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Results tagged with set-theory
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user 4753
forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.
5
votes
Examples of ZFC theorems proved via forcing
Another example is Solovay's result to partition any stationary subset $S \subset \kappa$ into $\kappa$ many stationary subsets of $S$. It is mentioned in Jechs book that Solovay first used a generic …
- 4,713
15
votes
0
answers
605
views
Cohen/Random reals over intermediate models in countable support iterations
Assume that the continuum is $\aleph_2$ in our ground model $V$. Suppose that $(P_n \, : \, n \in \omega)$ is a fully supported iteration of length $\omega$. Suppose further that the factors of the it …
- 1,615
4
votes
2
answers
613
views
Examples of stationary set preserving forcings that are not semiproper?
A notion of forcing $P$ is called stationary set preserving iff each stationary subset of $\omega_1$ remains stationary in $V^P$. It is standard to show that semiproper (and of course proper) notions …
- 1,808
7
votes
3
answers
730
views
A result of Shelah about the nonstationary ideal
Suppose that $\kappa$ is a regular cardinal and let $NS$ be the ideal of its nonstationary subsets. One can consider the Boolean algebra $P(\kappa) /NS$ and say that (if $\lambda$ is another cardinal) …
- 1,808
6
votes
1
answer
268
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?
Th …
- 2,381
8
votes
What ccc forcings add a Suslin tree?
It is consistent that the answer is no.
If we start with $L$ as our ground model then whenever $T$ is a Suslin tree, the forcing $\mathbb{P}_T$ which shoots a branch through $T$ will always introduce …
- 2,180
3
votes
Accepted
Preservation of Woodinness when it overlaps the active extender
The Woodiness of $\delta$ in $\mathcal{J}^{\mathcal{M}}_{lh(E)}$ is witnessed by a bunch of extenders, which are either on the $\vec{E}$ sequence of $\mathcal{M}$ or are definable from elements of $\v …
- 2,180
5
votes
Accepted
Showing that $\alpha$ isn't a cardinal in $J_{\alpha+1}^{\vec E}$ for a fine extender sequen...
We know that $\alpha = (\nu^{+})^{Ult(J^{\vec{E}}_{\alpha}, E_{\alpha})}$ and that $i_{E_{\alpha}} (\kappa) > \nu$, where $i_{E_{\alpha}}$ denotes the ultrapower embedding. Thus working in $Ult(J^{E_{ …
- 2,180
6
votes
1
answer
320
views
Elementary chains in forcing extensions of $M_1$
Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic fi …
- 39.8k
8
votes
The origins of forcing in mathematical logic and other branches of mathematics
One application I know is Scott's construction of forcing extensions of models of higher order theories of the Real numbers. Scott quickly after the invention of forcing, used a forcing argument to sh …
- 2,180
6
votes
2
answers
1k
views
An exercise in Jech's Set Theory
I had a hard time trying to solve exercise 7.24 in Jech's book (3rd edition, 2003) and finally came to the conclusion that the result there, which should be proved might be wrong. The claim goes like …
CommunityBot
- 1
8
votes
1
answer
466
views
Properness of quotient forcing
It is well known that if $P$ is a proper notion of forcing and $\Vdash_P \dot{Q} \text{ is proper}$ then the iteration $P \ast \dot{Q}$ is proper. Is the converse also true, i.e
Suppose that we have …
CommunityBot
- 1
2
votes
How do we avoid circularity when we build a structure for ZFC?
When set theorists investigate ZFC, they use sets (or classes) as models, and their existence is of course given by the axioms of ZFC again. This indeed seems at first sight circular, but in fact it i …
- 2,180
5
votes
2
answers
411
views
Question about prompt names of ordinals
I asked this question first on math SE and was told that it would better fit here. So:
The following concept is due to Shelah and I have some issues with a claim using this notion:
Suppose that $ …
- 73.2k
8
votes
Size of stationary sets
There exists a stationary subset of $P_{\omega_1} (\omega_2)$ of size $\aleph_2$. This is a result of Baumgartner and you can find a proof for this here: Why is this set stationary?
I don't dare to a …
CommunityBot
- 1