All Questions
80 questions
32
votes
2
answers
4k
views
Similarities between Post's Problem and Cohen's Forcing
Remark: I have since learned that G.H. Moore addresses this question in the third reference listed at the end of this post, beginning on p. 157 in which he cites a letter from Kreisel to Gödel dated 4/...
- 7,831
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 $A,B$ ...
- 21.2k
11
votes
2
answers
709
views
Which forcings preserve (some) determinacy?
The question is exactly as in the title. I'm interested in general in all questions of the form "which forcings preserve property P?" for any P, but determinacy assumptions occupy a special place in ...
- 21.2k
13
votes
2
answers
1k
views
Failure of diamond at large cardinals
What is known about the failure of $\Diamond_{\kappa}$ (diamond at $\kappa$) for $\kappa$ (the least) inaccessible, (the least) Mahlo and (the least) weakly compact.
Remark. The problem of forcing ...
- 32.1k
49
votes
1
answer
2k
views
Producing finite objects by forcing!
It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations,
where we can use forcing to prove the existence of finite objects with some ...
- 32.1k
20
votes
3
answers
2k
views
A limit to Shoenfield Absoluteness
Shoenfield's Absoluteness Theorem states that if $\phi$ is any $\Sigma^1_2$ sentence of second-order arithmetic, then $\phi$ is absolute between any two models of $ZF$ which share the same ordinals. ...
- 21.2k
17
votes
1
answer
2k
views
Forcing over set theory versus forcing over arithmetic
I've been trying to understand better some of the research on forcing over bounded arithmetic and its connections with lower bounds in complexity theory. For example, Takeuti and Yasumoto have some ...
- 82.7k
17
votes
2
answers
1k
views
Can measures be added by forcing?
The Lévy-Solovay theorem says that small forcings do not create measures. J.D. Hamkins has generalized this to a larger class of forcings called gap forcings. I would assume this cannot be ...
- 5,471
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
1k
views
Reverse-engineer forcing: am I reinventing the wheel?
In the course of a project I’m working on, I’ve started playing around with a sort of “reverse-engineering” forcing. It seems interesting, but
I have a sinking feeling I’m reinventing the wheel; does ...
- 21.2k
13
votes
4
answers
2k
views
Is it possible for countably closed forcing to collapse $\aleph_2$ to $\aleph_1$ without collapsing the continuum?
Suppose the continuum is larger than $\aleph_2$. Does there exist a countably closed notion of forcing that collapses $\aleph_2$ to $\aleph_1$, but does not collapse the continuum to $\aleph_1$? ...
- 1,174
13
votes
3
answers
2k
views
Forcing over an arbitrary model of ZFC
I’m learning set theory and forcing in the (french) book from Jean-Louis Krivine “Théorie des ensembles”.
Given a countable transitive model $\mathscr{M}$ of ZFC together with a poset $P$, he ...
- 3,033
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, ...
- 21.2k
12
votes
2
answers
1k
views
Questions about Prikry forcing and Cohen forcing
I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, \mathbb{P}...
- 32.1k
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
1
answer
744
views
Can we change the Lebesgue measure by forcing?
Suppose $M$ is a model of ZFC, and $\mu^M$ is the Lebesgue measure on $\mathbb R^M$ such that $\mu^M(\mathbb R^M)=1$. It is known that if $r$ is a Cohen real over $M$ and $N=M[r]$ then $\mu^N(\mathbb ...
- 39.7k
6
votes
1
answer
406
views
Consistency results using nonstandard models
Are there any consistency results in set theory (or in mathematics) that can be proved using nonstandard models of ZFC but not using transitive models of ZFC?
- 32.1k
65
votes
3
answers
6k
views
Forcing as a new chapter of Galois Theory?
There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
- 7,853
34
votes
5
answers
2k
views
Forcing as a replacement of induction and diagonal arguments
Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...
- 32.1k
26
votes
2
answers
1k
views
When does the choice of the generic matter?
It is a somewhat curious phenomenon that, in forcing arguments, one usually doesn't care about any particular properties of the generic filter being used (this isn't strictly true; there are cases ...
