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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.
4
votes
1
answer
239
views
Some consequences of internally approachable structures
I just read for the first time the definition of an internally approachable set, which says:
A set $N$ is internally approachable (I.A.) of length $\mu$ iff there is a sequence $(N_{\alpha} : \alph …
- 2,180
2
votes
Two questions about the boolean algebra $P(\kappa)/Cub^*$
Second question: You pick a stationary $S \subset \kappa$. Define a sequence recursively like this:
$S_0 := S$, $S_{\alpha +1}:= S_{\alpha} - A_{\alpha}$ where $A_{\alpha}$ is defined as the set of t …
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4
votes
1
answer
250
views
Another question on stationarity
Let $\kappa < \lambda$ be regular cardinals and let $A \subset \lambda$ be such that each $\alpha \in A$ has cofinality $\gamma < \kappa$. Then the following should hold:
$A$ is stationary iff $ \lbr …
- 2,180
2
votes
2
answers
297
views
Is $sat(I)$ always a regular cardinal?
Let $I$ be an ideal and let $I^+$ denote its complement (the so-called $I$-positive sets). Now we say that $I$ is $\lambda$-saturated iff each antichain in $I^+$ has size less than $\lambda$. Further …
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2
votes
Set theoretic question about real valued functions
EDIT: The following answer was not correct, as pointed out in the comments (though the false part is now deleted). At least it shows that there cannot be a finite collection of functions with the dema …
- 2,180
4
votes
2
answers
426
views
A characterization of stationarity?
I just read a proof and, after struggling some time with a mental leap, I think that it uses tacitly the following:
Let $\kappa$ be a regular cardinal, $\theta > \kappa$ a regular cardinal too then:
…
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24
votes
2
answers
4k
views
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
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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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 …
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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 …
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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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2
votes
Normal measures and Elementary Embeddings
I just wanted to fix my answer, which I couldn't do yesterday as it was already midnight and I was too tired (nevertheless the answer already given by Amit is elegant and true)
As $D$ is normal $\kap …
- 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 $ …
- 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 …
- 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 …
- 2,180