All Questions
Tagged with set-theory model-theory
401 questions
0
votes
0
answers
95
views
How near are a groupoid and its 'preorderification'?
As remarks, a groupoid is a category with only (categorical) isomorphisms as its morphisms and a preorder is a category only having one morphism between each object. If we choose one isomorphism by ...
1
vote
0
answers
99
views
Is this theory synonymous with ZF + Global Choice?
$\textbf{Logic:}$ Mono-sorted first order logic with equality.
$\textbf{Extralogical Primitives: } <, \in$
Define: $x > y \iff y < x \\ x \leq y \iff x < y \lor x=y \\ x \not > y \iff \...
- 11.3k
1
vote
0
answers
89
views
About synonymy relationships around these two theories?
The following question is about patterns of synonymy relationships around two theories, $T^+$ and $\sf PA$.
For purposes of self inclusiveness I'll re-iterate $T$ and its extensions.
$\textbf{Logic:}$ ...
- 11.3k
4
votes
1
answer
144
views
Stably embedded clone
Let $M$ be a first-order structure, considered as an element of the ambient set-theoretic universe $V$.
Clearly, for any $L_{\infty,\infty}$-formula $\varphi(\bar x)$ (with $\bar x$ finite, say) in ...
- 12.5k
6
votes
0
answers
251
views
Whence compactness of automorphism quantifiers?
The following question arose while trying to read Shelah's papers Models with second-order properties I-V. For simplicity, I'm assuming a much stronger theory here than Shelah: throughout, we work in $...
- 21.2k
6
votes
0
answers
287
views
Do maximal compact logics exist?
By "logic" I mean regular logic in the sense of abstract model theory (see e.g. the last section of Ebbinghaus/Flum/Thomas' book). My question is simple:
Is there a logic $\mathcal{L}$ ...
- 21.2k
3
votes
0
answers
117
views
Topologically symmetric models of $\mathsf{ZFA}$
The standard construction of permutation models (i.e. models of $\mathsf{ZFA}$ involves choosing some collection of atoms $A$, a group $G$ of permutations on these atoms, and then a normal filter $\...
- 131
3
votes
0
answers
211
views
Intuitionistic set-theoretic geology
Work in ZF, if there are proper class many supercompact cardinals, then all grounds are uniformly definable. Hence under reasonable assumption, we can have choiceless set-theoretic geology.
But can we ...
- 63.9k
13
votes
1
answer
2k
views
Shelah's book on "Classification Theory"
As we know one of the most important and fundamental books in stability, simplicity, forking and ... classification theory, is Shelah's "Classification Theory" where lots of original ideas ...
- 241
14
votes
2
answers
1k
views
If every definable class admits a group structure, must global choice hold?
It is a remarkable fact, due to Hajnal and Kertész and explained very well in this MathOverflow answer by user Ashutosh, that the axiom of choice is equivalent to the assertion that every nonempty set ...
- 1,851
1
vote
1
answer
138
views
Is there inconsistency with having countable models of Z with these internalizing properties?
Is there a clear inconsistency with the following?
There exists a countable transitive model of Zermelo set theory, such that for all external bijections between sets the images and preimages of sets ...
- 236k
3
votes
0
answers
99
views
Counting Eilenberg/Schutzenberger-type definitions of pseudovarieties
See Eilenberg/Schutzenberger, On pseudovarieties for background on pseudovarieties. I've phrased things in terms of pairs-of-sets to avoid some annoying language about multisets. Also, I'm aware that ...
- 21.2k
33
votes
15
answers
7k
views
What's a magical theorem in logic?
Some theorems are magical: their hypotheses are easy to meet, and when invoked (as lemmas) in the midst of an otherwise routine proof, they deliver the desired conclusion more or less straightaway&...
- 103
21
votes
1
answer
864
views
Is there a minimal (least?) countably saturated real-closed field?
I heard from a reputable mathematician that ZFC proves that there is a minimal countably saturated real-closed field. I have several questions about this.
Is there a soft model-theoretic construction ...
- 6,453
9
votes
2
answers
426
views
Can local $0^\#$ exists in L?
Assume $0^\#$ exists and there is an inaccessible cardinal.
Are there two transitive sets $M,N$ s.t. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 21.2k
8
votes
1
answer
222
views
Mostowski's absoluteness theorem and proving that theories extending $0^\#$ have incomparable minimal transitive models
This question says that the theory ZFC + $0^\#$ has incomparable minimal transitive models. It proves this as follows (my emphasis):
[F]or every c.e. $T⊢\text{ZFC\P}+0^\#$ having a model $M$ with $On^...
- 1,097
9
votes
1
answer
578
views
Are there ill-founded "maximally wide" models of $\mathsf{ZFC}$?
