Tagged Questions
0
votes
0answers
30 views
Adding sets with forcing
Let $M$ be a c.t.m. of ZFC. Let $\mathbb{P}$ be the poset $Fn(\lambda \times \kappa, 2, \kappa) = \{p : |p| < \kappa \wedge p$ is a function $\wedge dom(p) \subseteq \lambda\times\kappa \wedge ...
4
votes
0answers
69 views
Half of a Cohen real
I recently heard from a friend that Zapletal gave a talk at Toronoto where he constructed a proper forcing which adds a real which infinitely often equals every ground model real but doesn't add a ...
3
votes
1answer
62 views
Existence of a probability measure under which a subset of $[0,1]$ cannot be approximated by a Borel set
Is there some probability measure, $p$, on $\left(\left[0,1\right],2^{\left[0,1\right]}\right)$ relative to which some $D\subseteq\left[0,1\right]$ cannot be approximated by a Borel set, i.e. if $E$ ...
5
votes
1answer
89 views
Powers of infinite cardinals in the generic extension
Let $\kappa, \lambda,\theta$ be infinite cardinals of $M$ where $M$ is a transitive model of ZFC. Let $\mathcal{P}=Fn(\kappa\times \omega,2)$ (finite functions from $\kappa\times \omega$ to 2) and G ...
5
votes
1answer
68 views
Does forcing need a countable transitive model
Does forcing need a countable transitive model or can forcing be done using any model $M$ of $\mathsf{ZFC}$?
Also: what about standard vs. non-standard $M$? (here standard just means that the ...
3
votes
1answer
67 views
Factoring out a Cohen forcing
Suppose that in a forcing extension $V[G]$ by some ccc forcing $P$ there is a Cohen real over $V$. By a general argument we can factor $P$ into an iteration $P=\mathrm{Add}(\omega,1)*\dot{Q}$ for some ...
8
votes
3answers
251 views
Do we usually assume the consistency of ZFC when we use forcing?
I used to think that to avoid philosophical problems in forcing one can assume consistency of ZFC. Then one obtains a set model $M$ contained in the von Neumann universe $V$. And then the generic ...
2
votes
1answer
44 views
Isomorphism of Forcing Posets
Let $\text{Col}(\alpha, \beta)$ be the forcing poset that collapses $\beta$ to $\alpha$, let $Q$ be any old forcing poset, with $|Q| = \lambda$.
Why is $Q \times \text{Col}(\omega, \lambda) \cong ...
4
votes
2answers
69 views
A reference that forcing with a poset of cardinality less than $\kappa$ preserves supercompact cardinals
I'm looking for a reference for the fact that if $\kappa$ is supercompact and $Q$ is a poset of cardinality less than $\kappa$, then $V^Q\models \check{\kappa} \ \text{is supercompact}$.
Apparently ...
3
votes
1answer
58 views
Exercise in Just/Weese (amoeba forcing) (1/2)
I solved the following exercise (18.3):
Can you tell me if I got it right? Thanks:
18.3.(a) The inclusion "$\supseteq$" follows immediately from the definition. For "$\subseteq$" let $U \in ...
2
votes
2answers
54 views
How to find a condition that “decides” the value of a name for an ordinal
In a paper I'm reading involving forcing, the following is used without proof, and it seems to be stated as if it is "obvious," but I don't see why. It's probably either easy or known, but I might ...
8
votes
1answer
110 views
Failure of Choice only for sets above a certain rank
Let $\alpha$ be an ordinal. How can we show that the following theory is consistent?
$\mathrm{ZF}$ + "there exists a set with rank greater than $\alpha$ that is not well ordered" + "every set of rank ...
1
vote
1answer
38 views
Adding a surjection $\omega \to \omega$ by Levy forcing
I'm trying to understand the Levy collapse, working through Kanamori's 'The Higher Infinite'. He introduces the Levy forcing $\text{Col}(\lambda, S)$ for $S \subseteq \text{On}$ to be the set of all ...
2
votes
1answer
128 views
Did large cardinals exist before 1963?
I'm curious to know the history of the interaction between large cardinals and traveling to (creating) universes through forcing. The question arose because I understand that Peano Arithmatists ...
2
votes
1answer
56 views
Proof of levy forcing and cardinal collapse
Collapsing a cardinal to $\omega$: $P$ is the set of all finite
sequences of ordinals less than a given cardinal $\lambda$. If
$\lambda$ is uncountable then forcing with this poset collapses
...
1
vote
1answer
35 views
Proving the role of countable chain condition - what is this quote saying?
