All Questions
Tagged with model-theory set-theory
686
questions
3
votes
1
answer
396
views
Does Löwenheim-Skolem require Foundation in any way?
As title states, I'm curious whether Löwenheim-Skolem (in either of its upward or downward versions) necessitates some implicit use of Foundation. The usual presentation makes quite clear the reliance ...
- 512
3
votes
0
answers
62
views
Two families of isomorphic structures have isomorphic ultraproduct.
I am trying to prove the following result:
Let $(\underline{M}_i)_{i\in I}$, $(\underline{N}_i)_{i\in I}$ be two families of structures such that, for all $i\in I$, $\underline{M}_i \cong \underline{...
- 103
4
votes
1
answer
87
views
Why are extensions of countable models of ZFC better behaved than extensions of arbitrary models of ZFC?
This answer hints that certain kinds of extensions are only guaranteed to exist for countable models of ZFC. Why?
One intuitive reason i can think of is that the metatheory might not have enough new ...
- 457
1
vote
1
answer
62
views
Need to check this proof that the class of models of ZC that fail replacement is not axiomatisable.
Here ZC is ZFC minus Axiom of Replacement.
My proof is as follows:
Suppose $M$ was axiomatized by a theory $H$.
For non-zero limit ordinal $\alpha$, let $T_\alpha$ be the set of the replacement axioms ...
- 335
2
votes
1
answer
113
views
In what sense is forcing "impossible" in $L$?
I just saw an interesting video from Hugh Woodin about Ultimate $L$. In it, he says one of the reasons $L$ is so interesting is because it not only settles many natural set theory questions, but is ...
- 8,000
0
votes
1
answer
177
views
Is there a set of all set-theoretical truths?
Does Tarski's undefinability theorem implies that there cannot be a set of all set-theoretical truths? Or can there be such a set (although undefinable)?
- 53
3
votes
1
answer
110
views
Introduction to Proper forcing
I’m trying to understand how proper forcing was introduced and the feeling of the model theoretic equivalence
Could you please recommend a great introductory text? I have read the Jech’s Multiple ...
- 103
5
votes
1
answer
127
views
Why do 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^...
- 1,049
3
votes
1
answer
125
views
Löwenheim number ℓ(L) ≤ Hanf number h(L)?
For an arbitrary set of L-sentences T $\subseteq$ L[$\tau$] (where $\tau$ is the vocabulary we are working in), T 'pins down the cardinal' $\kappa$ iff T has a model of cardinality k but does not have ...
- 77
1
vote
0
answers
76
views
Can a Grothendieck universe be defined as a set which is a model of ZFC?
Let us assume for this question that a Grothendieck universe always contains the set of natural numbers.
We have the following fact: If $\mathbb{U}$ is a Grothendieck universe, then we get a model of ...
- 254
0
votes
0
answers
31
views
Possible validity of transfinite model construction for $\mathsf{ZFC}^- + \mathsf{AFA}$
I'm attempting to construct a model for $\mathsf{ZFC}^- + \mathsf{AFA}$ (Zermelo-Fraenkel set theory + Choice, minus Foundation and together with Aczel's Anti-Foundation Axiom) off an arbitrary model ...
- 512
1
vote
0
answers
84
views
Measure Problem and ZFC, is the existence of a measure on a set provable?
Let PMT be the statement "There exists a set on which there exists a measure" which is an affirmative answer to the Measure Problem.
We know that the existence of inaccessible cardinals is ...
- 45
-1
votes
1
answer
115
views
Can we sheaf-theoretically force a violation of the continuum hypothesis in a (nice) topos which is *bicomplete*?
$\newcommand{\p}{\mathcal{P}}$I recently dug through the exercises and details in Mac Lane and Moerdijk's book "Sheaves in Geometry and Logic" which concern themselves with (a baby version ...
- 42.7k
6
votes
1
answer
167
views
Is the class of models of ZFC minus Replacement for which Replacement fails axiomatisable?
Is there a set of first-order sentences axiomatising exactly the class of models of ZFC without replacement for which one of the replacement axioms fails? Perhaps there is a proof of this with the ...
- 335
1
vote
0
answers
66
views
Does every uncountable model of $\mathsf{ZFC}$ have an infinitely descending chain?
Let $\langle M, \in_M \rangle$ be a countable model of $\mathsf{ZFC}$. By Löwenheim-Skolem there is some elementary extension $\langle N, \in_N \rangle$ of our original model of, say, continuum ...
- 512