Throughout assume $\mathsf{ZFC}$ + "There is a proper class of inaccessible cardinals." I'm also happy to strengthen the large cardinal hypothesis if that would help.
Say that a model $M\...
- 17.7k
5
votes
1
answer
220
views
What is the theory of computably saturated models of ZFC with an *externally well-founded* predicate?
For any model of $M$ of ZFC, we can extend it to a model $M_{ew}$ with an "externally well-founded" predicate $ew$. For $x \in M$, We say that $M_{ew} \vDash ew(x)$ when there is no infinite ...
- 47.3k
3
votes
0
answers
133
views
Comparing two fragments of SOL with the downward Lowenheim-Skolem property
For $S$ a set of (parameter-free) second-order formulas and $\mathfrak{A},\mathfrak{B}$ structures, write $\mathfrak{A}\trianglelefteq^S\mathfrak{B}$ iff $\mathfrak{A}$ is a substructure of $\mathfrak{...
- 21.2k
4
votes
1
answer
176
views
Can we have external automorphisms over intersectional models?
Is the following inconsistent:
By "intersectional" set I mean a set having the intersectional set of every nonempty subset of it, being an element of it.
$\forall S \subset M: S\neq \...
- 236k
6
votes
1
answer
571
views
Parameter-free effective cardinals
In the paper "Effective cardinals and determinacy in third order arithmetic" by Juan Aguilera, effective cardinals is defined.
I'm curious about its little variation, parameter-free ...
- 2,031
13
votes
1
answer
2k
views
Are some interesting mathematical statements minimal?
Gödel's set $\mathrm{L}$, of constructible sets, decides many interesting mathematical statements, as the Continuum hypothesis and the Axiom of Choice.
Are some interesting mathematical questions, ...
- 236k
5
votes
1
answer
597
views
The "first-order theory of the second-order theory of $\mathrm{ZFC}$"
$\newcommand\ZFC{\mathrm{ZFC}}\DeclareMathOperator\Con{Con}$It is often interesting to look at the theory of all first-order statements that are true in some second-order theory, giving us things like ...
- 236k
2
votes
1
answer
973
views
Compactness and completeness in Gödel logic
The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in first-...
- 4,219
8
votes
2
answers
1k
views
Is there one binary operation foundational for set theory?
The membership relationship "$\in$" is foundational for set theory, in the sense that the axioms of any set theory are formulated in the language of "$\in$". Naturally, the ...
- 48.8k
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 ...
- 18.6k
4
votes
1
answer
515
views
Truth Values of Statements in non-standard models
Excuse me, if the question sounds too naive.
Non-standard models of PA will have statements of non-standard lengths, basically infinite. And it is also true that every statement of a theory will have ...
- 2,167
8
votes
1
answer
1k
views
Worst of both worlds?
It's well known that $\mathsf{AC}$ implies the existence of non-measurable sets. And it's also true that, if all sets are measurable, then $|\mathbb{R}/\mathbb{Q}| > |\mathbb{R}|$. But is there a ...
- 3,042
3
votes
0
answers
143
views
Lindström's theorem part 2 for non-relativizing logics
By "logic" I mean the definition gotten by removing the relativization property from "regular logic" — see e.g. Ebbinghaus/Flum/Thomas — and adding the condition that for every ...
- 21.2k
10
votes
1
answer
354
views
Elementary equivalence between $n\mapsto n+1$ and its inverse on the Stone-Čech remainder?
Consider structures $(A,f)$ encoding a Boolean algebra $A$ endowed with an automorphism $f$. There is an obvious notion of isomorphism between such structures.
Consider the endomorphism $\hat{\Phi}$ ...
- 18.5k
-1
votes
1
answer
750
views
Is theory with domain of interpretation in second order objects a First Order Theory?
Thank everybody for answering my previous questions: first, and second.
Here I would like to ask about some important thing which I do not understand clearly.
Is it necessary for theory to have given ...
- 1,446
1
vote
1
answer
213
views
Minimal models in strong set theories - pt. 2
This is a follow-up to this question.
So, as Noah elucidated (thanks Noah!), whenever $T$ is r.e., $M(T) < \sigma$ ($\sigma$ is the least stable ordinal, i.e. $L_\sigma\prec_{\Sigma_1}L$).
In ...
- 2,031
6
votes
1
answer
937
views
Smallest ordinal modelling $\aleph_1$?
Let $X_1$ be the class of all ordinals $\alpha$ such that there exists a transitive model $M$ of ZF(C) such that $M$ thinks that $\alpha$ is $\aleph_1$.
Every class of ordinals has a minimum element (...
- 2,031
39
votes
3
answers
3k
views
Can one show that the real field is not interpretable in the complex field without the axiom of choice?
We all know that the complex field structure $\langle\mathbb{C},+,\cdot,0,1\rangle$ is interpretable in the real field $\langle\mathbb{R},+,\cdot,0,1\rangle$, by encoding $a+bi$ with the real-number ...