The importance of antichains in forcing is that for most purposes,
dense sets and maximal antichains are equivalent. A maximal antichain
$A$ is one that cannot be extended and still be an ...
3
votes
2answers
101 views
Use of forcing to real line to make elements countable
Can we use forcing techniques to force the set of elements of the real line to be countable? If not can anyone show why it is not possible?
0
votes
1answer
35 views
What is Fin(E,2)?
It is easy to see that Bor(I) satisfies the c.c.c., because the measures add up to at most 1. Fin(E,2) is also c.c.c., but the proof is more difficult. (Countable chain condition, wikipedia)
So ...
4
votes
3answers
80 views
Countable transitive model of ZFC?
If there is countable transitive model of ZFC, this model cannot capture all ordinals of ZFC. But we use it for stuffs like forcing.
But this seems to violate ZFC's axioms - for example, power set ...
2
votes
2answers
115 views
Forcing Question about a sequence of functions from $\aleph_{0}$ into $2 = \{0, 1\}$
I'm trying to go through Timothy Chow's A Beginner's Guide to Forcing (arxiv pdf).
My particular question is about the first paragraph of Section $5$ (on p. $7$ of the above pdf).
First, my vague ...
1
vote
1answer
59 views
Why is the ground model set of reals nonmeager in the Cohen extension?
To be more precise, I want to add a Cohen real to a ground model $V$ of ZFC and then show that for each open interval $(a,b)$, the set $V \cap (a,b)$ is nonmeager in the extension.
Thanks in advance.
...
5
votes
1answer
114 views
Forcing and generic extensions
The following is from Halbeisen, page 289, on generic extensions:
This leads to one of the key features of forcing: By knowing whether a
certain condition $p$ belongs to $G \subseteq P$, people ...
2
votes
1answer
68 views
Questions on: $G$ a generic ultrafilter on $B$ if and only if $G$ is a generic filter on $(B+,<)$?
Here are the definitions, and then I shall explain which parts of the implication I understand, and which parts I don't, which are the questions. The definitions are from Jech, as well as the ...
10
votes
1answer
142 views
Allowing addition of ordinals in forcing
I'm reading Paul Cohen's "The Discovery of Forcing", this is a question related to my previous question:
Why did Cohen require forcing to be such that no new ordinals are added in the process? Or, ...
3
votes
1answer
85 views
Adjoining new ordinals to a model — a question about one of Cohen's articles
The following is an excerpt from Paul Cohen's "The Discovery of Forcing", pp 1091, in which he explains why we do not want to add new ordinals to a countable transitive model $M$ when extending it ...
2
votes
1answer
112 views
Infitive distributive law in boolean valued models
I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
3
votes
2answers
106 views
Question about passage in Halbeisen's book
I am looking at the following passage in Halbeisen's book "Combinatorial Set Theory" (p 260 at the bottom):
What is the role of $\Phi$? It seems to me that a finite fragment is the same as a ...
1
vote
0answers
48 views
A sheaf of cumulative hierarchies
I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on ...
5
votes
2answers
121 views
Trying understand a move in Cohen's proof of the independence of the continuum hypothesis
I've read a few different presentations of Cohen's proof. All of them (that I've seen) eventually make a move where a Cartesian product (call it CP) between the (M-form of) $\aleph_2$ and $\aleph_0$ ...
4
votes
1answer
114 views
Adding Cohen reals one at a time
We know that if we start with a ctm $\mathbb{B}$ and force with the poset of finite functions from $\omega$ to $2$, we add a single Cohen real. We also know that if we force with the poset $\mathbb{P} ...
1
vote
3answers
116 views
Question about models, cardinalities and collapsing
I have a (seeming) contradiction and I can't seem to figure out the (obvious) mistake in my reasoning. The following (related to my previous question):
(1) $\omega$, defined to be the least infinite ...
9
votes
1answer
136 views
Atoms necessary for the existence of a generic filter?
I'm reading Some applications of the method of forcing by Todorchevich and Farah and one of the exercises seems to be wrong to me.
Definitions
We fix some partially ordered set $(P,\preceq)$.
A ...
3
votes
0answers
78 views
A question about the proof that forcing extensions don't add ordinals
I've been reading Halbeisen's Combinatorial Set Theory: with a gentle introduction to forcing (well, more like skimming for now...) and I've stumbled into an apparent problem with a proof of the ...
11
votes
3answers
282 views
A nice introduction to forcing
I want to get acquainted with forcing, along with a few friends, and I'm looking for a text to introduce the basic notions (pardon the pun :) ).
The point is to study a text (or texts, if they can be ...
