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Results tagged with set-theory
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user 20597
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.
45
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1
answer
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Hilbert's alleged proof of the Continuum Hypothesis in "On the Infinite"
As is known, Hilbert attempted a proof sketch of the Continuum Hypothesis in the latter part of his paper, "On the Infinite". It is also known that it is false.
Has there ever been a published analy …
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21
votes
3
answers
2k
views
In What Way are Set Theorists' 'Experiences' in the CH Worlds Flawed, if Any?
This is in regards to Joel David Hamkins' new paper "IS THE DREAM SOLUTION OF THE CONTINUUM HYPOTHESIS ATTAINABLE?" (look under title in arXiv). I quote from the last paragraph of his paper:
"My chal …
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18
votes
2
answers
1k
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What sort of cardinal number is the Löwenheim–Skolem number for second-order logic?
In their paper “On Löwenheim–Skolem–Tarski numbers for extensions of first order logic”, Magidor and Väänänen make the following statement:
“For second order logic, $\mathrm{LS}(L^{2})$ [the Löwenheim …
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18
votes
2
answers
696
views
Category-theoretic characterization of $L$
Does there exist a characterization of Goedel's constructible universe $L$ in purely category-theoretic terms, or is constructibility an 'artifact' of material set theory? If, in fact, constructibili …
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13
votes
1
answer
2k
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Hausdorff and Naive Set Theory
Erhard Scholz, in his article "Felix Hausdorff and the Hausdorff edition" writes the following:
"Hausdorff considered the contemporary attempts to secure axiomatic foundations for set theory as prema …
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11
votes
Vopěnka's Principle for non-first-order logics
Since the title of your question is, "Vopenka's Principle for non-first-order logics", this, from Magidor's and Vaananen's paper "On Lowenheim-Skolem-Tarski numbers for extensions of first order logic …
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9
votes
1
answer
999
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How are material set theory and structural set theory related from the point of view of cate...
In his comments to both cody and Nik Weaver regarding his answer to user7280899's mathoverflow question "What kind of foundation are mathematicians using when proving metatheorems?", Mike Shulman writ …
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9
votes
Why should we care about "higher infinities" outside of set theory?
You might take a look at the following preprint of A.D.R. Mathias: "Strong Statements of Analysis". This paper deals with the same concerns you seem to have regarding the applicability of higher infi …
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8
votes
On independence and large cardinal strength of physical statements
Though this does not directly answer your question, here is a foundational paper that might help one derive results that might answer your question:
Marian Boykan Pour-El and Ian Richards: "Nonco …
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8
votes
1
answer
476
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A Question Regarding the Relation Between 0-sharp and Koepke's Bounded Truth Predicate.
In Jech's SET THEORY (a very early edition to which I have access), it is shown that the existence of 0-sharp implies the existence of a truth definition for the constructible universe L. Does the co …
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7
votes
Accepted
A question about how much set theory can be developed based on the "subset" relation rather ...
Though Hamkins and Kikuchi show that $\in$ is not definable from $\subseteq$ and that the theory of $($$V$, $\subseteq$$)$ is decidable, they also show the following:
What we should like to observ …
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7
votes
Belief in consistency of extremely large cardinals
There is another path that can be used to justify the existence of very large cardinals. Consider, for example, the abstract to Magidor's paper, "On the Role of Supercompact and Extendible Cardinals …
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7
votes
How (non-)computable is set theory?
For what it's worth, you might also consider the sets computable by Ordinal Turing Machines (see Koepke's paper, Turing Computations on Ordinals, arXiv:math/0502264v1 [math.LO] 13 Feb 2005). A centra …
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7
votes
Why hasn't mereology succeeded as an alternative to set theory?
Considering the nature of your question, you might be interested in the following paper by Geoffrey Hellman and Stewart Shapiro:
"The Classical Continuum without Points", The Review of Symbolic Logic …
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6
votes
2
answers
367
views
The role of the rigid relation principle ($RR$) in the Kunen inconsistency
Consider the rigid relation ($RR$) principle, i.e.
"every set admits a rigid binary relation", that is,"that for every set $A$ there is a binary relation $R$ on $A$ such that the structure $(A,R)$ is …
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