- 2,389
24
votes
1
answer
1k
views
Forcing and Family Contentions: Who wins the disputes?
The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible ...
20
votes
4
answers
3k
views
A New Continuum Hypothesis (Revised Version)
Define $N_n$ as $n$ th natural number: $N_0=0, N_1=1, N_2=2, ...$.
What happens after exponentiation?
We have the following equation: $2^{N_n}=N_{2^{n}}$.
(Which says: For all finite cardinal $n$ ...
user42090
18
votes
1
answer
2k
views
What is the modal logic of outer multiverse?
The mathematical multiverse could be viewed as a gigantic Kripke model with models of $ZFC$ as possible worlds connected to each other via a certain accessibility relation.
The modal logic associated ...
18
votes
3
answers
2k
views
Scott-Solovay unpublished paper on ``Boolean valued models of set theory''
I have read some papers from 1970$^{th}$, and in some of them, the paper of Scott and Solovay on ``Boolean valued models of set theory'' is given as a main reference, with many references to the ...
- 32.1k
16
votes
1
answer
694
views
Is every class that does not add sets necessarily added by forcing?
We know there are many situations in which we can force over a model $M$ of GBC to add a class $G$ without adding any sets. That is, the extension $M[G]$ satisfies GBC and has the same sets as $M$. ...
- 1,146
16
votes
2
answers
2k
views
Two versions of "absolutely ccc"
I have recently been slogging my way through Shelah's "Large continuum, oracles". Essentially from the start there has been a question needling me which I cannot seem to answer.
In the paper, Shelah ...
- 666
16
votes
1
answer
751
views
Is it consistent with ZFC that no nontrivial forcing notion has automatic mutual genericity?
A nontrivial forcing notion $\newcommand\Q{\mathbb{Q}}\Q$ exhibits
automatic mutual genericity, if whenever $G,H\subseteq\Q$ are
distinct $V$-generic filters (existing, say, in some forcing
extension ...
- 236k
16
votes
1
answer
1k
views
Can there be a global linear ordering of the universe without a global well-ordering of the universe?
This question arose in the answers to Asaf Karagila's
question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...
- 236k
15
votes
3
answers
1k
views
What did Paul Cohen mean by saying that generic sets of natural numbers have "no asymptotic density?"
In Paul Cohen's original 1963 paper on forcing, The independence of the Continuum Hypothesis, published in PNAS, he gives his general proof sketch of how he intends to create a model of ZFC that doesn'...
- 4,897
14
votes
2
answers
982
views
Can we collapse $\omega_1$ to $\omega$ without adding a dominating real?
(Disclaimer: This question was also asked at MSE (https://math.stackexchange.com/questions/71020/can-we-collapse-omega-1-without-adding-a-dominating-real). I'm posting it here because, when I asked it,...
- 21.2k
14
votes
2
answers
2k
views
Condensed / pyknotic sets in terms of forcing over Boolean-valued models of set theory / multiverse concepts?
Here is one way of saying what a pyknotic set is. Fix an inaccessible cardinal $\kappa$, and let $Proj_\kappa$ be the category of $\kappa$-small, extremally disconnected compact Hausdorff spaces. ...
- 63.9k
14
votes
3
answers
866
views
Is Prikry forcing minimal?
Let $M$ be a model of $\sf ZFC$ in which $\kappa$ is a measurable cardinal, and $\cal U$ is a normal measure on $\kappa$. We can define the Prikry forcing (the most simple one) as the poset: $$\Bbb P=\...
- 39.7k
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 $\varphi$ ...
- 21.2k
13
votes
4
answers
1k
views
Cantor-Bernstein for notions of forcing
For forcing notions $\mathbb{P}$ and $\mathbb{Q}$ let us write $\mathbb{P}\triangleleft\mathbb{Q}$ if forcing with $\mathbb{Q}$ always adds a $\mathbb{P}$-generic filter over $V$. In other words, $\...
- 2,701
13
votes
1
answer
672
views
Forcing PFA with ccc forcing
Is it consistent (from suitable large cardinals) that there is a ccc poset which forces PFA?
This seems quite implausible to me. If we could force PFA via ccc forcing, the ground model would have to ...
