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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.
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A question regarding $ZFC^{-}$
Given $ZFC^{-}$, that is, ZFC-Powerset+Collection+Separation, is there a set of alternative axioms $X$ (other than the trivial one, namely, {Powerset}) that, when added to $ZFC^{-}$, allow one to deri …
- 6,132
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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4
votes
Axiom of Symmetry, aka Freiling's argument against CH
Actually, in regards to your question B), there is a large cardinal axiom that implies $AS$. Your link to the wikipedia article regarding Freiling's axiom of symmetry states the following, in the sec …
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4
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3
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1k
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A question regarding a remark of John Horton Conway
In his book On Numbers and Games (pg. 38) John Horton Conway makes the following remark, "But the collection of all gaps is not even a Proper Class, being an illegal object in most set theories." Is …
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Why should we care about "higher infinities" outside of set theory?
"However, let's say you concluded that there were only three types of cardinality--finite, countably infinite, and uncountable."
So let's consider the countable ordinals, and certain combinatorial pr …
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0
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The universe of sets, existential quantification in set theory
Considering what you wrote in your slide presentation "On the definitional character of axioms.", you might be interested in the following preprint by John L. Bell (found on his Homepage) titled "SETS …
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0
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262
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Can Dedekind's 'proof' of the existence of infinite sets be properly formulated and carried ...
This question is related to Mikhail Katz's recent mathoverflow question, "Has Dedekind's proof of the existence of existence of infinite sets been analyzed by historians?". Dedekind's 'proof' seems ( …
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-2
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1
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279
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Critical points and the Foundation Axiom
(Note: This question is related to my previous mathoverflow question, "Critical Points in $ZF$ without Choice".)
In the Stanford Encyclopedia of Philosophy entry "Non-Wellfounded Set Theory" (Sectio …
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5
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320
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Can the Kunen inconsistency (or the existence of Reinhardt cardinals) be 'properly formulate...
In their paper "Generalizations of the Kunen Inconsistency" (arXiv:1106.1951v1 [math.LO]10 Jun. 2011), Hamkins, Kirmayer, and Perlmutter write the following:
The first [metamathematical issue--my …
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1
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A Question Regarding the Powerset Size Axiom
Consider the the Powerset Size Axiom, that is, the following:
(PSA) ($\forall$x,y) |x|$\lt$|y|$\Rightarrow$$2^{|x|}$$\lt$$2^{|y|}$.
Does there exist a class $\mathscr M$ of models of ZFC such that …
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Partial Universes and the Axioms of $ZF$ Set Theory Without Choice
In his Senior Thesis, Samuel Coskey answered the question of which axioms of $ZFC$ hold at each stage of the cumulative hierarchy. Here is the list of his results:
Axioms that always hold: Extensio …
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0
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1
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255
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Real-valued measurability vs. Two-valued measurability in determining whether $CH$ holds or not
The following fact is known:
If there is a measurable cardinal, then there are only countably many constructible reals.
It is also known that if $ZFC$ + "There is a (two-valued) mesurable cardin …
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2
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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
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454
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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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0
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2
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214
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A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory
It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example:
MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) …
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