All Questions
Tagged with independence-results forcing
17 questions
5
votes
0
answers
212
views
Questions about very fat sets
If $\kappa$ is a regular uncountable cardinal, we call a set $S\subseteq\kappa$ fat if for every $\alpha<\kappa$ and every club $C\subseteq\kappa$, there is a closed subset of $S\cap C$ of ...
- 1,799
11
votes
3
answers
994
views
Why can we assume a ctm of ZFC exists in forcing
Following Kunen's book, it makes clear that countable transitive models (ctm) exist only for a finite list of axioms of ZFC. So, why can we assume a ctm of the whole ZFC axioms exists and use it as ...
- 111
4
votes
1
answer
194
views
Is every first-countable Lindelof space of cardinality $<\mathfrak c$ a $Q$-space under MA?
Definition. A topological space $X$ is a $Q$-space if every subset of $X$ is of type $G_\delta$.
It is clear that every $Q$-space has countable pseudocharacter (= all singletons are $G_\delta$) and is ...
- 41.8k
11
votes
1
answer
704
views
Is $\mathfrak j_{2:1}=\mathfrak{j}_{2:2}$ in ZFC?
A function $f:\omega\to\omega$ is called
$\bullet$ 2-to-1 if $|f^{-1}(y)|\le 2$ for any $y\in\omega$;
$\bullet$ almost injective if the set $\{y\in \omega:|f^{-1}(y)|>1\}$ is finite.
Let us ...
- 41.8k
3
votes
0
answers
122
views
The existence of $T$-ultrafilters in ZFC
Looking at this MO-problem, my collegue Igor Protasov suggested to ask on Mathoverflow his old question on $T$-ultrafilters hoping that somebody on MO can solve it.
First I recall the necessary ...
- 41.8k
11
votes
1
answer
628
views
A new cardinal characteristic (related to partitions)?
In this post I will discuss some cardinal characteristic of the continuum, related to partitions of $\omega$ and would like to know if it is equal to some known cardinal characteristic.
By a partition ...
- 41.8k
6
votes
0
answers
375
views
How bad a proper forcing of size $\aleph_1$ can be?
This question concerns proper forcings of size $\aleph_1$. In the context of
$\rm ZFC+\neg CH$, I couldn't find any counter example to the following property. Suppose $\mathbb P$ is a proper forcing ...
- 2,381
6
votes
1
answer
626
views
Can $H_{\omega_1}$ and $H_{\omega_2}$ be in bi-interpretation synonymy?
This question concerns the possibility of the bi-interpretation synonymy of the structure
$\langle
H_{\omega_1},\in\rangle$, consisting of the hereditarily countable sets, and the structure $\langle ...
- 236k
5
votes
1
answer
207
views
Examples of independent $\Sigma_4^1$ statements
As in the title, I'm looking for examples of $\Sigma^1_4$ (preferably complete) sentences which are independent from ZFC in both ways, namely given a model $V$ we can extend it to $V'$ where such a ...
6
votes
1
answer
273
views
$\omega_2$-sequence of Suslin trees
Is it possible to have an $\omega_2$-length sequence of ($\omega_1$-)Suslin trees such that if one builds the product of finitely many trees in that sequence, one ends up with a Suslin tree again?
...
- 2,180
9
votes
2
answers
1k
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 ...
- 2,240
7
votes
1
answer
425
views
On Consistency of an Existence
Let $\omega \leq \kappa <2^{\omega}$ , $\omega \leq\lambda \leq \kappa$ and $D(\kappa, \lambda)$ be the statement:
For all $ \mathfrak{B} \subseteq \mathbf{P}(\omega)$ with $|\mathfrak{B}|=\kappa$...
- 2,381
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
7
votes
6
answers
2k
views
Formal proof of Con(ZFC) => Con(ZFC + not CH) in ZFC
Is it possible to prove $Con(ZFC) \rightarrow Con(ZFC + \neg CH)$ purely within ZFC? To prove this (using forcing) one seems to need a countable transitive model of ZFC. The texts I am reading avoid ...
- 365
8
votes
3
answers
745
views
Natural statements independent from true $\Pi^0_2$ sentences
I am looking for sentences in the language of first order arithmetic ($0,1,+,\cdot,\leq$) which are independent from $\Pi^0_2$ consequences of true arithmetic $\Pi^0_2\text{-}\mathsf{Th}(\mathbb{N})$. ...
- 5,502
7
votes
2
answers
844
views
Statements forced by one condition of a poset, but not the whole thing
In order to get the relative consistency of some statement, it suffices to find a notion of forcing, and a condition $p$ in that forcing, such that $p$ forces the desired statement. It seems to be ...
- 3,982
3
votes
2
answers
688
views
A problem about posets similar to Suslin's problem
Suslin's problem is:
Given a complete dense linear order without endpoints, if it has the ccc must it be isomorphic to $\mathbb{R}$?
The answer is that it's independent of ZFC. The related ...
- 3,982