6
votes
1answer
69 views

Proper Forcing and Sequence of Names for Reals.

I read something that seems to suggest the following is true: If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, ...
8
votes
1answer
130 views

Why is ZF favoured over NBG

So why is ZF favoured over NBG? Is it historical, but I've read after Gödel's monograph was published, NBG was more prominent. Or is the reason that NBG gets the cold shoulder to do with forcing and ...
0
votes
1answer
18 views

Redundancy in definiton for forcing poset

The following definition appears in Kunen (2nd edition): For any sets $I,J$ and cardinal $\lambda$: $\text{Fn}_{\lambda}(I,J)$ is the set of all $p\in{[I\times{J}]}^{<\lambda}$ such that $p$ is ...
2
votes
1answer
36 views

What is the definition of the Feferman-Levy model?

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.
6
votes
1answer
110 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
1
vote
0answers
52 views

Existence of a generic ultrafilter over constructible universe

I am referring to the Jech's "Set Theory" book. He wants to show that, if $(P,<)$ is the notion of forcing defined as: $p \in P$ iff $p$ is a finite sequence of ordinals less than $\omega_{1}^{L}$ ...
0
votes
0answers
62 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
3
votes
0answers
49 views

Exercise on preservation of cardinals

I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the ...
2
votes
0answers
38 views

The notations $ M[X] $ and $ V[A] $ and intermediate models

I am confused about Jech's notations $ M[X] $ and $ V[A] $. Let us begin with exercise 13.34 (see [Jec02, p. 199]). Let $ \mathbf{M} $ be a transitive model of $ \mathsf{ZF} $ containing all ...
4
votes
1answer
122 views

cov(meager) strictly between $\aleph_1$ and $2^{\aleph_0}$

Is is consistent that $\aleph_1 < \text{cov(meager)} < 2^{\aleph_0}$? I can only seem to find references for results that assert it is consistent that it (or other cardinal characteristics) is ...
2
votes
0answers
62 views

Proof of PFA from Supercompact

In Jech's proof (in chapter 31) that the consistency of a supercompact cardinal implies the consistency of PFA, he needs the following fact: Let $\mathbb{P}_\kappa$ be the countable support forcing ...
4
votes
2answers
55 views

Example of a non-proper product of two proper forcing notions

I am looking for an example of a proper forcing notion $ \mathbb{P} $ such that $ \mathbb{P} \times \mathbb{P} $ is not proper. Maybe someone knows an obvious example or can give a reference to such ...
1
vote
0answers
105 views

Does the Math-tea Argument Have Any Relevance to the Method of Forcing?

The Math-tea Argument (i.e. the argument that, for example, there must be real numbers that we cannot describe or define, because there are only countably many definitions, but uncountably many reals) ...
2
votes
2answers
80 views

An exercise in Kunen

The following exercise appears in Kunen; In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, ...
2
votes
1answer
108 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
2
votes
1answer
71 views

Some Questions Regarding Pointwise Definable Models of ZFC

In their paper "Pointwise Definable Models of Set Theory" Hamkins, Linetsky, and Reitz prove the following theorem: "Every countable model of ZFC has a pointwise definable class forcing extension." ...
2
votes
1answer
40 views

Adding subsets of regular cardinals (Jech p. 226)

On p. 226 of his ${\it Set}$ ${\it Theory}$, Jech considers adding $\lambda$ many subsets of $\kappa$ to a ground model $M$. He outlines a suitable partial order on the assumption that $M$ satisfies ...
1
vote
1answer
76 views

On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
10
votes
1answer
73 views

Martin's Axiom and products of c.c.c. spaces

A topological space $X$ has the c.c.c. if every family of pairwise disjoint non-empty open subsets of $X$ is countable. Consider the following statement: $(P)$ The product of two c.c.c. spaces is ...
3
votes
2answers
88 views

Introduction to Proper Forcing Reference

What is a good introduction to proper forcing? I am aware of Shelah Proper and Improper Forcing, but I heard this book may be somewhat challenging to read. There is also Devlin's The Yorkshiremen's ...
2
votes
1answer
66 views

