Questions tagged [multiverse-of-sets]

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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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15 votes
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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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11 votes
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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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24 votes
1 answer
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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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36 votes
7 answers
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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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22 votes
4 answers
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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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2 answers
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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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3 votes
3 answers
480 views

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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9 votes
2 answers
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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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61 votes
3 answers
5k views

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 ...
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8 votes
2 answers
594 views

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-...
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10 votes
1 answer
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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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7 votes
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Maps between forcing posets

We all know that forcing can be seen (if you like things that way) as a category of sheaves over the poset of forcing conditions equipped with the double negation Grothendieck topology. As such it is ...
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34 votes
3 answers
3k views

The set-theoretic multiverse as a (bi)category

Joel Hamkin's The set-theoretic multiverse has featured in MO questions before, e.g., here and here. But I was wondering about the best category theoretic angle to take on it. In the paper Joel ...
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18 votes
2 answers
3k views

Universe view vs. Multiverse view of Set Theory

Here I refer to Hamkins' slides: http://lumiere.ens.fr/~dbonnay/files/talks/hamkins.pdf particularly, to the "Universe view simulated inside Multiverse", p. 22. My question is: is it very unsound ...
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9 votes
2 answers
1k views

Using the multiverse approach to decide the law of the exluded middle?

Recently, in response to deciding the Continuum Hypothesis $CH$, Hamkins and Gitman have proposed consider a multiverse of set-theoretic universes, some in which $CH$ is true, some in which $\neg CH$ ...
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