Questions tagged [multiverse-of-sets]
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19 questions
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Multiverse view of set theory and finitary statements
I think almost all mathematicians would agree that the finitary statements (like those expressed in PRA) have defnite realist objective truth values. E.g. either there is a finite string of bits (an ...
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Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
To be clear, I am not a mathematics educated student and I can not follow the details of the technicality of the forcing extension, but I feel that I have a good understanding of the big picture (of ...
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Omega logic, V = Ultimate L and hierarchy of laws collapse
I would like to ask a question about the omega conjecture and its relationship with the V = Ultimate L axiom.
In his lecture Prof. Hugh Woodin have stated that "Assuming the omega conjecture, ...
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An axiomatic approach to the multiverse of sets
Work in a theory where the primitives are classes $X,Y,Z,\dots$, and class membership $X\in Y$, and add an individual constant $\mathcal{M}$ called 'the multiverse'. Classes $V$ which are members of ...
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Toposophy vs Set theoretical multiverse philosophy
Johnstones classic topos theory book talks at some length in its introduction about how category theory/topos theory suggest that we view the 'universe' in which mathematics takes place as consisting ...
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Set-theoretical multiverses and their representation as functors? Why *the* multiverse?
In some related MO questions like The set-theoretic multiverse as a (bi)category it is discussed how one might represent the multiverse (see The set-theoretic multiverse) in a category theoretic way, ...
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Forcing and Family Contentions: Who wins the disputes?
The famous game-theoretic couple, Alice & Bob, live in the set-theoretic universe, $V$, a model of $ZFC$. Just like many other couples they sometimes argue over a statement, $\sigma$, expressible ...
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Is V, the Universe of Sets, a fixed object?
When I first read Set Theory by Jech, I came under the impression that the Universe of Sets, $V$ was a fixed, well defined object like $\pi$ or the Klein four group. However as I have read on, I am ...
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Set theoretical multiverse and truths
In J.D. Hamkins' multiverse view of set theory, every universe has an ill-founded $\mathbb{N}$ from the perspective of another universe. Does this mean that every proof in our universe can be seen as ...
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Is platonism regarding arithmetic consistent with the multiverse view in set theory?
A "truth" platonist for arithmetic believes, given a statement in the language of arithmetic, that the problem whether the statement is true has a definite answer.
Prof. Hamkins has argued for a ...
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The Set-Theoretic Multiverse and Joint Embeddings
I am curious whether or not the following axiom is independent of Hamkins's axioms for the Set-Theoretic Multiverse. Hamkins's axioms can be found here on pages 1-2 and here on pages 24-26.
Consider ...
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Is the Mostowski collapse natural?
The Mostowski collapse lemma (see here for a quick ref) is one of the key basic tools in the set-theory arsenal. I wonder if the collapse is natural, in the functorial sense.
More precisely, is ...
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Forcing as a new chapter of Galois Theory?
There is a (very) long essay by Grothendieck with the ominous title La Longue Marche à travers la théorie de Galois (The Long March through Galois Theory). As usual, Grothendieck knew what he was ...
- 8,058
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Does the class category of ZF-algebras satisfy the Multiverse axioms?
I read with interest both Hamkins Multiverse Axioms and Joyal and Moerdijk's algebraic set theory. Both of these perspectives takes a set-theory universe as an object, and consider collections of set-...
user2529
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Set-theoretical multiverse and foundations
I just had a look to the article The set theoretical multiverse by (mo user) J.D.Hamkins. Not being a logician and not knowing forcing techniques, I couldn't fully appreciate the mathematical ideas, ...
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