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Results tagged with set-theory
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user 11145
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.
124
votes
17
answers
18k
views
Pressure to defend the relevance of one's area of mathematics
I am a set theorist. Since I began to study this subject, I became increasingly aware of negative attitudes about it. These were expressed both from an internal and an external perspective. By the “ …
35
votes
8
answers
7k
views
Why not adopt the constructibility axiom $V=L$?
Gödelian incompleteness seems to ruin the idea of mathematics offering absolute certainty and objectivity. But Gödel‘s proof gives examples of independent statements that are often remarked as having …
29
votes
2
answers
3k
views
Who introduced direct limits?
The general notion of a direct limit of a commuting system of embeddings, indexed by pairs in a directed set, has seen heavy use in set theory. It is the same notion as in category theory. I was sur …
- 18.7k
28
votes
3
answers
3k
views
Construction of nonmeasurable sets
I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et …
- 18.7k
23
votes
Accepted
Can GCH fail everywhere every way?
No. An early nontrivial constraint on the $\beth$ function comes from Kőnig's Theorem, that for all infinite $\kappa$, $\mathrm{cf}(2^\kappa)>\kappa$. This implies that we cannot have $\beth_\alpha …
- 18.7k
19
votes
1
answer
1k
views
A game on sets of reals
A 2 player game on $\mathcal{P}(\mathbb{R})$: Players take turns playing uncountable sets of reals. Each play must be a subset of the previously played set. Player 1 wins if the intersection of all t …
- 18.7k
18
votes
2
answers
2k
views
A “paradox” about the inner model problem
As stated in Woodin, Davis, and Rodriguez - The HOD dichotomy, a longstanding open problem in set theory is to construct a canonical inner model for supercompactness. In general there are various way …
- 18.7k
17
votes
What is so special about set theory anyway?
It is my understanding that set theory's interpretive power is a quasi-empirical fact, and that at present there is no grand theoretical explanation of the phenomenon. By "set theory," I mean the mat …
- 18.7k
17
votes
6
answers
1k
views
Strategic vs. tactical closure
The Banach-Mazur game on a poset $\mathbb P$ is the $\omega$-length game where the players alternate choosing a descending sequence $a_0 \geq b_0 \geq a_1 \geq b_1 \geq \dots$. Player II wins when th …
- 18.7k
16
votes
Accepted
Does stationary reflection imply Mahloness?
Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength:
Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ …
- 18.7k
15
votes
1
answer
973
views
Does inner model theory seek canonical models for large cardinals?
Like the author of this question, I have heard that a main goal of inner model theory is building canonical inner models for large cardinals. My questions are: (a) Is this accurate? (b) If so, in wha …
- 18.7k
15
votes
2
answers
540
views
capturing small sets in small factors
Suppose $\kappa$ is a regular cardinal and $P$ is a $\kappa$-c.c. partial order. I want to know when are small sets added by subforcings of size $<\kappa$. The following seems well-known:
Fact: …
- 18.7k
15
votes
2
answers
802
views
Who needs RCS iterations?
According to this paper of Chaz Schlindwein, any countable support iteration of semi-proper forcings is semi-proper. This seems like a breakthrough simplification, and I wonder why it is not more wel …
- 18.7k
15
votes
3
answers
1k
views
Singularizing forcing of "small" cardinality?
Can there be a large cardinal $\kappa$ and a forcing of size $\kappa$ that makes $\kappa$ a singular cardinal? The motivation is that the standard Prikry forcing does not have a dense set of size $\k …
- 18.7k
14
votes
3
answers
483
views
For ideals, does normal imply countably complete?
The following little question has bugged me for a while.
Suppose $Z \subseteq \mathcal P(X)$. We say an ideal $I$ on $Z$ is normal when it is closed under diagonal unions, which means that if $\{ A_x …
- 18.7k