**3**

votes

**0**answers

111 views

### Silver's unpublished work on reverse Easton iteration

Silver was the first person who used the method of reverse Easton iterations in connection with large cardinals, and used it to force the failure of $GCH$ at some measurable cardinal.
At most papers ...

**6**

votes

**1**answer

259 views

### A “good scale” that is not really a scale

I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.
Let $\lambda$ be a ...

**-10**

votes

**0**answers

283 views

### Who know about Rumek proof [on hold]

Rumek has actually shown that the usual ZFC axioms
for mathematics are in fact
inconsistent. For example, Rumek has been able to prove from the ZFC
axioms that both 2+2 = 4 and 2+2 = 5.
...

**21**

votes

**3**answers

473 views

### Is fixed point property for posets preserved by products?

Recall that a Partially Ordered Set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorm : Suppose $P$ and $Q$ are posets ...

**6**

votes

**1**answer

74 views

### $RUCar^{V}$-semiproperness implies properness

This is a claim in Shelah's Proper and Improper Forcing, more specifically Claim 2.3(1) of Chapter X (p. 484). The proof of the claim is "Easy" but I cannot quite figure it out. There must be some ...

**14**

votes

**1**answer

294 views

### Three old questions on the Sacks forcing

I came across the two following Qs in 1970.
Find reals $a,b$ such that $a$ is Sacks over $L[b]$ and vice versa $b$ is Sacks over $L[a]$. Note that a Sacks $\times$ Sacks generic pair definitely does ...

**10**

votes

**1**answer

307 views

### Erdős cardinals and ineffable cardinals

In Cantor's Attic it is stated that an $\omega$-Erdős cardinal is a stationary limit of ineffable cardinals, and Jech book is given as a reference, but I cannot find this result in that book. I have ...

**8**

votes

**0**answers

176 views

### Can Suslin lines ever be orderings of abelian groups?

I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of ...

**19**

votes

**2**answers

661 views

### construction of nonmeasurable sets

I have a history question for which I've had trouble finding a good answer.
The common story about nonmeasurable sets is that Vitali showed that one existed using the Axiom of Choice, and Lebesgue et ...

**5**

votes

**0**answers

95 views

### A variant of Chang's model with choice

Let $M_n$, $n < \omega$, be a models of $ZFC$ with the same ordinals, closed under countable sequences. Let $\alpha_n$ be an ordinal which is a regular cardinal in $M_n$.
Question: Is it possible ...

**-4**

votes

**0**answers

58 views

### Paritial order help? [closed]

Let (A,≼A) and (B,≼B) be partially ordered sets. Define C = A×B and define the relation ≼' on C by (a,b)≼'(a′,b′) ⇐⇒ (a≼A a′)∧(b≼B b′).
(a) Prove that ≼' is a partial order on C.
$[(a,b) ...

**9**

votes

**2**answers

186 views

### What is the maximal number of distinct values of the product of n permuted ordinals

Because addition and multiplication of two order types are non-commutative operations, we have that for every integer n, given n ordinals, there are at most n! distinct possible values for the sum (or ...

**5**

votes

**2**answers

306 views

### Exponentiation and Dedekind-finite cardinals

It is known that the sum and the product of two Dedekind-finite cardinals are also Dedekind-finite cardinals. What about cardinal exponentiation ?
Question: Let A and B be two Dedekind-finite ...

**8**

votes

**0**answers

124 views

### Which forcing types preserve the axiom of determinacy?

Do we have some rudimentary understanding of some properties that a forcing can have in order to guarantee that it doesn't violate the axiom of determinacy?
To be more specific, in Which forcings ...

**5**

votes

**1**answer

208 views

### Which axioms of ZF are used for finite choice?

Apologies if this is a silly question, not an expert in set theory but just wondering about it.
ZF implies finite choice. But let's suppose one wanted to work without it. The thinking here is being ...

**4**

votes

**0**answers

114 views

### Ultracoproducts of C(X)-algebras

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**6**

votes

**1**answer

136 views

### Slim Kurepa tree at a singular strong limit cardinal of uncountable cofinality

For a strong limit cardinal $\kappa$ the notion of $\kappa$-Kurepa tree is trivial: the full binary tree is a $\kappa$-Kurepa tree. Accordingly, we consider the following strengthening:
A slim ...

**7**

votes

**0**answers

189 views

### A question about cardinal numbers when the Axiom of Choice is absent

The Axiom of Choice constrains every set of cardinal numbers which is linearly ordered by size to be well-ordered. By contrast, does ZF-without the Axiom of Choice (but with the Axiom of ...

**10**

votes

**0**answers

204 views

### Is there a “hereditary” construction for $L$?

Recall that $L$, Godel's constructible universe is constructed by defining the following hierarchy:
$L_0=\varnothing$, for a limit ordinal $\delta$, $L_\delta=\bigcup_{\alpha<\delta}L_\alpha$, and ...

