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 ...

- 198k

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 ...

- 198k

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:
...

- 2,167

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 ...

- 7,281

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 ...

- 198k

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-...

- 198k

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### Scott-Solovay unpublished paper on ``Boolean valued models of set theory''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

- 6,307

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.
...

- 198k

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 ...

- 198k

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 ...

- 198k

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 $G$ ...

- 5,426

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{...

- 20.9k

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 ...

- 14.7k

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 ...

- 30.5k

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 ...

- 5,426

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 ...

- 2,769

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 ...

- 2,260

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 ...

- 68.9k

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 ...

- 1,722

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 ...

- 30.5k

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

forcing × 790set-theory × 740

lo.logic × 404

large-cardinals × 139

reference-request × 104

model-theory × 39

descriptive-set-theory × 35

axiom-of-choice × 33

infinite-combinatorics × 32

gn.general-topology × 26

boolean-algebras × 24

continuum-hypothesis × 21

ultrafilters × 19

inner-model-theory × 19

independence-results × 16

ct.category-theory × 15

computability-theory × 14

ho.history-overview × 11

posets × 11

measure-theory × 10

determinacy × 9

cardinal-characteristics × 9

cardinal-arithmetic × 9

inner-models × 8

soft-question × 7