All Questions
48 questions
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
2
votes
0
answers
211
views
Some questions about the Hyperuniverse Program
The Hyperuniverse Program, founded by Sy D. Friedman, intends to produce new second-order axioms of set theory which appropriately formalize "the universe is maximal" in one of a few ways. A ...
- 2,031
3
votes
0
answers
157
views
Systems of elementary embeddings
Recently I've been thinking about elementary embeddings, partition cardinals, etc. as part of my never-ending quest for understanding of consistency strength :p
I came up with this idea, called I* ...
- 704
3
votes
1
answer
154
views
If a theory speaks of sets that cannot be forced to be parameter free definable, then does this entail a large cardinal property?
If we say that an effectively generated first order theory $\sf T$ extends $\sf ZF$, such that every countable model of $\sf T$ doesn't have a class forcing extension that is pointwise definable. ...
- 11.3k
2
votes
0
answers
195
views
"Very $L$-like" models, part 2: combinatorics
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ and having the finite use property and the strong downward Lowenheim-Skolem property together with, for each finite ...
- 21.2k
5
votes
0
answers
192
views
"Very $L$-like" models, part 1: large cardinals
(The original version of this question was much narrower and less natural; but see the edit history if interested.)
Say that a good logic is a regular logic $\mathcal{L}$ containing $\mathsf{FOL}$ ...
- 21.2k
8
votes
1
answer
396
views
On the strength of higher-logic analogues of $\mathsf{ZFC}$ + Montague's Reflection Principle
Throughout, I work in $\mathsf{MK}$ in order to be able to conveniently quantify over logics; if one prefers, we can restrict attention to (say) $\Sigma_{17}$-definable logics and work in $\mathsf{ZFC}...
- 21.2k
5
votes
1
answer
204
views
Upwards-fragility of inaccessibles (again)
Although self-contained, this question is a follow-up to this earlier one. Also, thanks to Fedor Pakhomov for fixing a trivial early version of this question!
Work in $\mathsf{ZFC}$ + "There is a ...
- 21.2k
5
votes
1
answer
183
views
Fragility of large cardinals with respect to transitive end extensions
To motivate things, let me start with a special case of the question I'm interested in. Let $\mathsf{In}(x)\equiv$ "$x$ is an inaccessible cardinal."
Question 1: Is it consistent with the ...
- 21.2k
5
votes
0
answers
265
views
How strong is this "modal definability principle"?
Throughout, we work in the class theory $\mathsf{MK}$ (although I'm open to tweaking this), "logic" means "set-sized logic whose semantics is definable over $V$," and "$\...
- 21.2k
6
votes
1
answer
245
views
Can there be no complexity bound on the definable elementary $V\rightarrow M$?
This starts with a vaguely-recalled result (which may be false!): that if $\mathcal{U}$ is a measure on the least measurable cardinal $\kappa$, then every elementary $j: L[\mathcal{U}]\rightarrow M$ ...
- 21.2k
3
votes
1
answer
331
views
Why does the second smallest worldly cardinal believe the smallest worldly cardinal is worldly?
It is known that the property of being a worldly cardinal is not absolute (a cardinal $\kappa$ is worldly iff $V_{\kappa} \vDash \textsf{ZFC}$). See here and here for more.
This said, it is the case ...
- 161
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. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 1,490
4
votes
0
answers
151
views
How big a "scaffold" does second-order logic need to detect its own equivalence notion?
(Previously asked and bountied at MSE:)
Let $\Sigma$ be the language consisting of a single binary relation symbol. Second-order logic can "detect" second-order-elementary-equivalence of $\...
- 21.2k
3
votes
1
answer
356
views
Dehornoy's proof that the application of two elementary embeddings is an elementary embedding
What is meant by the statement and the proof of Lemma 3.2 in Chapter XII of Dehornoy's book Braids and Self-Distributivity?
That lemma states "Assume that $j_1$ and $j_2$ are elementary ...
- 245
3
votes
0
answers
191
views
A restricted form of the inner model hypothesis
Previously asked and bountied at MSE, with slight difference. To keep things relatively simple I'm presenting a somewhat-butchered version of the IMH; for more details, see S.-D. Friedman, Internal ...
- 21.2k
7
votes
1
answer
367
views
Compatibility of Łośian phenomena in second-order logic
(Throughout, all ultrafilters are nonprincipal.)
Given a property $P$ - really, a sentence in some appropriate logic - say that a ultrafilter $\mathcal{U}$ on a cardinal $\kappa$ averages $P$ iff for ...
- 21.2k
4
votes
1
answer
411
views
Compactness number for a fragment of second-order logic
Previously asked and bountied without response at MSE.