- 21.2k
13
votes
1
answer
571
views
Is there a complete uncountable theory with two countable models?
This is a question originally asked at MSE a few years ago; the original poster hasn't been active in a while, so I'm taking the liberty of asking it here:
Is there a complete first-order theory $T$ ...
- 655
16
votes
4
answers
1k
views
Can Suslin (or Aronszajn) lines ever be orderings of abelian groups?
I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of (...
- 4,413
2
votes
1
answer
333
views
Is elementary equivalence absolute?
Assume we have two objects $M_1$ and $M_2$ models of respective $L_{\omega_1,\omega}$-sentences $\Sigma_1$ and $\Sigma_2$.
Assume $M_1$ and $M_2$ are elementarily equivalent in some model of set ...
- 236k
12
votes
4
answers
1k
views
Universal order type
Every countable order type, such as the countable ordinals, $\mathbb Z$, etc. can be embedded in $\mathbb Q$, so it is universal for countable order types. Is there a universal space for all linear ...
- 4,713
4
votes
1
answer
205
views
If a theory has many mutually non-embeddable countable models can it have a countable $\omega$-saturated model?
A theory can have $2^\omega$-many non-isomorphic countable models but has a countable $\omega$-saturated model. (https://math.stackexchange.com/questions/305602/if-a-theory-has-a-countable-omega-...
- 236k
6
votes
5
answers
2k
views
Standard models of N and R: An Alice/Bob approach
This is a question about a comment in a recent publication by Roman
Kossak. Kossak wrote:
"Nonstandardness in set theory has a different nature. In
arithmetic, there is one intended object of ...
- 2,281
24
votes
4
answers
3k
views
A Löwenheim–Skolem–Tarski-like property
I am interested in the following Löwenheim–Skolem–Tarski-like property.
Given a cardinal $\kappa$, what (if any) is a property $\phi(x)$ such that if $\phi(\kappa)$ holds, then we can prove the ...
- 343
3
votes
1
answer
240
views
Is Morse-Kelley set theory with Class Choice bi-interpretable with itself after removing Extensionality for classes?
Let $\sf MKCC$ stand for Morse-Kelley set theory with Class Choice. And let this theory be precisely $\sf MK$ with a binary primitive symbol $\prec$ added to its language, and the following axioms ...
- 236k
3
votes
1
answer
950
views
Is there a concept of limit of formulas
I wonder if there is a notion like the limit of formulas (and structures) because I believe it is important in describing countable structures (from finite structures). (For more detail, see this ...
- 663
7
votes
0
answers
296
views
A system with distinct infinite cardinalities but no "best" version of $\mathbb{N}$
Let $\mathfrak{S}=(M_z,U_z)_{z\in\mathbb{Z}}$ be a sequence such that for each $z\in\mathbb{Z}$ we have
$M_z\models\mathsf{ZFC}+$ "$U_z$ is a nonprincipal ultrafilter on $\omega$" (so in ...
- 21.2k
7
votes
0
answers
185
views
Interest in the size of ultrapowers in model theory
It seems that in the 60s (at least), there was interest in computing the size of ultrapowers by countably incomplete ultrafilters. For example, given an ultrafilter $U$ on some relatively small set ...
- 18.6k
8
votes
1
answer
591
views
Is there an abstract logic that defines the mantle?
It is a known result by Scott and Myhill that the second-order version of $L$ yields $\mathrm{HOD}$.
Recently, Kennedy, Magidor, and Väänänen (Inner models from extended logics: Part I and II) ...
- 1,446
2
votes
1
answer
271
views
I have a problem about elementary submodels of ZFC
So suppose $\kappa$ inaccessible so that $V_\kappa$ is a model of ZFC, using Skolem and the Mostoswky collapse we have a countable elementary submodel $M$ of $V_\kappa$. This implies that for any ...
- 3,065
4
votes
0
answers
247
views
Cantor-Bernstein phenomena for structures (and a "moderate zigzag" property)
My favorite proof of the Cantor-Bernstein theorem is the one that argues by "histories" - given injections $f:A\rightarrow B$ and $g:B\rightarrow A$, we identify each element of $A$ as ...
- 21.2k
10
votes
2
answers
470
views
Is the set of permissible numbers of models of various cardinalities computable?
This question arose in the comments to this question.
Let $X$ be the set of pairs $(m,k)$ such that there is some (consistent complete countable first-order) theory $T$ with exactly $m$ models of size ...
- 5,031
6
votes
1
answer
247
views
Morley Phenomena for Special Families of Reals
Vaught's Conjecture is a dual form of Continuum Hypothesis in model theory. It asserts that for each complete consistent theory $T$ in a countable language if $I(T,\aleph_{0})>\aleph_{0}$ then $I(T,...
