All Questions
Tagged with set-theory forcing
825 questions
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
56
votes
2
answers
3k
views
How to add essentially new knots to the universe?
A knot is an embedding of a circle $S^{1}$ in $3$-dimensional Euclidean space, $\mathbb{R}^3$. Knots are considered equivalent under ambient isotopy. There are two different types of knots, tame and ...
49
votes
4
answers
7k
views
Sheaf-theoretic approach to forcing
Inspired by the question here, I have been trying to understand the sheaf-theoretic approach to forcing, as in MacLane–Moerdijk's book "Sheaves in geometry and logic", Chapter VI.
A general ...
- 21.3k
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
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
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
29
votes
2
answers
5k
views
What is the dimension of the mathematical universe?
Forcing construction in set theory leads to a new understanding of the mathematical (multi)universe by providing a machinery through which one can construct new models of the universe from the ...
27
votes
1
answer
2k
views
How hard is it to destroy a diamond? (with a real)
If we start with $V\models\lozenge$, it is not hard to force the failure of diamond. You can blow up the continuum, or destroy all the Suslin trees. You can blow up the continuum of $\aleph_1$, and ...
- 39.7k
26
votes
1
answer
3k
views
How far wrong could the Continuum Hypothesis be?
I hear it's consistent with ZFC to have
$$ 2^{\aleph_0} = \aleph_n $$
for any $n = 1, 2, 3, \dots $. How much worse can it get?
More precisely: are there models of ZFC with $2^{\aleph_0} \gt \aleph_n$...
- 22.3k
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
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 ...
- 21.2k
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 ...
23
votes
5
answers
4k
views
What is the generic poset used in forcing, really?
I'm not a set theorist, but I understand the 'pop' version of set-theoretic forcing: in analogy with algebra, we can take a model of a set theory, and an 'indeterminate' (which is some poset), and add ...
- 35.5k
22
votes
1
answer
1k
views
When will the real numbers be Borel?
In set theory Borel sets are important, but we don't actually care about the sets. We can about the Borel codes. Namely, the algorithm to generate a given Borel set starting with the basic open sets (...
- 39.7k
22
votes
2
answers
1k
views
How "much" does (Grigorieff) forcing destroy an ultrafilter?
Introduction. I recently revisited Shelah's model without P-points and I was wondering how "badly" Grigorieff forcing destroys ultrafilters, i.e., what kind of properties can survive the destruction ...
22
votes
1
answer
938
views
How badly can the GCH fail globally?
It's known that we can have global failures of GCH---for example, where $\forall \lambda(2^\lambda = \lambda^{++})$---given suitable large cardinal axioms.
My question is whether we can have global ...
- 1,198
22
votes
2
answers
1k
views
Gently changing measure
This question was asked and bountied on MSE without answer, so I'm porting it here:
There's an easy way to change the measure of a set of reals by moving to a larger universe: simply make $\mathbb{R}$...
- 21.2k
22
votes
1
answer
883
views
Is the axiom $\Diamond\Box\varphi\to\Box\Diamond\varphi$ in c.c.c. forcing potentialism equivalent to the productivity of c.c.c. forcing?
This question arose in connection with a lecture series on
Potentialism
that I have just completed here in Hejnice in the Czech Republic at
the Winter School 2018 (see
Slides). Several of us discussed ...
- 236k
21
votes
10
answers
3k
views
Examples of ZFC theorems proved via forcing
This is an old suggestion of Joel David Hamkins at the end of his answer to this question: Forcing as a tool to prove theorems
I just noticed it while trying to understand his answer. But indeed it ...
21
votes
3
answers
2k
views
In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?
This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My ...
- 6,132
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
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
19
votes
9
answers
5k
views
Forcing as a tool to prove theorems
It is often mentioned the main use of forcing is to prove independence facts, but it also seems a way to prove theorems. For instance how would one try to prove Erdös-Rado, $\beth_n^{+} \to (\aleph_1)...
- 3,804
19
votes
2
answers
2k
views
Woodin's unpublished proof of the global failure of GCH
An unpublished result of Woodin says the following:
Theorem. Assuming the existence of large cardinals, it is consistent that $\forall \lambda, 2^{\lambda}=\lambda^{++}.$
In the paper "The ...
