All Questions
Tagged with set-theory mathematical-philosophy
117 questions
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A Question Regarding Productive Sets in the Koepke-Koerwien System SO (Sets of Ordinals)
In their paper "The Theory of Sets of Ordinals" (arXiv), Koepke and Koerwien propose a theory SO axiomatizing the class of sets of ordinals in a model of ZFC and show that SO and ZFC are bi-...
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260
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A question regarding Koepke' s Ordinal Computability in HOD
Consider the following theorem of Koepke-Koerwien-Siders:
"A set x of ordinals is ordinal computable [either by ordinal Turing machines or ordinal register machines--my comment] if and only if it is ...
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Why are model theorists free to use GCH and other semi-axioms? [closed]
Looking into the open problem section of the book Model theory by Chang and Keisler, I noticed that many problems assumed semi-axioms like GCH. I talk about 'semi-axioms' because these "axioms" are ...
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Is a function needed here?
This question is related to my question Can we choose an element from a class?.
However, I decided to create a separate question.
Let $H$ be a complex Hilbert space and $H_1,\dotsc,H_n$ be closed ...
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Why do we try to encode every mathematical object as a set? [closed]
Probably everyone of us has seen set-theoretic encodings of mathematical objects which we wouldn't naturally consider to be sets. May it be the "definition" of a function from $A$ to $B$ as a relation ...
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Formalizing ontological optimism
Inform speaking ontological optimisms means that everything that possibly exists in the abstract reality actually exists. From this principle we (again informally) get the Axiom of infinity, the Power ...
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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 $...
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Axiom of Choice : Reasons behind using it or avoiding it? [duplicate]
I am a 1st year PhD student in Mathematics.
During my masters course in Ring theory, the proof of the result : Every commutative unital ring has a maximal ideal was done in the class. Every proof of ...
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'Maximising interpretative power entails maximising consistency strength'?
I'm hoping there is a clear mathematical answer to this question (hence asking it here) rather than anything more exegetical (in which case it's presumably not appropriate for this site).
In his paper ...
- 161
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Erotetic inference and extrinsic justification?
Gödel introduced his notion of what has come to be called extrinsic justification in the following terms:
Furthermore, however, even disregarding the intrinsic necessity of some new axiom, and even ...
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Tarski-Grothendieck in the cumulative hierarchy
How can the Grothendieck-Tarski axiom seen to be true in the cumulative hierarchy of sets?
What are intuitions that would convince us that this axiom is true?
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Is it natural to hold that Ur-elements, small & big sets and proper classes exists? [closed]
The topic of this post was shifted to
https://philosophy.stackexchange.com/questions/49504/is-it-natural-to-hold-that-big-sets-and-proper-classes-exist
Since it was deemed to be a philosophical ...
- 11.3k
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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" (...
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There is a typo in Stall's textbook on Set Theory: unable to prove the trichotomy of sets (m ∈ n or m = n, or n ∈ m) [migrated]
Here is the textbook, chapter 7, page 300. This lemma seems very of important, and I've spend about 8 hours trying to figure it out, but I'm unable to prove even the weaker version of the lemma (only ...
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What is the intuitive notion that ZF-Extensionality-Foundation+Collection can be said to capture? [closed]
This question has been moved to philosophy.stackexchange.com
I'll try to abbreviate it here: the question asks about the "informal notion" that the fragment of $\text{ZFC}$ that is axiomatized by ...
- 11.3k
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Can only the constructible sets be proven to exist in $ZF$ without benefit of extra assumptions? [closed]
I am interested in asking the following question:
What sets can be proven to exist in $ZF$ without the benefit of extra assumptions? (Thanks to Toshiyasu Arai for inspiring me to ask this variation ...
- 6,132
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Is the notion of measurable cardinal definable from the perspective of set-theoretical potentialism?
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 $\kappa$ ...
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