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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.
24
votes
2
answers
4k
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Montague's Reflection Principle and Compactness Theorem
Here's a question I can't answer by myself: The Reflection Principle in Set Theory states for each formula $\phi(v_{1},...,v_{n})$ and for each set M there exists a set N which extends M such that the …
- 2,180
23
votes
Set theory and Model Theory
I totally agree with the answers already given but I still want to say something to your question, which emphasizes probably the formalist side. To cut a long story short the foundation of model theor …
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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 …
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10
votes
Kunen's use of Countable Transitive Models
It is true that you can use Löwenheim Skolem to get a countable model $M$ of ZFC assuming $Con(ZFC)$. But to use Mostowski you need additionally the well foundedness of that model, which doesn't have …
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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 …
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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 …
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 …
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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 …
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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) …
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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 …
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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 …
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6
votes
2
answers
1k
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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 …
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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_{ …
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5
votes
1
answer
556
views
Why is this set stationary?
Hi
I really need a proof for the following statement by Baumgartner:
There exists a stationary subset of $[\omega_2]^{\omega}$ of size $\aleph_2$.
This is Exercise 38.15. in Jechs Book (2003) and …
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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 $ …
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