**1**

vote

**1**answer

150 views

### Is any axiom system for sets categorical? [closed]

$ZF$ define membership by conditions demanding the existence of some constructable right-side-terms $M $ ($x \in M$). Is it meaningsful to ask for a categorical axiom system here? Shouldn't it be ...

**10**

votes

**0**answers

257 views

### Ideas behind Gitik's solution of PCF conjecture

Recently Moti Gitik has refuted Shelah's PCF conjecture (see Short extenders forcings II ) by proving the following theorem:
Theorem. Assuming the consistency of infinitely many strong cardinals, one ...

**4**

votes

**4**answers

921 views

### How short can we state the Axiom of Choice?

How short can we state a principle which is equivalent with the Axiom of Choice under $ZF$? The principle should be a sentence in the language of set theory with only $\in$ and$=$ as extralogical ...

**2**

votes

**2**answers

192 views

### Are there fragments of set theory which are axiomatized with only bounded (restricted) quantifiers used in axioms?

Bounded quantifiers in set theory are represented as $(\forall x \in S)$ or $(\exists x \in S)$. But since the modifier "bounded" brings up an association with "bounded variable", I prefer the term ...

**5**

votes

**2**answers

171 views

### Iteration of Proper Forcing and Support of Master Conditions

Suppose $\mathbb{P}$ is a definable proper forcing (for instance Sacks forcing). Let $\alpha$ be some ordinal. Let $\mathbb{P}_\alpha$ be the countable support iteration of $\mathbb{P}$ of length ...

**6**

votes

**1**answer

273 views

### Does OCA imply $2^{\aleph_0}=\aleph_2$?

Is it known whether Todorcevic's Open Coloring Axiom implies $2^{\aleph_0}=\aleph_2$?
The only consistency proofs for OCA that I know are the following:
1) PFA implies OCA (and also ...

**5**

votes

**1**answer

226 views

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

The dependent choice principle ${\rm DC}_\kappa$ states that if $S$ is a nonempty set and $R$ is a binary relation such that for every $s\in S^{\lt\kappa}$, there is $x\in S$ with $sRx$, then there ...

**8**

votes

**2**answers

300 views

### Ultracoproducts and Cartesian products

Let $X$ be a metrizable compact topological space, let $\mathcal U$ be an ultrafilter, and denote by $X^{\mathcal U}$ the ultracopower of $X$ with respect to $\mathcal U$.
As a C$^*$-algebraist, I ...

**0**

votes

**0**answers

82 views

### A question about ordinal numbers and sub-theories of ZF

A number of set theories have been investigated which were obtained from ZF by restricting in various ways, or even deleting, some of the axioms of ZF-such as Power set, Aussonderung, Infinity, ...

**1**

vote

**1**answer

149 views

### Density with infinite cardinals [closed]

Let κ ≤ µ infinite cardinals.
and lat D(µ, κ) = min{|D| : D ⊆ [µ]^κ ∧ (∀y ∈ [µ]^κ)(∃x ∈ D)(x ⊆ y)}
D(µ, κ) is called the density of κ-sets of µ.
1) Suppose κ = cf(µ) < µ. prove that D(µ, κ) > ...

**2**

votes

**2**answers

232 views

### Is not SH + not CH consistent?

I guess that $\lnot$SH + $\lnot$CH is consistent, but I have not found this question discussed anywhere. Is there any relatively simple model of $\lnot$SH + $\lnot$CH?

**7**

votes

**1**answer

234 views

### Inaccessible becomes successor of singular

Is it possible, starting from any large cardinal assumption, to find a countably closed forcing $\mathbb{P}$ such that for some inaccessible $\kappa$, $\Vdash_\mathbb{P} "\kappa = \lambda^+$ and ...

**6**

votes

**1**answer

247 views

### Partitioning $\omega_1$-branching trees of size and height $\omega_1$

Is it possible, in ZFC, to find an $\omega_1$-branching tree $(T,\leq)$ of size and height $\omega_1$ such that whenever $T$ is partitioned into countably many sets $T=\bigcup_{n<\omega} T_n$ one ...

**0**

votes

**0**answers

130 views

### A not defined notion in Friedman's article about Generalized Fubini's Theorem

I intend to study Friedman's article, A Consistent Fubini-Tonelli Theorem for Nonmeasurable Functions (http://projecteuclid.org/download/pdf_1/euclid.ijm/1256047607). I think since I had a modern ...

**5**

votes

**2**answers

264 views

### non-Borel set which intersects every compact in a Borel set

I remember hearing some time ago that there is a locally compact Hausdorff space $X$ and a non-Borel subset $E$ which intersects every compact set in a Borel set. (This would contradict Lemma 13.9 of ...

