**7**

votes

**1**answer

352 views

### Demuth's theorem in set theory

I am quite sure the following fact must have been known for set theorists, though I could not find it anywhere.
If $r$ is random over $L$ and $x\in L[r]\setminus L$, then there must be some real ...

**10**

votes

**0**answers

290 views

### Short proof of $\frak p=t$

It is known for a while now that $\frak p=t$, as a result of Malliaris-Shelah. The original paper draws from model theoretic methods.
I've heard rumors that there was a proof which was purely set ...

**5**

votes

**2**answers

818 views

### Why can't mathematics be formalised in terms of classes rather than sets? [closed]

I've always been curious about the seeming compulsion to found mathematics upon sets, be it ZF(C) or some other system. Of course, there are other approaches these days like category theory and type ...

**1**

vote

**1**answer

145 views

### Possible no standard use of replacement axiom

The idea is to build in ZFC using replacement, a set REPLACEMENT(x ∈ A: TERM(x))
from a set and a term in the same way the set {x ∈ A: FORMULA(x)} is built
using specification from a set and a ...

**1**

vote

**1**answer

251 views

### Is there an analogue of Shoenfield's absoluteness theorem, but for $\mathrm{On}$?

From wikipedia:
Shoenfield's absoluteness theorem shows that $\Pi^1_2$ and $\Sigma^1_2$
sentences in the analytical hierarchy are absolute between a model $V$
of ZF and the constructible ...

**3**

votes

**1**answer

561 views

### Is there a “large powerset axiom” so extreme that it disproves the existence of strongly inaccessible cardinals?

If $\kappa$ is a strongly compact cardinal, then the singular cardinal hypothesis holds above $\kappa$. Hence the existence of large cardinals at the level of "strongly compact" or above is ...

**21**

votes

**1**answer

590 views

### Is there a model of ZF set theory with a set that does not inject into the cardinals?

Question. Is there is a model of ZF set theory with a set $X$ that does not inject into the cardinals?
I use the term "cardinal" here in the ZF sense, so they are not necessarily well-orderable.
To ...

**5**

votes

**1**answer

311 views

### A categorical characterization of ordinal numbers

It's rather easy to notice that the operation of join of categories reproduces the ordinal sum once restricted to act on (iso classes of) well-ordered set; it's rather easy to see that $\alpha\star ...

**1**

vote

**1**answer

222 views

### A conjecture about certain relations

In my research the following problem appeared (and if it is true, this solves positively several my conjectures):
Let $U$ be a fixed set (usually $U$ is infinite). Let $n$ be a fixed index set ...

**2**

votes

**1**answer

147 views

### Intermediate Extensions Determined by Reals

Does there exist a forcing $P$ which adds a generic real in the sense that $V[G] = V[x]$ for some $x \in ({}^\omega\omega)^{V[G]}$, and for all reals $y \in ({}^\omega\omega)^{V[G]}$, if $V[y] \neq ...

**6**

votes

**2**answers

225 views

### Only admissibles start gaps in clockable ordinals

This is a question about ITTM model introduced by Hamkins et al. In this paper it is proven that no admissible ordinal is clockable, so it either starts or lies within a gap in clockable ordinals. I ...

**3**

votes

**2**answers

334 views

### Is it consistent with ZFC that $\mathrm{dv}(\kappa) = \kappa$ for all infinite cardinal numbers $\kappa$?

Whenever $\kappa$ is an infinite cardinal number, write $L(\kappa)$ for the powerset of $\kappa$ ordered lexicographically. (Where the "$L$" stands for linear order.) Furthermore, write $B(\kappa)$ ...

**7**

votes

**1**answer

367 views

### Existence of infinite groups that are too reluctant to be topological

With ZFC, is there an infinite group $G$ such that there is no non-trivial non-discrete topology on $G$ with the functions $G\times G\to G,~~ (a,b) \mapsto ab$ and $G\to G,~~ a\mapsto a^{-1}$ ...

**2**

votes

**1**answer

221 views

### Absolutness of $\Pi_1^1$ statements

Shoenfield absoluteness is well known for $\Pi_2^1$-statements, but it does not hold between a countable transitive model of ZFC and the universe.
But it is also known that $\Pi_1^1$ statements are ...

**21**

votes

**2**answers

1k views

### Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,…)

Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many ...

**5**

votes

**0**answers

350 views

### Last Status of Feferman's Conjecture on Indefinite Value of Continuum

The "true" value of $2^{\aleph_0}$ is one of the most fundamental open questions of mathematics and its philosophy. Hundreds of set theoretic results during the last century don't say anything more ...

**5**

votes

**1**answer

272 views

### Ultraproduct of Forcing Extensions & Forcing Extension of Ultraproduct

Notation:
$M[{\mathbb{P}}:G]$ denotes the forcing extension of $M$ by $\mathbb{P}$-generic filter $G$.
$\prod_{\mathcal{F}}\langle M_i~|~i\in I\rangle$ denotes the ultraproduct of models using the ...

