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Results tagged with lo.logic
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user 2926
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.
44
votes
Accepted
Completion of a category
Yes, it's a general construction which is related to so-called Isbell conjugation.
Let $C$ be a small category. It is well-known that the free colimit cocompletion is given by the Yoneda embedding i …
- 53.3k
40
votes
Accepted
Is $π$ definable in $(\Bbb R,0,1,+,×,<,\exp)$?
It seems to me that Schanuel's conjecture (which is a kind of article of faith in transcendental number theory, but of course very far from proven itself) ought to imply that $\pi$ is not definable in …
- 53.3k
26
votes
Is the non-triviality of the algebraic dual of an infinite-dimensional vector space equivale...
This is a very partial answer (really in response to Thierry's question) which indicates that it is not provable in ZF that $V^\ast \neq \{0\}$ for every vector space $V$. This answer piggybacks on an …
- 53.3k
20
votes
Accepted
Where in ordinary math do we need unbounded separation and replacement?
I asked the same question about the replacement axiom not long ago at the $n$-Category Café, and the answer I got back from Mike Shulman is that it's used for example in the transfinite construction o …
- 53.3k
20
votes
What is neutral constructive mathematics
You'll probably have better luck with the phrase "intuitionistic higher-order logic" (IHOL). A good place to start is the book by Lambek and Scott, Introduction to Higher Order Categorical Logic. But …
- 53.3k
18
votes
Is there a natural bijection from $\mathbb{N}$ to $\mathbb{Q}$?
To my aesthetic sensibilities, the Calkin-Wilf tree response is pretty close to optimal, but I'll add some additional glosses. (I only noticed later that Vladimir Dotsenko wrote something similar befo …
- 53.3k
17
votes
Is there a constructive proof of Cantor–Bernstein–Schroeder theorem ?
If you accept that toposes are models of constructive set theory, then another way to answer the question is to give a (non-Boolean) topos where the CBS theorem fails; that would show that this theore …
- 53.3k
17
votes
Accepted
Coproducts of complete Boolean algebras
Chris Heunen's comment under the OP can be turned into a proof. Suppose the category of compact Hausdorff extremally disconnected spaces has binary products. Let $X \times Y$ denote the product in tha …
- 53.3k
15
votes
Accepted
Logical complexity of algebraically closed fields
From Dirk van Dalen's Logic and Structure: the theory of algebraically closed fields is not finitely axiomatizable (see page 109 and preceding).
- 53.3k
13
votes
How constructive is Doyle-Conway's 'Division by three'?
(I'm cutting and pasting and slightly modifying some comments taken from a discussion on this question currently taking place at the nForum. It's based on my memory of their paper, which I do not have …
- 53.3k
12
votes
What if Current Foundations of Mathematics are Inconsistent?
Voevodsky is not the only one who hopes for a proof of inconsistency (as mentioned in Dick Palais's answer): see Conway and Doyle's Division by Three, bottom of page 34, where they express the same ki …
12
votes
Logic in mathematics and philosophy
A more recently developed candidate might be Linear Logic, which is a successful formalization of modes of reasoning of considerable philosophic interest. I highly recommend Jean-Yves Girard's inimita …
- 53.3k
12
votes
Connections between ultrafilters in topology and logic
It's a multifaceted question, and answers will be multifaceted too.
At a simpler level, you know doubt know that an ultrafilter on a set $X$ can be identified with a Boolean algebra homomorphism $PX …
- 53.3k
11
votes
Accepted
Propositional logic with categories
Qiaochu, let me see if this answers your question:
Proposition: Suppose $B$ is a cartesian closed category with finite coproducts such that the canonical double dual embedding
$$b \to (b \Rightarr …
- 53.3k
11
votes
Is PA consistent? do we know it?
I am a little baffled by some of this discussion. It seems everyone agrees that consistency of PA is a theorem, if you accept some stronger system, such as ZFC. So, PA is consistent relative to ZFC. J …