- 2,389
13
votes
3
answers
796
views
How to make countably closed forcing "nice" without choice
When working over a model $V$ of $ZFC$, countably closed forcings are extremely nice:
If $\mathbb{P}$ is countably closed, then $V[G]$ has no new $\omega$-sequences of elements of $V$. In ...
- 21.2k
13
votes
1
answer
1k
views
What ccc forcings add a Suslin tree?
In a comment to Miha's question in Forcing PFA with ccc forcing, I suggested that if such situation is even possible, it might be achieved by screwing with PFA by some ccc forcing (e.g. adding a Cohen ...
- 39.7k
13
votes
1
answer
561
views
Iterating Neeman's forcing
In the paper, "Forcing with sequences of models of two types," (MR3201836), Neeman claims that, using a supercompact and a weakly compact above, one can force with his pure side conditions ...
- 18.6k
12
votes
1
answer
448
views
Comparing generic versions of $\mathbb{R}$
This question was previously asked and bountied at MSE, unsuccessfully.
I'm currently interested in the behavior of cardinalities in generic extensions of models of $\mathsf{ZF+\neg AC}$, and ...
- 21.2k
12
votes
1
answer
695
views
A new cardinality living in every forcing extension?
I'm broadly interested in notions of "generic presentability" - when a given object exists in every forcing extension of the universe by some fixed forcing, at least up to the appropriate ...
- 21.2k
10
votes
0
answers
514
views
Existence of a regular subposet which collapses everything except the top cardinal
Suppose $\delta$ is an inaccessible cardinal, and $\mathbb{P}$ is the Levy Collapse $\text{Col}(\kappa, \delta)$ which adds a surjection from $\kappa \to \delta$ (for some regular $\kappa < \delta$)...
- 2,231
11
votes
1
answer
812
views
Set-theoretic geology: controlled erosion?
I have to say that after the two last posts by Timothy Chow on Forcing I got so intrigued that I am trying to rethink the little I know about this formidable chapter of mathematics.
I have also to add ...
- 7,853
11
votes
3
answers
534
views
Who proved "sets in every generic are already in the ground model?"
Suppose $\mathbb{P}$ is a notion of forcing in the ground model $V$, and $X$ is a set which is in $V[G]$ for every $\mathbb{P}$-generic filter $G$. Then $X\in V$ already, by a fairly simple (if ...
- 21.2k
11
votes
3
answers
792
views
When are two forcing posets "the same"?
Let $B$ and $C$ be complete Boolean algebras. To avoid triviality I may also want them to be atomless. For $b\in B$ nonzero, denote $B\upharpoonright b=\{p\in B:p\leq b\}$, which can be viewed as a ...
- 877
10
votes
1
answer
580
views
Can you have many independent reals?
Working in $\sf ZFC$, is it provable, or at least consistent (say, over $L$), that you have $\aleph_1$ forcings, $\Bbb P_\alpha$ such that:
$\Bbb P_\alpha$ is c.c.c.
$\Bbb P_\alpha$ adds a real which ...
- 39.7k
10
votes
3
answers
1k
views
Philosophy of forcing and ctm
I asked a similar question on SE before and received an answer. Not completely convinced, I decided to ask it here with some modifications. Note: I understand how forcing works and how it proves ...
- 237
10
votes
3
answers
545
views
A model of CH +$\lnot \diamondsuit$
All of the models of CH which I know of also satisfy $\diamondsuit$. What is the easiest way to produce a model of CH wherein $\diamondsuit$ is false?
- 103
9
votes
0
answers
440
views
A new maximality principle and its consequences
Let us consider the following maximality principle:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$
and all trees of height and size $\kappa$ are specialized.
It ...
- 32.1k
9
votes
1
answer
765
views
Why relative consistency results by forcing arguments are provable in finitistic metatheory
It is claimed in many textbooks that relative consistency results, such as $\text{Con}(\text{ZFC})\rightarrow\text{Con}(\text{ZFC}+2^{\aleph_0}\geq\aleph_2)$, are provable in the finitistic metatheory....
- 779
9
votes
1
answer
559
views
Just a little absoluteness might be cheaper?
Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...
- 21.2k