6
votes
1answer
281 views

Different approaches to forcing

There are many different approaches to the forcing method, and I am looking for all known such approaches. So my question is: Question 1. Which different approaches to set theoretic forcing are ...
6
votes
0answers
206 views

Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal. At most papers ...
5
votes
1answer
227 views

Relationship between fragments of the axiom of choice and the dependent choice principles

The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...
4
votes
2answers
184 views

Forcing $\neg AC$

Sorry if this sounds like a silly reference request, but I wasn't able to track down any. I'm looking for proof, via forcing, that axiom of choice can fail in a model of $ZF$. All of papers I found ...
10
votes
1answer
301 views

Elements of the method of forcing in some papers of N. N. Luzin

In the paper Eléments de la méthode de forcing dans quelques travaux de N. N. Lousin. (French) [Elements of the method of forcing in some papers of N. N. Luzin] Amphora, 469–479, Birkhäuser, ...
9
votes
2answers
421 views

Origin of the term “generic” in set theory

In set theory, in particular the context of forcing, if $M$ is a model of $\sf ZFC$ and $P\in M$ is a partial order, we say that $G\subseteq P$ is a generic filter (or $M$-generic or generic over $M$) ...
10
votes
2answers
587 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}, ...
3
votes
1answer
246 views

What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech. Does anyone know when Hechler forcing was first used in a publication?
2
votes
1answer
208 views

Forcing Notions with Unknown Real/Cardinal Preserving Situations

Question. Is there any set forcing notion $\mathbb{P}$ in one of the following categories? (a) $\mathbb{P}$ preserves cardinals but it is still open whether $\mathbb{P}$ adds reals or not. (b) ...
4
votes
2answers
293 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 ...
1
vote
1answer
169 views

Different Methods for Forcing Total Failure of Generalized Continuum Hypothesis

It seems there are few papers on total failure of Generalized Continuum Hypothesis. As an example Merimovich in 2007 constructed a model of ZFC such that GCH fails everywhere assuming consistency of ...
8
votes
3answers
297 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 ...
2
votes
1answer
209 views

Forcing Language

I asked the following question (slightly paraphrased) about a week ago in Stack exchange but no one knew the answer to the particular question. I was hoping someone here might be able to help me. ...
4
votes
1answer
246 views

Consistency Strength of the Failure of Square on Singular Cardinals

Q1. What is the consistency strength of the failure of square on singular cardinals? Q2. What are known as partial results in this direction?
6
votes
2answers
236 views

Possible Choices for Cofinality of $\aleph_n$ without Choice

$\text{ZFC}$ proves that each $\aleph_{n}$ for $n\in \omega$ is a regular cardinal. But it seems without the Axiom of Choice there are many consistent possible choices for cofinality of such ...
7
votes
4answers
453 views

A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that: (1) $M\subseteq N$ (2) $\aleph_{1}^{N}=\aleph_{1}^{M}$ (3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$ Update: According to Peter's useful ...
3
votes
3answers
376 views

A Question on Special Forcings

The usual use of forcing begins with a "countable" and "transitive" ground model of $ZFC$ and reaches to a "countable" and "transitive" generic model of $ZFC$ with the "same ordinals". In a discussion ...
4
votes
2answers
274 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 ...
12
votes
6answers
1k 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 ...
4
votes
1answer
131 views

Amalgamation of two ccc algebras may collapse the continuum

The claim that appears in the title of this question is mentioned in the paper "On Shelah's amalgamation" by Judah and Roslanowski. I'd really like to see a proof of this fact, but unfortunately I ...
21
votes
2answers
1k 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 ...
6
votes
2answers
351 views

From the product lemma to to a result about powersets

Recall the product lemma from Easton's famous paper, which tells us something about when we have a forcing notion (which may be a proper class) that splits as a product with one factor $\lambda^+$-cc ...
5
votes
2answers
346 views

Suslin algebras

A Suslin algebra is a generalization of a Suslin tree: it is a ccc, $(\omega,\infty)$-distributive boolean algebra. Jech proved that every Suslin algebra has size at most $2^{\omega_1}$. A proof is ...
6
votes
2answers
346 views

Solovay's paper from AD+ that all sets are Ramsey

For my research I've been trying to locate Solovay's proof that AD+ implies that all sets are Ramsey (or completely Ramsey, depending on the terminology). I know that it uses Mathias forcing but I ...
4
votes
3answers
502 views

Background for classic forcing

When I learning forcing theory, I am surprised by various foring definition,I want to know the original purpose, why define partial order in such way and some background material which can help me ...
8
votes
2answers
309 views

$\mathfrak{c}$-universal linear order

I've been told once or twice that the following holds: There is a model of $ZFC+MA+\neg CH$ in which there is a $\mathfrak{c}$-universal linear order embedded in $(\omega^\omega, \le^\ast)$ ...
8
votes
2answers
435 views

Finite support iterations of $\sigma$-centered forcing notions

I am looking for a proof (or better, a reference) of the following fact: The finite support iteration of $\sigma$-centered forcing notions is again $\sigma$-centered, assuming we iterate less than ...
6
votes
1answer
353 views

GCH+ Kurepa Families

I have a couple of questions about known theorems for GCH+Kurepa families. Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ ...
11
votes
4answers
878 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 ...
6
votes
1answer
362 views

Symmetric extensions and class forcing

Suppose $V\models ZFC$ and $P\in V$ is a poset of forcing conditions. It is a basic theorem in forcing that $V[G]\models ZFC$ for any generic extension by a $V$-generic filter $G$. It is also known ...
5
votes
1answer
866 views

What is random real forcing?

Hi all, Can someone please explain the idea and the main steps in a random real forcing? - what makes it (the new real) different from adding a Cohen real? - is there a good reference for it? ...
11
votes
3answers
820 views

When can we detect forcing?

First, a rather broad question: has there been any work on what, given a model $M$ of set theory, we can say about those models of set theory $N$ and posets $\mathbb{P}$ such that $\mathbb{P}\in N$ ...