**2**

votes

**1**answer

189 views

### Measure of the same set in different models of ZF

Let $A$ be a definable subset of $\mathbb{R}$ in $\mathsf{ZF}$, and let $\mathcal{M},\mathcal{N}\models\mathsf{ZF}$ such that $A$ is lebesgue measurable in both models.
Is ...

**6**

votes

**1**answer

162 views

### continuum many mutually generic filters

Given a countable model $M$ of set theory and an atomless, separative partial order $\mathbb{P} \in M$, can we construct (in the real universe) $2^\omega$ many pairwise mutually $\mathbb{P}$-generic ...

**1**

vote

**0**answers

72 views

### Question on the consistency of Zermelo set theory minus specification and extensionality [closed]

Let $W=Z^{-}-Specification$ where $Z^{-}=Z-Extensionality$ and Z is Zermelo set theory. What is known about models of $W$ or $W^{+}=W+Extensionality$?

**5**

votes

**0**answers

103 views

### A question about ordinal definable sets of real numbers revisited [duplicate]

Citing (almost)
A question about ordinal definable real numbers
If ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) is consistent, does it remain consistent when the following statement is ...

**6**

votes

**0**answers

100 views

### Can $Ded(\kappa)$ be a supremum?

Definition If there is a dense linear order w/o endpoints of size $\lambda$ with a dense subset of size $\kappa$ then write $D(\kappa,\lambda)$. $Ded(\kappa)=\sup_\lambda \{D(\kappa,\lambda)\}$.
It ...

**1**

vote

**1**answer

202 views

### Problem of book Kunen [closed]

Suppose $P$ is a notion of forcing in $M$ such that $\left | P \right | \leq \omega_{1}$ and $P$ is ccc. Suppose further $\Diamond$ holds in $M$. How does one show that $\Diamond$ also holds $M[G]$?

**2**

votes

**1**answer

148 views

### The word problem of the free left distributive algebra on one generator

A left distributive algebra is a set $A$ together with a binary operation, $\cdot$, satisfying $a\cdot(b\cdot c)=(a\cdot b)\cdot(a\cdot c)$.
One important example of left distributive algebras arises ...

**2**

votes

**3**answers

271 views

### Cardinality of $C^*([0,1])$ [closed]

What is the cardinality of the continuous dual of $C([0,1])$ (the set of continuous functions from $[0,1]\to \mathbb{R}$)?

**3**

votes

**1**answer

88 views

### Boolean completion (of a forcing notion) isomorphic to each of its cones

Suppose $ \mathbb{P} := (P, {\leq_P}, 1_P) $ is a separative partial order. Let $ \mathbb{B} := \operatorname{RO}(\mathbb{P}) $ denote the Boolean completion.
Fix some dense embedding $ i \colon P ...

**4**

votes

**1**answer

309 views

### Set-theoretic tautologies

Let us consider unquantifed formulas of a set theory (for example, NBG), more precisely,
the formulas, constructed from variables and the constants $\emptyset, V$ (the empty set
and the class of all ...

**4**

votes

**1**answer

175 views

### forcing square with small conditions

In the paper, Large cardinals and definable counterexamples to the continuum hypothesis, Foreman and Magidor mention a way to force $\square_{\omega_1}$ with countable conditions. (This is used in ...

**4**

votes

**2**answers

162 views

### Borel Sets in Sacks Generic Extension

Let $\mathbb{S}$ denote Sacks forcing. This is forcing with perfect trees or equivalently forcing with uncountable Borel subsets of ${}^\omega 2$ with the relation $\subseteq$.
Let $G \subseteq ...

**6**

votes

**2**answers

402 views

### Can we define an “empirically generic” real number?

Summary: My question, in a nutshell, is how we should intuitively imagine a generic real number (as opposed to a random one), and whether we can construct numbers which empirically behave like generic ...

**5**

votes

**1**answer

307 views

### Embeddings of forcing notions - preserve properness?

Let $ M $ be a countable, transitive model for $ \mathsf{ZFC}^* $, i.e. for a sufficiently large finite fragment of $ \mathsf{ZFC} $. Suppose that $ \mathbb{P} := (P, {\leq_P}, \mathbb{1}_P) \in M $ ...

**7**

votes

**0**answers

110 views

### Without AC, which implications between the different definitions of amenability still hold?

More precisely, I would like to know which implications between the following definitions of amenability of a discrete countable (or even finitely generated) group can be proved to hold with only ZF ...

**4**

votes

**1**answer

148 views

### Is it compatible with ZF to assume that every amenable discrete group is finite?

The question is in the title, amenability being understood as the existence of a left-invariant finitely additive probability measure on the group of interest. The case of countable groups is treated ...