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120 votes
Accepted

Czelakowski's claimed proof of the Twin Prime Conjecture

The error in the paper is in the proof of Theorem 7.2. The proof of Theorem 7.2 is immediately suspicious because of how vague it is in places and because of how lofty the expository text before and ...
James E Hanson's user avatar
65 votes

Czelakowski's claimed proof of the Twin Prime Conjecture

The Editor-in-Chief has released a statement on behalf of the journal retracting the papers, as follows. Public announcement Recently two articles on the applications of Rasiowa-Sikorski Lemma to ...
57 votes

Czelakowski's claimed proof of the Twin Prime Conjecture

From my reading, the only facts about the concept of twin primes used in the argument are that there exist pairs of numbers $n$ and $n+2$ (in the discussion after equation (8.4)) and there exist ...
Will Sawin's user avatar
  • 138k
38 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 ...
Joel David Hamkins's user avatar
37 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 ...
Mike Shulman's user avatar
  • 65.4k
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 ...
Joel David Hamkins's user avatar
33 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: ...
Rodrigo Freire's user avatar
33 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 ...
Mirco A. Mannucci's user avatar
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'...
Mike Shulman's user avatar
  • 65.4k
23 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-...
Joel David Hamkins's user avatar
22 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 ...
Zhen Lin's user avatar
  • 14.9k
20 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 ...
Joel David Hamkins's user avatar
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. ...
Joel David Hamkins's user avatar
20 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 $G$ ...
Gabe Goldberg's user avatar
19 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?...
Asaf Karagila's user avatar
  • 37.9k
18 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 ...
Peter Scholze's user avatar
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{...
Noah Schweber's user avatar
18 votes
Accepted

Changing the cofinality of a regular cardinal without collapsing any cardinals?

The possibility of changing the cofinality of a regular cardinal without collapsing any cardinals is equiconsistent with a measurable cardinal: On one hand, if $\kappa$ is measurable, then by Prikry ...
Hannes Jakob's user avatar
  • 1,622
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 $M$ ...
喻 良's user avatar
  • 4,151
17 votes

May two Cohen reals collapse cardinals?

Here's a slightly different example. This example has the additional benefit that the analogous argument using measure shows that two random reals can also collapse cardinals. By Solovay's ...
Jason Zesheng Chen's user avatar
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 ...
Mohammad Golshani's user avatar
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 ...
Gabe Goldberg's user avatar
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 ...
Ingo Blechschmidt's user avatar
16 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 ...
Victoria Gitman's user avatar
15 votes

Sheaf-theoretic approach to forcing

I can't really answer your question, since they are outside my field of expertise. But until Mike and others come to answer, let me make a long comment about the following sentence: Note that in ...
Asaf Karagila's user avatar
  • 37.9k
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. ...
Timothy Chow's user avatar
  • 79.3k
15 votes
Accepted

Is there a more modern account of the main results of "Adding Dependent Choice" by D. Pincus?

No. Most of the work of Pincus that involved these sort of iterated constructions, where one "pushes the counterexamples out" has never been reworked into modern terms. In my Ph.D. one of ...
Asaf Karagila's user avatar
  • 37.9k
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 $MA + \neg CH$, ...
Simon Thomas's user avatar
  • 8,228
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 ...
Andreas Blass's user avatar
14 votes
Accepted

How badly can the GCH fail globally?

In the Foreman-Woodin model The generalized continuum hypothesis can fail everywhere. for each infinite cardinal $\kappa, 2^\kappa$ is weakly inaccessible. This answers your last question. The answer ...
Mohammad Golshani's user avatar

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