All Questions
Tagged with model-theory large-cardinals
51
questions
8
votes
1
answer
612
views
Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
- 183
23
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 ...
- 333
2
votes
0
answers
194
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 ...
- 1,696
3
votes
0
answers
151
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
152
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. ...
- 10.2k
2
votes
0
answers
191
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 ...
- 20.7k
5
votes
0
answers
185
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"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 $\mathsf{FOL}$ ...
- 20.7k
8
votes
1
answer
368
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}...
- 20.7k
5
votes
1
answer
199
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 ...
- 20.7k
5
votes
1
answer
174
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 ...
- 20.7k
5
votes
0
answers
252
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 "$\...
- 20.7k
6
votes
1
answer
237
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 $j: L[\mathcal{U}]\rightarrow M$ ...
- 20.7k
3
votes
1
answer
301
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
404
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. $M\in N,M\vDash ZF+V=L[0^\#],N\vDash ZF+V=L$?
- 1,480
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 $\...
- 20.7k