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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.
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 …
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1
vote
On the utility of Silver machines
To begin to answer your 'abstract question', "What are Silver Machines good for', one need only look at the title of Prof. Silver's unpublished manuscript
"How to eliminate the fine structure from th …
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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 …
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2
votes
Forcing and new ordinals
You might consider the following quote from Cohen's paper, "The Discovery of Forcing", Rocky Mountain Journal of Mathematics, Volume 32, number 4, 2002, pg. 1091 (found under title on the Web):
So …
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2
votes
What is induction up to $\varepsilon_0$?
The answer to your question can be found in Maria Hameen-Anttila's paper, Nominalistic Ordinals, Recursion on Higher Types, and Finitism, Bulletin of Symbolic Logic, 25(1), 101-124 (2019) doi:10.1017/ …
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-4
votes
2
answers
454
views
Is the notion of measurable cardinal definable from the perspective of set-theoretical poten...
Consider the definition of measurable cardinal (this definition was found in Neil Barton's paper, "Large cardinals and the iterative conception of set"):
Definition 8. A cardinal $\kappa$ is measura …
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2
votes
Is V, the Universe of Sets, a fixed object?
I believe the answer to your question revolves around correcting a subtle confusion between classes and sets in the Cumulative Hierarchy. This can be shown by reference to Samuel Coskey's Senior Thes …
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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 …
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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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2
votes
Can we add set complements on top of ZF?
I believe that an answer to your question [1] is the system that Dana Scott developed in his paper, "Axiomatizing Set Theory" found in Proceedings of Symposia in Pure Mathematics, Volume 13, Part II, …
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2
votes
Forcing over models without the axiom of choice
Here is an equivalence of a forcing principle to the Axiom of Choice, courtesy of Arnold Miller, found in his preprint, "The maximum principle in forcing and the axiom of choice":
(Abstract) In th …
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2
votes
0
answers
323
views
The universe and multiverse views of set theory from the perspective of $ZFC^2$
(Note: the 'Second-order $ZFC$' ($ZFC^2$) I am talking about is the theory [in the second order language of set theory consisting of a single non-logical symbol $\in$ ] consisting of the axioms Exte …
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4
votes
Set theoretical multiverse and truths
Since the Fundamental Theorem of Arithmetic is a theorem of $PA$, it holds for both standard and nonstandard models of $PA$. Since one can interpret $PA$ in both $ZFC$ and $GBC$ (e.g., for $ZFC$, it …
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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 …
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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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