All Questions
15 questions
22
votes
1
answer
1k
views
How strong is "all sets are Lebesgue Measurable" in weaker contexts than ZF?
Famously, Solovay showed that, if $\textrm{ZFC}$ plus $\textrm{IC}$ (the existence of an inaccessible cardinal) is consistent, then so is $\textrm{ZF}$ plus $\textrm{DC}$ (dependent choice) plus $\...
- 866
15
votes
3
answers
1k
views
A rigid type of structure that can be put on every set?
Call a type of structure rigid if any automorphism of such a structure is an identity. (This is a bit different from some other uses of the word, but hopefully I'll be forgiven.) For example, well-...
- 66.8k
11
votes
1
answer
670
views
Do all toposes satisfy the internal Zorn's lemma?
I came up with this question when trying to give a more detailed answer to a question by Tim Campion in a comment to Ingo Blechschmidt's answer to Examples of statements that are valid in every ...
- 18.4k
10
votes
1
answer
780
views
When does a topos satisfy the axiom of regularity?
In categorical set theory, we observe that certain topoi satisfy (suitable versions of) certain axioms from set theory. For example, Lawvere's $\mathsf{ETCS}$ asserts that $\mathbf{Set}$ is a well-...
- 788
9
votes
2
answers
2k
views
Truth in a different universe of sets?
I understand that provability and truth as different concepts.
Provability is syntactic, it only concerns whether the given
sentence can be derived by reiterating the inference rules over a
collection ...
- 5,230
9
votes
2
answers
2k
views
Using the multiverse approach to decide the law of the exluded middle?
Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
user2529
9
votes
0
answers
149
views
Are there good criteria for the topological models where BD-N and BD hold?
A (non-empty/inhabited) subset $S$ of $\mathbb{N}$ is said to be pseudo-bounded if for every sequence $x_n$ in $S$ we have
$\lim_{n\to \infty} \frac{x_n}{n} = 0$
Clearly all bounded subsets are pseudo-...
- 1,947
8
votes
1
answer
343
views
What topos-theoretic construction lies behind the “symmetric model” construction (used to refute AC) in Set Theory?
Suppose we want to prove that (classical!) $\mathsf{ZF}$ does not prove, say, “for every infinite set $A \subseteq \{0,1\}^{\mathbb{N}}$ there exists an injection $\mathbb{N} \to A$” (I take this ...
- 32.5k
7
votes
2
answers
957
views
Can models of set theory contain extra ordinals?
In the paper "Complete topoi representing models of set theory" by Blass and Scedrov, they consider a general notion of Boolean-valued model of set theory, and one of the conditions they impose is ...
- 66.8k
7
votes
1
answer
1k
views
Encoding fuzzy logic with the topos of set-valued sheaves
One of the canonical examples used by Barr & Wells in order to motivate the use of topoi is that we can construct a theory for fuzzy logic and fuzzy set theory as set-valued sheaves on a poset (...
- 2,537
7
votes
0
answers
156
views
Logical properties of realizability (topoi or McCarty models) defined by alpha-recursion on admissible ordinals
Setup: Let $\alpha$ be an admissible ordinal (viꝫ., one such that $L_\alpha$ is a model of Kripke-Platek set theory), identified as usual with the set of ordinals $<\alpha$. Then there is a ...
- 32.5k
7
votes
0
answers
234
views
How much is known about the consistency strength of toposes and topos-like categories?
It's a well-known fact that the theory of a well-pointed topos with a natural numbers object (NNO) has the same consistency strength as MacLane set theory (also known as bounded Zermelo). There are ...
- 12.5k
6
votes
0
answers
190
views
Is Vopěnka's principle inherited by Grothendieck topoi?
I call the Vopěnka's principle:
Every subfunctor of an accessible functor is accessible
but other formulations (which may lose equivalence in weak contexts?) are also interesting to me.
If this is ...
- 6,962
5
votes
1
answer
502
views
Failure of SVC in Grothendieck toposes
The axiom SVC (for "small violations of choice") asserts that there is a set $S$ such that for every set $X$ there is a choice set $A$ such that $X$ is a subquotient of (i.e. admits a surjection from ...
- 66.8k
2
votes
0
answers
148
views
Dedekind reals in heyting valued models
Let $V^{H}$ be a Heyting valued model of intuitionistic set theory. What conditions does $H$ have to satisfy in order for the following claim to hold? (where $\| \phi(u) \| \in H$ is the truth value ...
- 631