**5**

votes

**2**answers

168 views

### absorption of strategically closed posets

It is a (folklore?) fact that if $\kappa$ is a regular cardinal, and $\mathbb{P}$ is a $\kappa$-closed poset such that $\Vdash_\mathbb{P} |\mathbb{P}| = \kappa$, then $\mathbb{P}$ is equivalent to ...

**2**

votes

**1**answer

249 views

### Consistency of: “The continuum function is injective, and for all infinite cardinals $\kappa$ we have that $2^\kappa$ is weakly inaccessible.”

I asked here about "large powerset axioms" and to my delight, learned that such axioms are being taken seriously. I've been toying with them ever since. My favourite is: "The continuum function is ...

**5**

votes

**1**answer

184 views

### Scott sentence in models of set theory

Let $\mathfrak{M}$ be a countable transitive model of set theory.
Let $L$ be some countable language and $A$ be a countable (in $\mathfrak{M}$) $L$-structure.
My question is:
In $\mathfrak{M}$ can ...

**4**

votes

**1**answer

122 views

### Class theory with support for self-application of class functions?

To every natural number $n$, we can assign its Church numeral $\underline{n}.$ A formal definition would be:
$\underline{0}(f)=\mathrm{id}_{\mathrm{dom}(f)}$
$\underline{n+1}(f) = ...

**19**

votes

**1**answer

630 views

### Why isn't this a computable description of the ordinal of ZF?

In a previous MO question, I was told by several commenters that
(a) it's known that there exists a computable ordinal $\alpha_{ZF}$ that "encodes the strength of ZF set theory" (i.e., a least ...

**7**

votes

**1**answer

153 views

### Preservation of ultrafilters by Sacks forcing

It is well-known that, if $p$ is a Ramsey (selective) ultrafilter on $\omega$, then after adding a Sacks real $p$ remains an ultrafilter (well, it's really the upwards closure of $p$ the one that's an ...

**6**

votes

**0**answers

173 views

### Foundation scheme for $\Sigma_{n+1}$-formulas

I have trouble working out a proof in the second part of
Jean-Pierre Ressayre and Alex Wilkie. Modèles non standard en arithmétique et théorie des ensembles. Publications ...

**10**

votes

**0**answers

185 views

### Namba forcing and semiproperness

This question is the result of leaving "Proper and Improper Forcing" on my nightstand by accident.
Is the statement "Namba forcing is semiproper" known to be equiconsistent with some more standard ...

**5**

votes

**2**answers

449 views

### A question about “small” uncountable cardinal numbers

If $X$ denotes a set, let $C(X)$ denote its cardinal number and let $P(X)$ denote its power set. There is a school of thought which considers any set having the cardinal number ...

**5**

votes

**2**answers

260 views

### For which cardinal numbers $\kappa$ is it consistent with ZFC that $\kappa^{\mathrm{cf}(\kappa)} < \kappa^\kappa$?

ZFC proves that $\kappa^{\mathrm{cf}(\kappa)} \leq \kappa^\kappa$ for all infinite cardinal numbers $\kappa$. Further, it is consistent with ZFC that we always have equality (e.g. assume GCH).
...

**0**

votes

**1**answer

81 views

### Question about the consistency of assuming (via axiom) that $\kappa < \nu$ for certain pairs of cardinal numbers provably satisfying $\kappa \leq \nu$

Call an ordered pair of formulae $\langle P(\kappa,\tilde{a}), Q(\nu,\tilde{a})\rangle$ in the language of $\{\in\}$ unproblematic iff
ZFC proves that for all $\tilde{a}$ and all cardinal numbers ...

**4**

votes

**1**answer

139 views

### Absoluteness between $L_\kappa$ and $L$

Working in $L$, suppose $L \models \kappa$ is a cardinal and $(\mathbb{P}, <) \in L_\kappa$. Let $\varphi(x)$ be a $\Sigma_1^1$ formula. Let $\tau \in L_\kappa$ be a $\mathbb{P}$-name for an ...

**5**

votes

**3**answers

707 views

### Why isn't there more interest in “large powerset axioms”?

By a large powerset axiom, let us mean informally an axiom that says that for some cardinal numbers $\kappa$, we have that $2^\kappa$ is somehow "large" or "difficult to access from below." For ...

**2**

votes

**1**answer

164 views

### Some random questions about forcing

Are there more general forms of forcing, in any of the following senses?
1) The forcing adds new ordinals to $M[G]$.
2) The forcing is developed on a less or more restrictive form of $\mathbb{P}$ ...

**4**

votes

**1**answer

175 views

### Can a Measureable Cardinal Become the Least Weakly Compact Cardinal in a Forcing Extension?

I am trying to establish whether it is consistent that some property holds at the least weakly compact cardinal. I know that the property holds at measureables.
Hence (hoping everything else goes ...

