All Questions
38 questions
26
votes
0
answers
1k
views
Where do uncountable models collapse to?
Suppose $T$ is a complete first-order theory (in an finite, or at worst countable, language). Given any model $\mathcal{M}\models T$ of cardinality $\kappa$, we can ask whether $\mathcal{M}$ can be ...
- 20.5k
17
votes
3
answers
1k
views
Minimum transitive models and V=L
Is there a c.e. theory $T⊢\text{ZFC}$ in the language of set theory such that the minimum transitive model of $T$ exists but does not satisfy $V=L$?
You may assume that ZFC has transitive models. ...
- 7,545
14
votes
1
answer
522
views
Is there an infinitary sentence which is absolutely not second-order expressible?
This is a "forcing-absolute" followup to this question, whose answer was largely unsatisfying. The question is:
Suppose $V=L$. Is there an $\mathcal{L}_{\infty,\omega}$-sentence $\varphi$ ...
- 20.5k
14
votes
0
answers
404
views
O-minimality and forcing
It is well-known that the structure $(\mathbb{R}, +, \cdot, <, 0, 1)$ is an o-minimal structure and hence the set of integers $\mathbb{Z}$ is not definable in it.
In an ongoing project with Will ...
- 32.1k
13
votes
2
answers
1k
views
The (non-)absoluteness of second-order elementary equivalence
Elementary equivalence is set-theoretically absolute between any two transitive models of set theory; this is also true for the infinitary logics - e.g., $\mathcal{L}_{\omega_1\omega}$ - at least, ...
- 20.5k
13
votes
1
answer
742
views
Is forcing computable?
By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...
- 28.2k
12
votes
3
answers
2k
views
Why do we need a transitive model in forcing arguments?
One major approach to the theory of forcing is to assume that ZFC has a countable transitive model $M \in V$ (where $V$ is the "real" universe). In this approach, one takes a poset $\mathbb{P} \in M$, ...
- 3,058
12
votes
2
answers
937
views
Is this definability principle consistent?
(Below I'm thinking only about computably axiomatizable set theories extending $\mathsf{ZFC}$ which are arithmetically, or at least $\Sigma^0_1$-, sound.)
Say that a theory $T$ is omniscient iff $T$ ...
- 20.5k
12
votes
1
answer
648
views
Can $\mathcal{L}_{\omega_1,\omega}$ detect $\mathcal{L}_{\omega_1,\omega}$-equivalence?
Roughly speaking, say that a logic $\mathcal{L}$ is self-equivalence-defining (SED) iff for each finite signature $\Sigma$ there is a larger signature $\Sigma'\supseteq\Sigma\sqcup\{A,B\}$ with $A,B$ ...
- 20.5k
11
votes
3
answers
994
views
Why can we assume a ctm of ZFC exists in forcing
Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
- 111
10
votes
7
answers
1k
views
Applications of forcing in model theory
What are the major applications of (set theoretic) forcing in model theory?
user39869
10
votes
1
answer
435
views
When are generic models not too wild?
This is a question related to ideas raised in http://arxiv.org/abs/1410.1224 and http://arxiv.org/pdf/1405.7456.pdf. Basically, the idea is the following:
Suppose I have a first-order theory $T$. ...
- 20.5k
9
votes
2
answers
777
views
A "suitably generic" set of Cohen reals without forcing?
I was reading a paper by David Marker, whose main theorem was that if $T$ is a first-order theory which is not small, then $F_2\leq_B \cong_T$. That's not especially relevant to the question at hand, ...
- 1,979
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
8
votes
1
answer
480
views
Intuition behind Boolean-valued models of set theory
$\DeclareMathOperator\Card{Card}$The book Forcing Eine Einführung in die Mathematik der Unabhängigkeitsbeweise by Hoffmann provides an intuition behind boolean valued models of set theory which I will ...
- 123
7
votes
2
answers
683
views
Can second-order logic identify "amorphous satisfiability"?
Recall that a set is amorphous iff it is infinite but has no partition into two infinite subsets. I'm interested in the possible structure (in the sense of model theory) which an amorphous set can ...
- 20.5k
7
votes
1
answer
291
views
Is there a forcing closure?
The main theorem of forcing says that for any c.t.m of $ZFC$ like $M$ and for all partial order $\mathbb{P}$ and $\mathbb{P}$-generic $G$ over $M$, there is a c.t.m of $ZFC$, like $N$ such that $N$ is ...
user36136
7
votes
1
answer
401
views
How hard is it to get "absolutely" no amorphous sets?
A beautiful and surprising (to me at least) result around the axiom of choice is that, while full $\mathsf{AC}$ is preserved by forcing, a model of $\mathsf{ZF}$ + "There are no amorphous sets&...
