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Results tagged with lo.logic
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user 22131
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.
11
votes
Accepted
Why is it not possible to define the necessity operator internally $\Box: \Omega \to \Omega$...
As said in the comment, I'm not sure what to add to the paragraph, the point is that in a topos there is no functions $\Omega \to \Omega$ that has the property expected of a neccessity operator except …
- 12.9k
18
votes
Is Bauer–Hanson’s result “there is a topos where the Dedekind reals are countable” novel?
A first big difference between Brauer & Hansen's result and the one you are talking about is that CZF is a predicative theory (it doesn't have power set/power object) so consistency with CZF doesn't …
- 35.5k
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 …
- 22.3k
9
votes
Accepted
Do all toposes satisfy the internal Zorn's lemma?
Assuming the law of excluded middle internally your formulation of Zorn's lemma is equivalent to the axiom of choice by the usual argument. Now there are Boolean Grothendieck topos in which the axiom …
- 42.4k
7
votes
Accepted
Are lists in homotopy type theory free $A_\infty$-spaces?
In an informal sense, the answer "should be yes", in the sense that if one ignore type theory and work with an $\infty$-topos one can make sense of the construction $List(A)$ either by the usual unive …
- 42.4k
6
votes
Assuming decidable equality but not LEM in HoTT
Decidability of every set implies the law of excluded middle as soon as there is a "subobject classifier".
Indeed, for every proposition $U$, the fact that "$U = \mathsf{True}$ or $U \neq \mathsf{True …
- 204
4
votes
1
answer
142
views
Decomposition of an ultrafilter on the fibers of a map
Short version: If I have a map $f:Y \to I$, and $\mu$ an ultrafilter on $Y$, under what condition can $\mu$ be written as a limit/sum/integral of ultrafilters on the fibers of $f$ along the ultrafilte …
- 73.2k
6
votes
1
answer
341
views
How to build large recursive ordinals using Dillator and/or Ptykes?
I've only recently learned about Girard's theory of Dilators and Ptykes, and I find this theory very elegant, but it is not clear at all to me whether/how it can be used to produce ordinal notations f …
- 5,831
3
votes
Accepted
Products of double-negation sublocales (and probability distributions on them)
For your first question, if $X$ and $Y$ are two boolean locale then $X \times Y$ is boolean only if $X$ or $Y$ is discrete. So unless $\neg \neg A$ or $\neg \neg B$ are discrete, $\neg \neg A \times \ …
- 50.6k
7
votes
Accepted
Comparing Kripke-Joyal semantics of toposes to model-theoretic satisfaction
As you observe yourself, the question does not quite make sense as $\phi$ in 1. is a formula in the first order language of a category and in 2. $\phi$ is a formula in higher order logic (something li …
- 42.4k
13
votes
Accepted
Alternatives to "Sketches of an Elephant" Volume 3
As I said in the comment, this would involve a very large number of different references! (almost one by subsection...)
But to some extent, contributors to the nLab already started doing that and it i …
- 103
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
8
votes
Characterization of 'canonical' natural numbers objects
I suspect there is no good answer to the question:
The type theoretic results you are mentioning definitely have a category theoretic interpretation in fact their proof using gluing is already very ca …
- 42.4k
4
votes
Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / mul...
So, I wanted to say something, I don't think this answer the questions, but that was definitely too long for a comment. In short, this is just the result of me trying to make sense of this idea:
Comin …
- 42.4k
10
votes
Accepted
Is there an equivalent of the incompleteness theorems/halting problem in category theory?
There is a category theoretic version of the incompleteness theorem originally due to André Joyal that has been unavailable for a long time, but has been written up not so long ago by Joost van Dijk a …
- 42.4k