**0**

votes

**1**answer

270 views

### On the consistency of ZFC (and ZF)

Let ZF be the Zermelo-Fraenkel set theory, ZFC be ZF with choice, con(ZF) be the consistency of ZF and con(ZFC) be the consistency of ZFC. Let IN be the hypothesis "There exists one (strongly) ...

**6**

votes

**1**answer

216 views

### Can we have more malleable proper classes without sacrificing conservativity?

NBG is a conservative extension of ZFC that includes a concept of "proper class." Now I like the conservativity, since it means anytime I want to prove something in ZFC, I am free to work in NBG. ...

**10**

votes

**1**answer

327 views

### Does small forcing preserve CH?

Suppose CH holds and $\mathbb{P}$ is a poset of size $\omega_1$, such that forcing with $\mathbb{P}$ preserves $\omega_1$. Does forcing with $\mathbb{P}$ preserve CH? If $\mathbb{P}$ is proper then ...

**2**

votes

**0**answers

84 views

### Do there exist projective realcompact covers?

In 1958 Gleason [1] constructed projective covers in the category of compact Hausdorff spaces. These may be characterized in many ways. One description that is most interesting to me: $p:EX\to X$ is a ...

**2**

votes

**3**answers

570 views

### The Hidden Aspect of Set Theory [closed]

This question is inspired by a similar question at the beginning of Kunen's new book, "Set Theory".
Many mathematicians believe they are exploring a "real" universe. In such a Platonic point of ...

**14**

votes

**6**answers

2k views

### What “forces” us to accept large cardinal axioms?

Large cardinal axioms are not provable using usual mathematical tools (developed in $\text{ZFC}$).
Their non-existence is consistent with axioms of usual mathematics.
It is provable that some of ...

**0**

votes

**1**answer

124 views

### A question about sentences in the language of first order ZFC which assert the existence of cardinal numbers

These sentences are usually of two kinds. The first kind are actually theorems of ZFC asserting the existence of various cardinal numbers, and their negations are inconsistent with ZFC. The second ...

**3**

votes

**2**answers

212 views

### Sufficient Condition for Defining $\in$

Consider the first order language $\mathcal{L}=\{\in,\in'\}$ with two binary relational symbols $\in , \in'$ and $ZFC$ as a $\{\in\}$-theory. If we define $\in'$ using $\{\in\}$-formula $\varphi(x,y)$ ...

**1**

vote

**1**answer

220 views

### A Special Pair of Formulas

Consider the first order language $\mathcal{L}=\{\in,\subseteq\}$ and $\{\in\}$-theory $\text{ZFC}$. Is there a formula $\psi (x,y) \in \{\subseteq\}-Form$ with the following ...

**27**

votes

**1**answer

816 views

### Is there a computable ordinal encoding the proof strength of ZF? Is it knowable?

In comments on Quora (see, for example, here, here, here), Ron Maimon has repeatedly expressed the strong opinion that Hilbert's program was not killed by Gödel's results in the way typically ...

**10**

votes

**2**answers

595 views

### Questions about Prikry forcing and Cohen forcing

I have some questions.
The first one is about the product of Prikry's forcing.
Let $\kappa$ be a measurable cardinal, $U_1, U_2$ be normal measures on $\kappa$ and let $\mathbb{P}_{U_1}, ...

**-1**

votes

**1**answer

113 views

### ($^{\omega}2$,<) is not well-order. [closed]

Let < be a lexicographic order on $^{\omega}2$ or in other words given distinct functions $f,g$ from $\omega$ to 2, let $f<g$ if and only if $f(n)=0$ and $g(n)=1$, where $n$ is the lease ...

**6**

votes

**1**answer

217 views

### Sets of cardinalities of bases without choice

For a vector space $V$, let $BS(V)$ be the set of cardinalities (not necessarily $\aleph$s) of bases of $V$. Of course, in ZFC each $BS(V)$ is a singleton, but supposing the axiom of choice fails, it ...

**9**

votes

**1**answer

292 views

### Inner model in which every uncountable cardinal is large

The following is known:
$(*)$ If $0^\sharp$ exists, then any uncountable cardinal is is an inaccessible cardinal (and even more) in $L$.
My question is that:
Are there any large cardinal ...

**2**

votes

**2**answers

458 views

### Have axioms / axiom schemata of this flavour been proposed or otherwise considered?

With the exception of a few miscellaneous cases, the axioms (and/or schemeta) of ZFC can roughly be divided into two kinds:
Those that guarantee the existence of more complicated sets, given that ...

**10**

votes

**2**answers

238 views

### Sets that are not $\infty$-Borel

I have seen a few techinques for proving that certain sets of real numbers are $\infty$-Borel (definition) but it just occurred to me that I don't know of any way to prove that a set of real numbers ...

**3**

votes

**1**answer

269 views

### Suslin lines hereditarily Lindelof

I need to prove that every suslin line is hereditarily Lindelof. Any idea will be helpfull.

**3**

votes

**1**answer

250 views

### What year was Hechler forcing created?

Hechler forcing is described on page 278, Jech.
Does anyone know when Hechler forcing was first used in a publication?