This question is a companion to this one, about a tame(?) fragment of second-order logic with the standard semantics, $\mathsf{SOL}$, motivated ...
- 21.2k
6
votes
1
answer
371
views
How strong is "all up-classes are infinitarily definable"?
Working in MK (or some other not-too-strong class theory if you prefer), say that an up-class is a class of structures $\mathfrak{X}$ which is definable in $V$ (allowing parameters from $V$) and such ...
- 21.2k
7
votes
1
answer
314
views
Lowenheim-Skolem numbers for SOL + correctness quantifiers
For a logic $\mathcal{L}$, say that a cardinal $\kappa$ is $\mathcal{L}$-correct iff every satisfiable $\mathcal{L}$-theory of size $<\kappa$ has a model of size $<\kappa$. First-order ...
- 21.2k
4
votes
0
answers
198
views
Elementary self-embeddings conservative over ZFC
Question: Is the following theory conservative over ZFC? And if not, what is its strength?
Language: $∈$, $j$ (unary function symbol)
Axioms:
1. ZFC (without separation and replacement for formulas ...
- 7,545
13
votes
1
answer
505
views
Does the statement 'there exists a first-order theory $T$ with no saturated models' have any set theoretic strength?
Exercises 6, 7, and 8 in section 10.4 of Hodges' big model theory textbook contain an outline of a proof of the consistency of the following statement
There exists a countable first-order theory $T$...
- 12.5k
5
votes
0
answers
472
views
The surreal numbers under a change of universe
Suppose we start with a model $\mathcal{M}$ of $ZFC$ (or $GBC$ or $MK$ if you prefer), and let $N_0^\mathcal{M}$ denote the surreals in $\mathcal{M}$. If we add some large cardinal assumptions $\{\...
- 10.1k
3
votes
0
answers
138
views
If any satisfiable $\mathcal{L}_{κ,κ}(Q_{=κ})$-theory remains satisfiable when replacing $Q_{=κ}$ with $Q_{=μ}$, is $κ$ huge?
Recently, I have asked a model-theoretic question concerning a weakening of different forms of compactness. I now present another model-theoretic question as a weakening of hugeness.
If any ...
- 1,252
4
votes
1
answer
545
views
A weakening of cardinal compactness - is it equivalent?
I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak ...
- 1,252
13
votes
1
answer
448
views
Does this consequence of measurability in terms of games of length $\omega+1$ imply measurability?
For any two structures $\mathcal{M}$ and $\mathcal{N}$ in the same first-order language $\mathcal{L}$ and any ordinal $\theta$, let $G_\theta(\mathcal{M},\mathcal{N})$ be the two-player game of ...
- 5,471
7
votes
2
answers
773
views
Which large cardinals have a Matryoshka characterization?
What on Earth do Russian Matryoshka dolls have in common with large cardinal axioms?! Well, the answer lies in Jónsson algebras! Here is how:
As illustrated in the pictures, a Matryoshka set is a self-...
4
votes
1
answer
247
views
Weaker forms of Vopěnka's principle (using Indiscernables and other forms of Elementarity): How weak are they?
Vopěnka's principle is commonly used (or at least it was for me) as an intuitionistic approach to large cardinal axioms; that is, there is much intuition to it. This intuition is that for any proper ...
- 1,252
6
votes
1
answer
505
views
Upward Löwenheim–Skolem theorem for well-ordered models with/without measurable cardinals
Consider a complete first order theory $T$ whose language contains a binary predicate $\leq$. Assume that $T$ has an uncountable model that is well-ordered by $\leq$ so that this question isn't stupid ...
- 12.5k
3
votes
0
answers
117
views
Examples of algebras of inner elementary embeddings in model theory (as opposed to set theory)
The algebras of elementary embeddings have been studied from a set theoretic perspective and an algebraic perspective. I wonder is there is a purely model theoretic approach to the self-distributive ...
12
votes
1
answer
583
views
Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$
Suppose $\kappa$ is a measurable cardinal and $j:V\to M$ is the ultrapower by a normal measure on $\kappa$. Let's say, for instance, that $2^\kappa=\kappa^{++}$ (note that this assumption has ...
- 2,240
6
votes
1
answer
358
views
Are there first order theories of interest to an algebraist or at least a model theorist of large cardinal consistency strength?
I am wondering if there are some first order theories of algebraic structures or structures of interest to model theorists of large cardinal consistency strength or at least unexpectedly high ...
6
votes
1
answer
487
views
What is known about the large cardinal strength of Shelah's categoricity conjecture?