Equivalent (?) definitions of Axiom A

(Why) Are the following definitions of Axiom A equivalent? Soft question: Which one is more common, natural, or usually easier to verify? What was Baumgartner's original definition? ( B ...
1
vote
1answer
92 views

A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
0
votes
1answer
43 views

Existence of Certain Names in Iterated Forcing

Suppose $\mathbb{P}$ is a forcing. Let $\dot{\mathbb{Q}}$, $\dot{<_\mathbb{Q}}$, and $\dot{1}_\mathbb{Q}$ be a name such that $1_\mathbb{P} \Vdash_\mathbb{P} ``\langle \dot{\mathbb{Q}}, ...
2
votes
0answers
102 views

How large or small can the gap in Shelah's main gap theorem be, up to consistency?

Shelah's main gap theorem in model theory says: For each first order complete theory $T$ in a countable language if $I(T,\kappa)$ denotes the number of its models of size $\kappa$ then one of ...
1
vote
1answer
98 views

Is there any relevance between Boolean Algebras and Fields?

In some sense Boolean Algebras and Fields have same operators and constants. In both structures there are operators addition ($+$ , $\vee$), multiplication ($\times$ , $\wedge$), inverse with respect ...
2
votes
2answers
104 views

Coding Forcing Notions by Ordinal Numbers: A Possible Approach to Shelah-Foreman-Magidor Conjecture

Forcing notions are partial orders. In some sense each partial order is a "combination" of some well-orderings and each well-orderings is isomorphic to a unique ordinal number. Thus in some sense a ...
2
votes
1answer
97 views

What does it mean to say that a forcing “collapses cardinals”?

I hear the following terminology a lot: "So-and-so forcing collapses cardinals." Does this just mean that certain cardinals in the ground model are no longer cardinals in the forcing extension? If ...
2
votes
0answers
33 views

Strategic closure of quotients

Is there an example of a poset $P$ that is a regular suborder of $Q$ such that $Q$ is $\omega_2$-strategically closed, but the quotient forcing $Q/P$ fails to be $\omega_1$-strategically closed? To ...
6
votes
1answer
84 views

Terminology in forcing

In the context of forcing one reads the relation $p \leq q$ in a poset $P$ as "$p$ extends $q$". A typical example is the poset $P$ of finite partial functions, where one defines $p \leq q$ when $q ...
4
votes
2answers
51 views

countably closed forcing cannot add a branch to a $\aleph_2$-tree if $\neg\mathsf{CH}$

I'm reading this survey. In it the author states the following result (fact 5.3) which is attributed to Silver: If $2^{\aleph_0}>\aleph_1$, countably closed forcing cannot add a new branch to ...
3
votes
1answer
62 views

Truth values in Boolean valued models

In Devlin's "The Joy of Sets" the author introduces the Boolean valued model $V^{{\mathcal B}}$ based on a given Boolean algebra ${\mathcal B}$ and describes how to assign a truth value ...
2
votes
1answer
34 views

Do Different Generic Filters Give Different Generic Extensions?

Let $\mathbb{P}$ be a forcing. If $G \subseteq \mathbb{P}$ and $H \subseteq \mathbb{P}$ are two $\mathbb{P}$-generic filters over $V$ and $G \neq H$, does this imply that $M[G] \neq M[H]$. If $G$ ...
5
votes
2answers
168 views

About proving that the Continuum Hypothesis is independent of ZFC

In Mathematical Logic, we were introduced to the concept of forcing using countable transitive models - ctm - of $\mathsf{ZFC}$. Using two different notions of forcing we were able to build (from the ...
9
votes
0answers
74 views

Choice in a Particular HOD Type Model

Let $V \models ZFC$. Let $P$ be a forcing and $G \subseteq P$ be generic over $V$. Let $x \in V[G]$. Let $M$ be the class of set that are hereditarily definable (in $V[G]$) using as parameters $x$ ...
1
vote
0answers
20 views