- 20.5k
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
6
votes
2
answers
495
views
Forcing for Arbitrary First Order Theories
Forcing is a relative model construction method for models of $ZF$ as a particular first order theory using models of another first order theory (forcing companion) that in this case is the theory of ...
user45939
6
votes
1
answer
227
views
How often are forcing extensions of countable computably saturated models of $\mathsf{ZFC}$ computably saturated?
Recall that given a finite language $\mathcal{L}$, we say that an $\mathcal{L}$-structure is computably saturated (or recursively saturated) if for any computable set $\Sigma(\bar{x},y)$ of $\mathcal{...
- 12.5k
6
votes
2
answers
342
views
Categoricity of the complex field in the generic extensions
Let $V[G]$ be a generic extension of $V$ by adding a new Cohen real (or generally a generic extension which adds new reals and do not blow up the power of the continuum). Working in $V[G],$ we can ...
- 32.1k
5
votes
1
answer
273
views
Can the forcing-absolute fragment of SOL have a strong Lowenheim-Skolem property?
Previously asked and bountied at MSE:
Let $\mathsf{SOL_{abs}}$ be the "forcing-absolute" fragment of second-order logic - that is, the set of second-order formulas $\varphi$ such that for ...
- 20.5k
5
votes
1
answer
248
views
Highly improper forcings
The following question comes from a typo in an old notebook of mine (I changed what I was calling my forcing notion partway through writing the definition of properness):
Say that a forcing $\mathbb{P}...
- 20.5k
5
votes
1
answer
327
views
Constructing a model of $\mathrm{DCF}_0$ via forcing
As is mentioned in the introduction of this paper of Spodzieja there is a lack of 'natural' examples of differentially closed fields. The immediate naive guesses, namely the field of germs of ...
- 12.5k
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 ...
- 1,338
4
votes
2
answers
625
views
The category of Boolean-valued models associated to a model of ZFC
This is related to my previous question here, and indirectly motivated by Andreas Blass intriguing answer therein:
start from a ZF transitive model $M$, and consider the category $CBA(M)$ of ...
- 7,853
4
votes
0
answers
182
views
Generic two-cardinal behavior of first-order sentences
This is a hopefully improved version of a question I asked before and then deleted because it was based on some fundamentally incorrect assumptions.
Some first-order theories are able to control the ...
- 12.5k
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
3
votes
2
answers
185
views
Joint Forcing Extension Property
Definition 1: A class $\mathcal{K}$ of countable transitive models of $\text{ZF}$ has strong "joint forcing extension property" (JFEP) iff for all $M,N\in \mathcal{K}$ there are forcing notions $\...
user42090
3
votes
1
answer
419
views
Forcing, a technical detail
In the snippet below from Shelah's book P&I Forcing,
in the definition 5.2(2)
I do not follow why in this sentence [naturally extended to include $N\prec (H(\mu^\dagger),\epsilon),\mu\in N$] $N$ ...
- 307
3
votes
1
answer
299
views
A kind of saturation property related to forcing notions
Forcing is typically done over well-founded models. There are lots of good reasons for this, but it can feel confining at times. Fortunately, we can equally well force over non-well-founded models! It ...
- 20.5k
3
votes
2
answers
557
views
Can we force with Fraisse filters to solve Vaught's conjecture?
Around the classic Fraisse amalgamation theorem in model theory we have the following notions:
Definition (1): If $M$ be an $\mathcal{L}$-structure then define:
$age(M):=\lbrace N~|~N~\text{is ...
user36136
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 ...
- 695
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
2
votes
0
answers
676
views
Boolean-Valued Models vs. the Infinite-valued Logic of Lukasiewicz and set theory
Is anyone familiar with an old paper of C.C. Chang entitled "The Axiom of Comprehension in Infinite-Valued Logic" which shows that the Axiom of Comprehension without parameters is consistent in the ...
- 6,132
1
vote
1
answer
147
views
Pullback a model in generic extension to a model in ground
Let $M\prec H_\theta$ be countable and $\mathbb P\in M$, then clearly $M[G]\prec H_\theta[G]$ for every $\mathbb P$-generic filter $G$ over $V$. then actually there exists a $\mathbb P$-name $\dot N$ ...
- 2,381
1
vote
0
answers
219
views
What can be said about a Boolean-valued structure from what the Boolean-valued forcing extension thinks about it?
Suppose that $\phi$ is a formula in the language of set theory such that
there are some $n_{1},...,n_{k}$ such that if $V\models\phi(x)$, then $x=(X,R_{1},...,R_{k})$ and $\mathrm{Eq}:X^{2}\...
