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Results tagged with set-theory
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user 57583
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.
3
votes
What is the best way to construct an Aronszajn Tree?
There is also Shelah's very enjoyable construction using descending sequences of infinite subsets of $\omega$, close in spirit to Aronszajn's tree of rational sequences, and described in Judith Roitma …
- 2,111
1
vote
Accepted
Diagonalizing against a non stationary set of functions
Sorry this is late and you may already have it all sorted out more attractively. I think an argument might go as follows. It rests on K. Devlin, Variations on $\diamondsuit$, JSL 44 (1979), modified f …
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6
votes
Why should we care about "higher infinities" outside of set theory?
In line with Joel's answer and the theme that stronger set theories permit finer analysis of higher infinities, an example from commutative algebra suggesting the desirability of distinguishing more t …
- 2,111
4
votes
Accepted
Is it consistent that $\frak{d} < 2^{\aleph_0}$?
Theorem 5.1 in Eric van Douwen's paper "The integers and topology" (Handbook of Set-theoretic Topology) is an old reference for a positive answer to your question and will provide a fuller explanation …
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7
votes
0
answers
433
views
$\delta$-strong compactness and generalized strong tree properties
Are there non-trivial equivalent characterizations of $\delta$-strongly compact (and almost strongly compact) cardinals in terms of generalized tree properties?
Recall the definitions as per Joan Ba …
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5
votes
Forcing as a tool to prove theorems
An example from elementary geometry is the very simple forcing argument to establish the non-axiomatizability (in infinitary logic) of Sperner spaces, due to Blass and Pambuccian: Blass, A.; Pambuccia …
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10
votes
Accepted
Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?
The answer is negative apparently. It is consistent relative to ZFC that all Aronszajn trees are
special and that the club principle holds:
http://home.mathematik.uni-freiburg.de/mildenberger/postin …
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5
votes
Accepted
References for Forcing with Side Conditions
S. Todorcevic, Notes on Forcing Axioms, Chapter 7.
Itay Neeman, Forcing with side conditions. Oberwolfach, 2011. http://www.math.ucla.edu/~ineeman/
B. Velickovic, G. Venturi, Proper forcing remastere …
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4
votes
If every nonseparable metric space contains a sequence of subsets with no convergent subsequ...
This answer is the proof given by Ashutosh, but formulated in terms of the splitting number.
Proposition
If the splitting number $s$ is $\aleph_{1}$, then every nonseparable metric space contains a s …
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12
votes
Accepted
On a weak tree property for inaccessible cardinals
The following theorem of Kurepa, proved in his thesis of 1935, seems to address your question.
Theorem Suppose that $\kappa = cf(\kappa) > \gamma$, and $(T, <_{T})$ is a $\kappa$-tree each of whose l …
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6
votes
2
answers
581
views
If every nonseparable metric space contains a sequence of subsets with no convergent subsequ...
If every nonseparable metric space contains a sequence of subsets with no convergent subsequence, does the Continuum Hypothesis hold?
The answer is negative, and in the interests of self-contained it …
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11
votes
0
answers
295
views
Preserving Jonsson cardinals
I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ever …
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7
votes
1
answer
474
views
Under $\neg CH$, have countable unions of rationally independent numbers inner measure zero?
In their 1943 paper On non-denumerable graphs, Erdos and Kakutani suggest as likely the following proposition.
(EK*) Suppose CH fails and $\lbrace M_n : n \in \omega \rbrace$ is a countable family of …
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7
votes
Examples of ZFC theorems proved via forcing
The Gitik-Shelah theorem is also perhaps an example, first proved with forcing by its discoverers, and then without by Anastasis Kamburelis and David Fremlin independently:
Moti Gitik, Saharon Shela …
7
votes
Accepted
Complete resolutions of GCH
One candidate answer scheme might be the following: if $F$ is any (sufficiently absolute) definable function on the class of regular alephs such that $\kappa < \lambda \Rightarrow F(\kappa) \leq F(\la …
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