Intermediate Extensions and Coding Sets by Ordinals [duplicate]

Lemma 15.43 in Jech's "Set Theory" states that if $V \subseteq M \subseteq V[G]$ where $G \subseteq \mathbb{P}$ is some $V$-generic filter and $M$ is a transitive models of ZFC, then $M = V[D \cap G]$ ...
4
votes
1answer
102 views

A question regarding the status of CH in the Gitik model

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be ...
2
votes
1answer
41 views

Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
1
vote
0answers
58 views

Quotient Forcing in Iterations

I am trying to understand a proof of a lemma used to prove a preservation theorem for $^\omega \omega$-bounding for countable support iterations. In that quotient forcing is used to get a certain ...
1
vote
1answer
37 views

Defining forcing relation in base transitive model $M$

In page 177 of Set Theory for the Working Mathmatician, on chapter forcing it says: Theorem 9.2.7 For every formula $\varphi(x_1,..., x_n)$ of set theory there exists another formula ...
2
votes
1answer
79 views

Viewing forcing as a result about countable transitive models

I think of forcing in the following context: Truth and Definability Lemmas. So forcing is a schema in the meta theory. Now in his Set Theory book (the first edition), Kunen claims that setting up the ...
2
votes
1answer
71 views

Can set-theoretic forcing exist without law of excluded middle?

Of course the law of excluded middle is accepted by almost every mathematician except a few constructivists, but then I was wondering if set-theoretic forcing can exist without law of excluded middle. ...
-1
votes
1answer
70 views

P-generic filter [closed]

Let $M$ be a countable transitive model of $ZF$ and let $P\in M$ be a partial order then how can we see If $P$ is non atomic partial order and $G$ is a P-generic filter over $M$, then $G\notin M$. ...
0
votes
2answers
29 views

Boolean-valued model and the use of generic ultrafilter in ZFC

So I asked the question about generic filter; but I was also reading http://math.mit.edu/~tchow/forcing.pdf which is a forcing (in ZFC) guide for dummies. Then I was struck with the part where it ...
3
votes
2answers
62 views

How generic ultrafilter is used in forcing

So I just learned what ultrafilter is and generic filter is. As all maths are for beginners, it just looks like pure concepts, and I don't see how they are going to be applied in forcing. I am looking ...
4
votes
2answers
244 views

Does the Laver real determine the generic filter?

Let us concern the Laver forcing $ \mathbb{L} $. Let $ G $ be $ \mathbb{L} $-generic over a c.t.m. $ M $ for ZFC. Let $$ x_G := \bigcup \{ \operatorname{stem}(p) : p \in G \} $$ be the Laver real ...
3
votes
1answer
57 views

Absoluteness of $\mathbb{P}$-names

In Kunen's book he says that $\mathbb{P}$-names are absolute for a transitive models of ZFC using a theorem to the effect that functions defined by recursion are absolute, i.e; Let $R$,$A$,$G$ be ...
2
votes
1answer
96 views

Inaccessible cardinals

Let $M[G]$ be the full Solovay model, and let HOD be the model of hereditarily ordinal definable sets in $M[G]$. Is it possible for HOD not to have an inaccessible cardinal? Does HOD satisfy GCH?
1
vote
1answer
43 views

Describing two-step iteration in terms of complete Boolean algebras.

Suppose $B$ is a complete Boolean and let $\dot{C}\in V^B$ be such that $$\|\dot{C}\text{is a complete Boolean algebra}\|_B=1.$$ Let us consider all $\dot{c}\in V^B$ such that ...
4
votes
0answers
48 views

Full Theory of Structure of Set Theory and Generic Extensions

Let $V$ be a model of $ZFC$. Let $M \in V$ be (a possibly not transitive) countable model of large fragments of $ZFC$ (for example countable substructures of large $H_\Theta$). If $M$ models enough of ...
8
votes
0answers
173 views

How did Cohen invent forcing?

A couple of popular maths book, I forget which stated that Cohen invented Forcing. Now, generally I've noticed that there is a history which allows one in hindsight to show that how certain ...