All Questions
Tagged with topos-theory set-theory
36 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
21
votes
1
answer
2k
views
Surmounting set-theoretical difficulties in algebraic geometry
The category $\text{AffSch}_S$ of affine schemes over some base affine scheme $S$ is not essentially small. This lends itself to certain set-theoretical difficulties when working with a category $Sh(\...
- 3,019
21
votes
1
answer
1k
views
Logical endofunctors of Set?
What set-theoretic assumptions are necessary and sufficient to ensure the existence of a nontrivial (i.e. not isomorphic to the identity) endofunctor of the category Set which is logical (i.e. ...
- 66.8k
20
votes
1
answer
686
views
A nice subcategory of the category of measurable spaces
Is there some notion of "nice" measurable spaces and "nice" maps between them which satisfies the following properties?
The real line equipped with the Lebesgue $\sigma$-algebra is nice.
Any ...
- 301
19
votes
1
answer
2k
views
How much do universes matter in topos theory?
Suppose we fix two Grothendieck universes $\mathcal{U} \in \mathcal{V}.$ Then one has that $\mathcal{U}$-$\mathbf{Set},$ the category of $\mathcal{U}$-small sets, is a locally $\mathcal{U}$-small, $\...
- 15.5k
19
votes
3
answers
1k
views
Set-theoretic forcing over sites?
All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...
- 66.8k
18
votes
3
answers
2k
views
What's an example of a locally presentable category "in nature" that's not $\aleph_0$-locally presentable?
Recall the notion of locally presentable category (nLab): $\DeclareMathOperator{\Hom}{Hom}$
Definition: Fix a regular cardinal $\kappa$; a set is $\kappa$-small if its cardinality is strictly less ...
- 54.6k
17
votes
0
answers
2k
views
Toposophy vs Set theoretical multiverse philosophy
Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
- 10.1k
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
12
votes
1
answer
878
views
Higgs paper ``A category approach to Boolean valued set theory''
As Philip Scott says
about Denis Higgs:
In category theory, he wrote an influential and beautiful long paper, "A
category approach to Boolean valued set theory", which initiated many
early students ...
- 32.1k
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
10
votes
0
answers
222
views
Is any choice axiom other than WISC inherited by Grothendieck topoi?
It is well known that even if one works with say ZFC as a base theory, Grothendieck topoi do not in general satisfy even fairly weak axioms like countable choice or small violations of choice and one ...
- 1,947
10
votes
0
answers
391
views
How do properties of a partial order $\mathbb{P}$ affect the logic of the functor category $\mathsf{Set}^\mathbb{P}$?
$\DeclareMathOperator\true{\mathsf{true}}$I am very suspicious the answer to this (family of) question(s) is well-known, but I couldn't find anything after a bit of searching so I'll ask anyway.
I am ...
- 613
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
1
answer
310
views
Does a tight apartness relation on a subobject classifier imply the elementary topos is Boolean?
Given a set $S$, a tight apartness relation on $S$ is a relation $\#$ which is tight, irreflexive, symmetric, and weakly linear, or more specifically, a relation $\#$ such that
for all elements $a \...
- 2,624
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
7
votes
0
answers
370
views
Maps between forcing posets
We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is ...
- 35.5k
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
6
votes
0
answers
186
views
Does a well-pointed topos with enough projectives satisfy the internal axiom of chioice?
If yes, then I am also wondering if being well-pointed can be weakened to boolean (i.e. this is in the context of using Set as our metalogic so that well pointed Topoi are boolean). If not, then any ...
- 1,947
5
votes
2
answers
463
views
Subobjects as an object in a topos
Forgive me if this question turns out to be too elementary-then feel free to move it to stack exchange. I believe that this should be very basic fact from topos theory nevertheless being not familiar ...
- 9,330
5
votes
1
answer
409
views
Does the 2-category of topoi have exponential objects?
Does the 2-category of Grothendieck topoi have exponential objects?
There are size issues: Since Grothendieck topoi are supposed to have a small set of generators, the collection of objects of a ...
- 153
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
4
votes
1
answer
345
views
Under what assumptions does an elementary topos (+infinity) exist?
To prove there is an elementary topos with natural numbers object, it should be sufficient to assume ZF has a model. Probably ZF by itself, or IZF, is already sufficient. And probably even this is not ...
- 935
4
votes
0
answers
140
views
Is well-pointedness the reason that the internal/external distinction seems not to apply to $\mathbf{Set}$?
When reasoning about the category of sets, we usually don't have to worry about the internal/external distinction. For example, if $f : X \rightarrow Y$ is a morphism of sets, then $f$ is either ...
- 3,793
3
votes
0
answers
231
views
Clarification on the relationship of dream mathematics to ZFC and its potential as a synthetic measure theory
I'm interested in dream mathematics (https://ncatlab.org/nlab/show/dream+mathematics) as a foundation of "synthetic measure theory" in a similar vein as synthetic differential geometry, but ...
2
votes
0
answers
156
views
Do Grothendieck topoi with enough points satisfy the fan theorem internally?
Fourman and Hylland proved in the 80s that all spatial topoi satisfy the full fan theorem internally, while there are examples of localic topoi that do not satisfy it.
This leads one to conjecture a ...
- 1,947
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
1
vote
1
answer
112
views
Strong extensionality of 'membership' relation defined on the set of all morphisms of a well-pointed topos
We use the notion of category $C$ with one set of objects $\mathrm{Ob}(C)$ and one set of morphisms $\mathrm{Mor}(C)$, with source function $s:\mathrm{Mor}(C) \to \mathrm{Ob}(C)$ and target function $...
- 29
1
vote
0
answers
195
views
A categorial PCF theory?
I'm not an expert in PCF theory, so please forgive me if this question makes no sense.
I'm looking for a categorial version of PCF theory.
Specifically, if we replace $Set$ with another category, ...
- 773