- 32.1k
19
votes
2
answers
855
views
Do choice principles in all generic extensions imply AC in $V$?
It's well-known that not all choice principles are preserved under forcing, e.g. in this answer https://mathoverflow.net/a/77002/109573 Asaf shows the ordering principle can hold in $V$ and fail in a ...
- 4,316
19
votes
3
answers
1k
views
Set-theoretic forcing over sites?
All texts I have read on set-theoretic independence proofs consider several different sorts of constructions separately, such as Boolean-valued models (equivalently, forcing over posets), permutation ...
- 66.8k
19
votes
1
answer
815
views
If all reals are generic, is the set of reals generic?
Let $W\subseteq V$ be two models of $\sf ZFC$ with the same ordinals. Is the following situation consistent:
For every $x\in\Bbb R^V$ there is some $P_x\in W$ such that for some $G\subseteq P_x$ ...
- 39.7k
19
votes
0
answers
905
views
What examples of existence forcing proofs are there?
Forcing proofs tend to be fairly constructive, in the sense that if I claim that there is a forcing that does something, I usually prove this by constructing that forcing.
There are only a handful of ...
- 39.7k
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
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
2
answers
630
views
Is the notion of fixed point property for topological spaces an absolute notion?
Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...
- 32.1k
18
votes
1
answer
554
views
When can we add choice to a model of ZF
For countable transitive models of ZF, is existence of a ZFC extension with the same height a first order property?
In other words, is there a statement $τ$ (in the language of set theory) such that ...
- 7,545
18
votes
1
answer
871
views
Three old questions on the Sacks forcing
I came across the two following Qs in 1970.
Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...
- 2,281
17
votes
6
answers
1k
views
Strategic vs. tactical closure
The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when ...
- 18.6k
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
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
5
answers
2k
views
Forcing over models without the axiom of choice
In the vast majority of papers forcing is always developed over ZFC.
Not surprisingly too, since infintary combinatorial principles are often used to prove results based on properties such as chain ...
- 39.7k
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
2
answers
962
views
Namba forcing and semiproperness
This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...
- 7,125
17
votes
1
answer
1k
views
When can power sets be limit cardinals?
My original question (posted in here at the Math.SE) was:
Is it possible to create a model of ZFC, so that the cardinality of each power set is a limit cardinal (as opposed to GCH where they are ...
- 529
17
votes
0
answers
558
views
Gitik's work on Shelah's weak hypothesis
It seems that Moti Gitik has recently refuted some variants of Shelah's weak hypothesis. For this see the title and abstract of his talk at the Set Theory, Model Theory and Applications conference.
I ...
- 32.1k
17
votes
0
answers
908
views
Souslin trees and weakly compact cardinals
In Souslin trees on the first inaccessible cardinal it is asked if it is consistent that there are no $\kappa-$Souslin trees at the least inaccessible cardinal $\kappa$.
In this question I would like ...
- 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
1
answer
2k
views
Impact of discrepancy between Kunen's and Jech's definition of iterated forcing on full support iterations
One of the first things we usually learn when we study iterated forcing is that we can force over a model of ZFC + GCH to make the continuum function ($\lambda \mapsto 2^{\lambda}$) restricted to some ...
- 2,762
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
Kaplansky's conjecture and Martin's axiom
Recall Kaplansky's conjecture which states that every algebra homomorphism from the Banach algebra C(X) (where X is a compact Hausdorff topological space) into any other Banach algebra, is ...
- 32.1k
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
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
2k
views
May two Cohen reals collapse cardinals?
My question is the following:
Let $M$ be a c.t.m. of $\mathsf{ZFC}$. Are there two reals $r_0,r_1 \in \mathbb{R}$ such that $r_i$ is Cohen over $M$ for $i=0,1$ and such that $\omega_1^M$ is countable ...
- 2,286
15
votes
6
answers
2k
views
The origins of forcing in mathematical logic and other branches of mathematics
As everyone knows, forcing was created by Cohen to answer questions in set theory.
Question 1. What are the first applications of set theoretic forcing in other branches of mathematical logic, like ...