Shelah's categoricity conjecture states that for every Abstract Elementary Class $\mathcal{K}$ there is a cardinal $\mu$ depending only on $\operatorname{LS}(K)$ (i.e. the Löwenheim–Skolem number of $\...
user82740
3
votes
1
answer
802
views
Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2
While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms tend to be somewhat arbitrary (e.g. adding an ...
- 95
16
votes
2
answers
1k
views
When does Vopěnka's principle hold?
Vopěnka's principle (VP) is the statement that, given any proper class $\{\mathcal{A}_\eta: \eta\in ON\}$ of first-order structures in the same language, there are some $\alpha\not=\beta$ with $\...
- 21.2k
2
votes
3
answers
408
views
Can the structure of an ultrafilter determine the structure of its ultrapower?
Usually we work with ultrafilters as pure sets without any structure.
Q1. Is there any important notion of structure on an ultrafilter?
Q2. Is there any non-trivial notion of structure on ...
- 21
3
votes
1
answer
303
views
$(\kappa,\lambda)$ - Minimal Models & Stronger Version of Rowbottom's Theorem
Definition 1: Let $M$ be a $\mathcal{L}$ - structure and $A\subseteq Dom(M)$. Define:
$Def_{A}(M):=\{X\subseteq Dom(M)~|~\exists n\in \omega~~\exists \varphi (x,y_1,...,y_n)\in \mathcal{L}-Form~~\...
user42090
3
votes
1
answer
408
views
$(\kappa , \lambda)$ - Minimal Models of $\text{ZF}$
The notion of minimality in model theory is related to the existence of a gap in the size of definable subsets of a model. Now consider the following generalization:
Definition 1: Let $M$ be a $\...
user42090
5
votes
1
answer
316
views
What are the Possible Large Cardinals of $L[X]$?
I've been doing some basic reading in inner model theory, and I'm at the point where I've seen the definition of things like Martin-Steel and Mitchell-Steel inner models. I am wondering about the ...
- 711
3
votes
1
answer
297
views
Do inner models of unique measurable cardinals have a regular behavior? (Edited and Revised Version)
We know that if $\kappa$ is a measurable cardinal and $\mu$ be a two-valued non-trivial$\kappa$-additive measure on it then the corresponding inner model produced by ...
user36136
2
votes
1
answer
292
views
Can we flex the rigid models by enough power?
Definition (1): An $\mathcal{L}$ - structure $\mathcal{M}$ called "rigid" iff there is no non-trivial automorphism on $\mathcal{M}$.
Definition (2): An $\mathcal{...
user36136
5
votes
1
answer
306
views
Is there a truth approximation on a cumulative hierarchy?
Note to the following well known theorem:
Theorem (1): If $\kappa$ be a "measurable" cardinal and $\mathcal{F}$ be a "non-principal $\kappa$-complete normal" ...
user36136
7
votes
1
answer
950
views
Is there a monster behind the trees?
First Fix the following notation:
$\forall \kappa\in Card~~~Tp(\kappa):="\kappa~has~tree~property"$
The large cardinals as "monsters of heaven" live everywhere in the land of ...
user36136
16
votes
3
answers
1k
views
Vopěnka's Principle for non-first-order logics
(For simplicity, the background theory for this post is NBG, a set theory directly treating proper classes which is a conservative extension of ZFC.)
Vopěnka's Principle ($VP$) states that, given any ...
- 21.2k
18
votes
3
answers
1k
views
How badly does compactness fail in $\mathcal{L}_{\omega_1\omega}$?
I would like to get a better idea of how badly compactness fails in $\mathcal{L}_{\omega_1\omega}$.
Let $\Gamma$ be an arbitrary set of sentences from $\mathcal{L}_{\omega_1\omega}$. Let the ...
- 1,142
3
votes
3
answers
2k
views
Large Cardinals Imply a Model of ZFC
I've run across the statement, "The existence of a strongly inaccessible cardinal implies the consistency of ZFC" in several places (Cohen's Set Theory and the Continuum Hypothesis p. 80, for one). ...
8
votes
3
answers
1k
views
Tractability of forcing-invariant statements under large cardinals
It is usual to mention theorems of the kind:
Th. Assume there is a proper class of Woodin cardinals, $\mathbb{P} $ is a partial order and $G \subseteq \mathbb{P}$ is V-generic, then $V \models \phi \...
- 1,465
33
votes
6
answers
5k
views
Reasons to believe Vopenka's principle/huge cardinals are consistent
There are a number of informal heuristic arguments for the consistency of ZFC, enough that I am happy enough to believe that ZFC is consistent. This is true for even some of the more tame large ...
- 19.6k