2
votes
1answer
132 views
When collapsing a cardinal, what ordinal does it become?
Work in $V$. Let $P = \text{Col}(\omega, \omega_1)$ and suppose that $G$ is generic for $P$ over $V$. Then $V[G]\models |\omega_1^V|=\aleph_0$ and $\omega_2^V=\aleph_1$. In particular, ...
2
votes
1answer
62 views
How does one go to prove two forcing extensions are equivalent?
I want to know how does someone usually go, that is, what is the canonical way to go proving that having two notions of forcing $P$ and $Q$, the respective generic extensions $P[G]$ and $Q[H]$ are ...
4
votes
1answer
174 views
Why is this forcing notion closed?
I'm studying a forcing argument which produces a generic extension in which GCH holds, but I am, somewhat embarrassingly, stuck on a minor detail. I hope someone can point out the thing I'm missing.
...
2
votes
1answer
154 views
Forcing exercise
This exercise is from Kunen, Set Theory.
Let $M$ be a ctble transitive model of ZFC,$\kappa > \omega$, $\kappa$ regular, $P$ be a notion of forcing that is $\kappa$-closed (i.e. whenever $\gamma ...
1
vote
0answers
54 views
$\Delta_0$-formulas in the Boolean-valued model $V^B$
I want to show that $\Vert \check x = \check y \Vert = 1$ implies $x = y$, where $\check x$ is the canonical name for $x$ in $V^B$.
I'd like try induction over the ranks of $\check x$ and $\check y$ ...
2
votes
1answer
155 views
Forcing questions
I have been looking at a proof of a technical forcing lemma, and I have a couple of questions.
Here is the setup: $(N, \epsilon) \prec (\mathbf{H}( \chi ), P, \epsilon ) $ is a countable submodel, ...
3
votes
1answer
251 views
Can the real numbers be forced to have arbitrary cardinality?
Let $\mathbb R$ be the real numbers in a given model of set theory.
Given an arbitrary cardinal number $\kappa$, does forcing produce a larger model in which the cardinality of $\mathbb R$ is ...
4
votes
1answer
105 views
Generic reals in forcing iterations
Suppose that $(\mathbb{P}_{\alpha}, \underset{\sim}{\mathbb{Q_{\alpha}}} : \alpha<\beta)$ is a forcing iteration, and that for each $\alpha$, there is a name $\underset{\sim}{\eta_{\alpha}}$ for ...
3
votes
0answers
109 views
Question about prompt names of ordinals
The following concept is due to Shelah and I have some issues with a claim using this notion:
Suppose that $\nu$ is a limit ordinal and that $P_\nu$ is an iteration of forcing notions. We say that a ...
4
votes
2answers
135 views
Does forcing with a proper class of conditions require Global Choice?
When Jech, in his Set Theory, deals with forcing with a class of forcing conditions (with the aim of proving Easton's theorem), he starts with the assumption that there is a well-ordering of the ...
3
votes
1answer
172 views
How many non-isomorphic partial orders of a certain size?
I'm working on the consistency of Martin's axiom and I need some help counting. Assume we are in a universe where GCH holds and $\kappa$ is a regular cardinal. How many non-isomorphic partial orders ...
2
votes
2answers
333 views
A question regarding the Continuum Hypothesis (Revised)
Given that the order type of the reals $(\mathbb R,<)$ has no definable points, can it be proven that $|\mathbb R|= \aleph_1$ by virtue of the fact that every uncountable proper subset $S$ of ...
1
vote
1answer
108 views
Preorder of forcing notions
(Throughout, fix a universe of set theory $V$ to serve as a base.) We can define a fairly natural preordering on forcing notions as follows. Let $\mathbb{P}_1$, $\mathbb{P}_2$ be two forcing notions. ...
3
votes
2answers
210 views
Boolean-valued models have no essentially new ordinals
Since I still have some trouble with transferring theorems of ZFC into the Boolean-valued framework, would someone more at ease with this check the following short calculation?
This is a proposition ...
2
votes
1answer
130 views
Canonical name for the Boolean algebra in a Boolean-valued model
Following Jech's Set Theory, fix a complete Boolean algebra $B$ and form the Boolean-valued model $V^B$ of ZFC. We then have the name $\check{B}\in V^B$. Apparently, it is the case that ...
1
vote
1answer
107 views
Product forcing and generic objects
If we start with a model of ZFC, $M$ and $(P,\le)$ is a notion of forcing, $G\subseteq P$ a generic filter, then in $M[G]$ we can define some generic object from $G$. For example, if $P$ is the Levy ...
