31
votes
Set theoretical multiverse and truths
Thank you for your interest in my views on the set-theoretic
multiverse.
Yes, indeed, the well-foundedness mirage axiom you mention is
probably the most controversial of my multiverse axioms, and so
...
- 236k
26
votes
Accepted
Is V, the Universe of Sets, a fixed object?
As you noticed, the iterative conception of sets requires a pre-existing universe of sets, and ordinals with which we can label the stages. So if you work within ZFC itself, in other words within an ...
- 2,912
20
votes
Accepted
Forcing and Family Contentions: Who wins the disputes?
I like this question a lot. It provides an interesting way of talking about
some of the ideas connected with the maximality principle and the
modal logic of forcing.
Let me make several observations.
...
- 236k
20
votes
Is V, the Universe of Sets, a fixed object?
This seems like more of a philosophy of math question than a proper math question. However, in the past Mathoverflow has often been tolerant of such questions.
The basic concern is that the universe ...
- 42.8k
15
votes
Set theoretical multiverse and truths
There is a subtle issue here but it is not where the OP thinks it is. Any explicitly written integer is obviously "standard" whereas each new integer arising in the ultrapower of $\mathbb N$ ...
- 16.6k
10
votes
Set theoretical multiverse and truths
I think only Joel can answer this question, but I'd like to point out that there are some subtleties here that the commenters are missing. In Joel's multiverse conception, there is no "standard" $\...
- 42.8k
10
votes
Is V, the Universe of Sets, a fixed object?
First of all, I think that part of the confusion stems from talking about "models of ZFC." My recommendation, if you want to sort out what's going on, is to start by forgetting what a model of ZFC is....
- 82.6k
8
votes
Accepted
Set-theoretical multiverses and their representation as functors? Why *the* multiverse?
Of course we have been investigating a wide variety of
multiverse concepts, and in this sense, yes, we have not just one,
but many, multiverses.
But to be sure, much of this multiverse analysis has ...
- 236k
8
votes
Accepted
Concept of bedrock and mantle in the multiverse view in the philosophy of mathematics
The confusion seems to arise from your statement "the mantle of $V$ which is the smallest ground for $V$." But this not quite right.
In a bottomless model of ZFC, the mantle is not a ground. ...
- 236k
6
votes
Is V, the Universe of Sets, a fixed object?
For completeness, it should be mentioned that yet another answer could have been: yet, in some foundational approaches, $V$ literally is an object (albeit not a "fixed" one, whatever that means), ...
- 6,051
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 ...
- 6,132
4
votes
An axiomatic approach to the multiverse of sets
This might not precisely answer your question, but why not use topos theory? The category of (Grothendieck) topoi has as its objects generalized universes of sets whose internal logic does no not ...
2
votes
Is V, the Universe of Sets, a fixed object?
I come a little late, just to clearly state and sum up something that was began to be outlined by some here.
Namely, the syntacticalist view I have that nothing in math or formal sciences in general ...
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 ...
- 6,132
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