two kinds of answers as to what kind of category a "homotopy type theory" is the internal language of. On the one hand there is a kind of $(\infty,1)$-category that is the semantic object o …
- 66.8k
16
votes
Several Topos theory questions
Just to be clear: the category of sheaves on the big Zarsiki site is a
topos only if "big Zariski site" refers to the category of finitely
presented commutative rings, or else some other small subcate …
- 66.8k
16
votes
Taking a theorem as a definition and proving the original definition as a theorem
A Grothendieck topos can be defined either as the category of sheaves on a site, or as a category satisfying Giraud's axioms, or as an elementary topos that is bounded over $\rm Set$. I believe any o …
16
votes
How much of concrete mathematics can be expressed in the language of category theory?
I agree that the question is broad, but here's one sense in which the answer is "all of it". The functor $\rm Grp \to Set$ is represented by the object $\mathbb{Z}$ (the free group on one element); t …
16
votes
Variable-centric logical foundation of calculus
Here is another approach, which I believe I first learned from Toby Bartels. Suppose $X$ is an arbitrary differentiable manifold (think of the state space of some physical system), and define a varia …
- 66.8k
15
votes
Accepted
Category theorists stance on deductive systems
This sentence reads to me like "we will treat Xs as if they were special kinds of of Ys, which will make both X-theorists and Y-theorists unhappy because it is not true". Reminds me of the old joke w …
- 66.8k