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Results tagged with large-cardinals
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user 20597
6
votes
Reinhardt's ultimate classes
You can find Reinhardt's philosophy of set theory in
"Set existence principles of Shoenfield, Ackermann, and Powell", Fundamenta Mathematica, vol 84, pp 5-34 and
"Remarks on reflection principles, la …
- 6,132
1
vote
0
answers
313
views
Is there a second-order expression of '$\kappa$ is Reinhardt' in $NGB$, where $\kappa$ is a ...
In their paper, "Generalizations of the Kunen inconsistency", (Annals of Pure and Applied Logic 163 (2012) 1872-1890, doi:10.1016/j.apal.2012.06.001, arXiv:1106.1951), Hamkins, Kirmayer, and Perlmutte …
- 6,132
0
votes
1
answer
614
views
What are the difficulties involved in proving that the Kunen inconsistency holds in $NGB$
or (contrariwise) that $NGB$ + "There exists a Reinhardt cardinal" is consistent?
The question is partially in the title. $NGB$ is used for the reasons stated in the Hamkins, Kirmayer, and Perlmutte …
- 6,132
0
votes
1
answer
878
views
Forcing the existence of a weakly inaccessible cardinal in some strong set theory
Does the fact that, assuming the consistency of $ZFC$, no proof that the consistency of "$ZFC$ implies the consistency of '$ZFC$ + There exists a weakly inaccessible cardinal'" can be formulated in $Z …
- 6,132
5
votes
0
answers
940
views
Why are real-valued measurable cardinals never explicitly mentioned in Gödel's "What is Cant...
It is a matter of mathematical folklore that Gödel "entertained the idea of so called stronger axioms of infinity deciding $CH$...." (this quote from Radek Honzik's paper, "Large cardinals and the Co …
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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 …
- 6,132
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 …
- 6,132
5
votes
2
answers
493
views
Critical points in $ZF$ without Choice
Recall the definition of critical point for set theory:
A critical point of an elementary embedding of one transitive class into another transitive class is the smallest ordinal not mapped to itse …
- 6,132
1
vote
Are there first order theories of interest to an algebraist or at least a model theorist of ...
You might want to take a look at pp.1-2 (and the top quarter of pg. 3) of Harvey Friedman's paper "Restrictions and Extensions" (the rest of the paper (the paper is all of six pages) deals with the sy …
- 6,132
1
vote
1
answer
251
views
A question regarding models of $ZF+I_0$ [Revised]
In his answer to user42090's mathoverflow question"Minimal Generalized Contnuum Hypothesis & Axiom of Choice", Prof. Hamkins writes:
"...one can build the analogue of the symmetric models for $\lnot$ …
- 6,132
1
vote
2
answers
570
views
A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measu...
A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name):
"The naive probabalistic notion used by Freiling tacitly assumes that there …
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2
votes
1
answer
368
views
Class forcings and elementary embeddings
In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem:
"Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial eleme …
- 6,132
18
votes
2
answers
1k
views
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 …
- 6,132
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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1
vote
1
answer
444
views
A question regarding extendible cardinals and a result of M. Magidor
The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971):
"Definition: Logic is called $\k …
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