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37 votes
Accepted

How to add essentially new knots to the universe?

Yes, forcing can add fundamentally new knots, not equivalent to any ground model knot. Indeed, whenever you extend the set-theoretic universe to add new reals, then you must also have added ...
35 votes
Accepted

What is the dimension of the mathematical universe?

My co-authors and I introduced a notion of dimension for forcing extensions in the following paper: Hamkins, Joel David; Leibman, George; Löwe, Benedikt, Structural connections between a forcing ...
32 votes
Accepted

A better way to explain forcing?

I have proposed such an axiomatization. It is published in Comptes Rendus: Mathématique, which has returned to the Académie des Sciences in 2020 and is now completely open access. Here is a link: ...
31 votes

A better way to explain forcing?

Great Question! Finally someone asks the simplest questions, which almost invariably are the real critical ones (if I cannot explain a great idea to an intelligent person in minutes, it simply means I ...
30 votes

A better way to explain forcing?

This is an expansion of David Roberts's comment. It may not be the sort of answer you thought you were looking for, but I think it is appropriate, among other reasons because it directly addresses ...
  • 60.3k
29 votes

How far wrong could the Continuum Hypothesis be?

Solovay proved shortly after Cohen's result on the independence of CH that in any model of set theory $V$, if $\kappa^\omega=\kappa$, then there is a forcing extension in which $2^\omega=\kappa$. The ...
28 votes
Accepted

Sheaf-theoretic approach to forcing

Yes, this is a model of ETCSR. Unfortunately, I don't know of a proof of this in the literature, which is in general sadly lacking as regards replacement/collection axioms in topos theory. But here'...
  • 60.3k
21 votes
Accepted

What is the modal logic of outer multiverse?

I've noticed that recently you have asked a few questions about my work, and so let me thank you; you are kind to take an interest. This particular question can be seen as part of the subject of set-...
21 votes

Sheaf-theoretic approach to forcing

I think the language of classifying toposes is helpful in understanding this view of forcing. Let $P$ be a poset. The set theorists have the intuition that forcing over $P$ adjoins a generic filter of ...
  • 13.4k
20 votes

Scott-Solovay unpublished paper on ``Boolean valued models of set theory''

Scott was editor of the Oxford logic guides and was involved in the preparation of Set Theory: Boolean-Valued Models and Independence Proofs (Oxford Logic Guides). He wrote a forward to it and in this ...
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. ...
19 votes

Producing finite objects by forcing!

Here is another example. The theorem is: every finite partial order can be found as a suborder of the partial order of the Turing degrees. On the one hand, one can prove this by undertaking a ...
18 votes
Accepted

Why do we need a transitive model in forcing arguments?

Yes, one can undertake forcing without the transitivity assumption, and even the countability of the model is not important. One of the standard ways to do this is with the Boolean-valued model ...
18 votes

A better way to explain forcing?

This answer is quite similar to Rodrigo's but maybe slightly closer to what you want. Suppose $M$ is a countable transitive model of ZFC and $P\in M$. We want to find a process for adding a subset $G$ ...
18 votes

A better way to explain forcing?

I think there are a few things to unpack here. 1. What is the level of commitment from the reader? Are we talking about a casual reader, say someone in number theory, who is just curious about forcing?...
  • 35.6k
18 votes

Why can we assume a ctm of ZFC exists in forcing

Expositionally, forcing is (usually) easier to understand with a c.t.m. This does indeed lead to somewhat different results, such as $(*)\quad$ If there is a countable transitive model of $\mathsf{...
17 votes

When will the real numbers be Borel?

I think that Groszek and Slaman's result (see https://www.jstor.org/stable/421023?seq=1) gives a satisfying answer to your question. Groszek and Slaman's result says that given any inner model $M$ ...
  • 3,781
17 votes

Sheaf-theoretic approach to forcing

Thanks for all the enlightening answers! Let me summarize my understanding now. (Please correct me if I'm saying something stupid!) First, as explained by Mike Shulman in his answer, the answer to ...
16 votes

Connections between Complexity Theory & Set Theory

See Diagonalizations over polynomial time computable sets in which two types of genericity introduced with which it examines complexity properties provable by simple diagonalizations over $P$. See ...
16 votes

Forcings predicted by core model theory $+$$ZFC$ results proved by the method of core model theory

Question 1: A good example is Woodin's extender algebra. One reference describing the discovery of the extender algebra is the introduction to Neeman's book The Determinacy of Long Games. I am ...
16 votes
Accepted

Locales as spaces of ideal/imaginary points

I can only answer some of your questions. Yes, the Zariski locale is extensively studied. It's one of the ways of setting up scheme theory in a constructive context: Don't define schemes as locally ...
15 votes
Accepted

Who needs RCS iterations?

I think that this claim is false, at least in the way that I understand "countable support". The following was explained to me by Menachem Magidor in the 1990s; it may be folklore, and I suspect it ...
  • 13.7k
15 votes
Accepted

Is the Martin's axiom number $\mathfrak m$ regular

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, ...
15 votes
Accepted

What are examples of non-equivalent virtualizations of a large cardinal?

An important feature which separates the notion of virtual large cardinals from the related notion of generic large cardinals is that we only consider embeddings on set-sized structures. Since most ...
15 votes

Philosophy of forcing and ctm

It's not clear to me exactly what your question is. Certainly nobody literally thinks that if a statement can be forced then it is probably true, since we can force mutually contradictory statements. ...
  • 67.1k
14 votes

Adding a real with infinite conditions

While Prikry-Silver forcing satisfies Axiom A and hence does not collapse $\omega_{1}$, it is independent of $MA + \neg CH$ whether Prikry-Silver forcing preserves the continuum. Under $MA + \neg CH$, ...
  • 8,218
14 votes
Accepted

Does stationary reflection imply Mahloness?

Not necessarily. Here's a counterexample, but I'm sure it is a ridiculous overkill in consistency strength: Suppose $\kappa$ is the least inaccessible limit of supercompact cardinals. Then $\kappa$ ...
  • 16.6k
14 votes

Specific notions of forcing from the point of view of category theory

The topos version of forcing with a poset $P$ regards $P$ as a category, forms the topos of presheaves on it, and then passes to the subtopos of double-negation sheaves. The presheaf topos amounts to ...
14 votes
Accepted

Approximating a real in the ground model

The answer is no. Here is a counterexample: For definiteness, let's work with $\mathbb P$ equal to Sacks forcing, though the proof works verbatim for any reasonable forcing whose generic can be ...
14 votes

Non-set-theoretic consequences of forcing axioms

Indeed there is a vast of applications, for example: Using Martin's axiom, Shelah showed that there is a non-free Whitehead group. The book `` Consequences of Martin's Axiom'' contains many other ...

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