# Tag Info

## Hot answers tagged independence-results

37 votes
Accepted

### How undecidable is the spectral gap?

I haven't read the paper carefully, but this appears to be a standard undecidability result, of the sort of which there are dozens if not hundreds in the literature, of the same ilk as the ...
• 6,965
28 votes

### Should axiomatic set theory be translated into graph theory?

Although it may seem on the face of it that this proposal is just a question of terminology — yes, a model of set theory is a certain kind of acyclic digraph — nevertheless, my opinion is ...
25 votes

### In the two-person Killing the Hydra game, what is the winning strategy?

We can think of this as a game of "omega-nim;" to more precise since the game you are describing is impartial, operating under the normal play convention, and finite we have that the Sprague-Grundy ...
20 votes

### How undecidable is the spectral gap?

I interpret your question to be asking about the transition from computable undecidability to Gödelian or logical undecidability, and furthermore about the extent to which this logical ...
18 votes
Accepted

### Bidi: A new cardinal characteristic of the continuum?

Using a characterisation of $\min\{\mathfrak{d},\mathfrak{r}\}$ that comes from dualizing a result in Kamburelis' and Węglorz's paper called "Splittings": A. Kamburelis, B. Węglorz, Splittings, ...
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17 votes

• 21.5k
8 votes
Accepted

### Variants of reflection principle

The answer is "yes". The principle you call $\text{RP}^*$ is called "reflection to internally club sets" in the literature; as far as I know this terminology first appeared in Foreman-Todorcevic's ...
• 2,221
8 votes

### How undecidable is the spectral gap?

Does it represent a new way of proving independence results compared to forcing, etc.? In other words, is it an advance on Gödel sentences and the continuum hypothesis? No one's quite said it yet, so:...
• 107k
7 votes
Accepted

### Relationship between fragments of the axiom of choice and the dependent choice principles

The idea is to mimic the permutation models as given in Jech. One can then ask, "Well, in Jech he chooses some set of objects in the full universe, and shows it has a support. But in forcing we ...
• 35.4k
7 votes

### Examples of independent $\Sigma_4^1$ statements

As a starting point, think about the sentence "There is a nonconstructible real." This is $\Sigma^1_3$ and clearly not downwards-absolute. However, it is upwards-absolute. To get the desired ...
• 21.5k
7 votes
Accepted

• 6,940
6 votes

### Given a cardinal k, what's the biggest dense linear order with a dense subset of size k?

This is very late, but in 2012, Artem Chernikov, Itay Kaplan and Saharon Shelah posted on arXiv a paper titled "On non-forking spectra" ( http://arxiv.org/abs/1205.3101 ). They claim that it is ...
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6 votes
Accepted

### A topologically transitive dynamical system without dense orbits

The answer is yes: there is a topologically transitive dynamical system without dense orbits. Indeed, let X be a topological space that is not separable. Let $\ K=X^{\Bbb Z},\$ and let $\ G\$ be ...
• 3,439
6 votes
Accepted

Unfortunately, $\mathfrak r_{(2^\omega,f)}\ge\mathfrak r$. Indeed, let $\mathcal R$ be a family of infinite subsets of $\omega$ such that $|\mathcal R|=\mathfrak r_{(2^\omega,f)}$ and for any $x=(x_n)... • 2,717 5 votes Accepted ###$\omega_2$-sequence of Suslin trees The answer is yes, and indeed, one can even have that any countable number of the Suslin trees join to a Suslin tree. To see this, simply force with countable support to add$\omega_2$many Suslin ... 5 votes ### How undecidable is the spectral gap? I'm no expert, but the short version of the paper http://arxiv.org/pdf/1502.04135v1.pdf seems to address these questions in the conclusion: But what does it mean for a physical property to be ... • 8,968 5 votes ### What are some reasonable-sounding statements that are independent of ZFC? https://www.scottaaronson.com/blog/?p=2725 Busy Beaver$8000$is independent of ZFC. I think this is in the same spirit of this question. 5 votes Accepted ### Is$\mathfrak p=\omega_1$equivalent to the existence of a Hausdorff gap without infinite pseudointersection? Yes. This is a result due to Nyikos and Vaughan from 1983, appearing the paper Nyikos, Peter J.; Vaughan, Jerry E., On first countable, countably compact spaces. I: ((\omega_ 1,\omega^*_ 1))-gaps, ... • 6,940 5 votes Accepted ### Implications of the existence of a pair of surjective functions, without Axiom of Choice No, and here is a counterexample. Suppose that$|\Bbb R|<|[\Bbb R]^\omega|$, that is, there are more countable subsets of reals than reals. This is indeed possible, e.g. if all sets of Lebesgue ... • 35.4k 5 votes ### Dedekind-"finiteness" for arbitrary limit cardinals Yes, it is possible; one can see this by a variant of a standard proof of the consistency of ZF +$\neg$AC, as witnessed in symmetric submodels of forcing extensions. Start with universe$V=L$. Let$\...
• 5,044

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