**1**

vote

**2**answers

195 views

### Subsets of $\mathbb{N}$ whose lower density respects complements

The lower density of $A\subseteq\mathbb{N}$ is defined to be $\lambda(A)=\lim\text{inf}_{n\to\infty}\frac{|A\cap\{1,\ldots,n\}|}{n}$. We set $${\cal C} = \{A\subseteq \mathbb{N}: ...

**10**

votes

**1**answer

433 views

### A question of Erdos on entire functions

At the end of the following paper, Erdos asked if there is a family $F$ of entire functions of size continuum such that for every $z \in \mathbb{C}$, $\{f(z) : f \in F\}$ has size less than continuum. ...

**5**

votes

**0**answers

85 views

### Does the critical sequence for subalgebras of elementary embeddings with finitely many generators have order type $\omega$?

Suppose that $\lambda$ is a cardinal. Let $\mathcal{E}_{\lambda}$ be the set of all elementary embeddings from $V_{\lambda}$ to $V_{\lambda}$. If $j,k\in\mathcal{E}_{\lambda}$, then define
...

**-1**

votes

**0**answers

21 views

### Freedom of speech in scientific discussions - An invitation to more tolerance in Scientific debates [migrated]

I hope this post enjoys some tolerance, and don't get closed or put on hold immediately.
I believe that freedom of speech in scientific discussions is one of the key values which enriches the debates ...

**7**

votes

**0**answers

216 views

### preserving saturated ideals

A reliable source made the following claim:
Suppose CH there is an $\omega_2$-saturated ideal on $\omega_1$. Then this is preserved by $\mathrm{Add}(\omega_1,\omega_2)$.
Question 1: How do you ...

**11**

votes

**1**answer

493 views

### Can there be a global linear ordering of the universe without a global well-ordering of the universe?

This question arose in the answers to Asaf Karagila's
question Does ZFC prove that the universe is linearly orderable?. The answer there was that one can have a ZFC model with no global linear ...

**0**

votes

**0**answers

125 views

### If C is a closed, unbounded subset of Ord, are the sets generated by the new class of ordinals Ord restricted to C, a Universe?

Given any universe, and some closed unbounded subset of Ord, is it possible toy generate a universe in the following way:
Restrict Ord to a target club. Then generate all look the sets necessary to ...

**9**

votes

**2**answers

476 views

### Preserving $\omega_1$ is Inaccessible to the reals

$\omega_1$ is inaccessible to the reals if and only if for all $x \in {}^\omega\omega$, $\omega_1^{L[x]} < \omega_1$.
The question is if $\omega_1$ is inaccessible to the reals in $V$ and ...

**12**

votes

**1**answer

382 views

### Avoiding countable subgroups of a group homeomorphic to the Cantor space

The following question is motivated by the paper
[Brian, Mislove, Every compact group can have a non-measurable subgroup].
A positive solution to a variation of the following problem implies a ...

**6**

votes

**0**answers

107 views

### What is the significance of having Prime Ideal Theorem in models for failure of Axiom of Choice? [migrated]

Prime Ideal Theorem says:
PIT: Every ideal on a Boolean algebra can be extended to a prime ideal.
It follows from Axiom of Choice but is weaker than it.
In many cases I saw that people check ...

**4**

votes

**1**answer

201 views

### Ref. request: Additive probability measure on $\mathcal P({\bf N})$ supplies subset of $\mathbf R$ without Baire property

ZFC proves, among the other things, the existence of a (finitely) additive probability measure $\theta: \mathcal P(\mathbf N) \to \mathbf R$ on the power set of $\mathbf N$ such that $\theta(X) = 0$ ...

**11**

votes

**2**answers

515 views

### Open coloring axiom vs. CH

Is there a simple, direct proof that the open coloring axiom contradicts CH (straight from the definitions, no machinery allowed)? The separable metric spaces version of OCA, if that helps.
Edit: ...

**7**

votes

**5**answers

789 views

### Knaster Tarski theorem, example needed

http://en.wikipedia.org/wiki/Knaster%E2%80%93Tarski_theorem
Let $L$ be a complete lattice and let $f : L \to L$ be an order-preserving function. Then the set of fixed points of $f$ in $L$ is also a ...

**14**

votes

**2**answers

1k views

### Why does inner model theory need so much descriptive set theory (and vice versa)?

I am curious about how much descriptive set theory is involved in inner model theory.
For instance Shoenfield's absoluteness result is based on the construction of the Shoenfield tree which ...

**3**

votes

**0**answers

74 views

### Implications between different covering properties of spaces

Let $X$ be a set. A set ${\cal C}\subseteq {\cal P}(X)$ is said to be a cover of $X$ if $\bigcup {\cal C} = X$.
If ${\frak U}$ and $\frak{W}$ are collections of covers of a set, we define the ...

**5**

votes

**0**answers

141 views

### Sierpinski sets and extensions of Lebesgue measure

I am duplicating an old problem from stackexchange:
Suppose that every countably generated sigma algebra extending the Borel sigma algebra on $[0, 1]$ admits a measure extending the Lebesgue measure ...

**1**

vote

**1**answer

135 views

### $\mathbb{P}_{\kappa}$ forces non$(\mathcal{M})$=cov$(\mathcal{M})=\kappa$

Let $\kappa$ be an uncountable regular cardibnal. Consider the finite support iteration $(\langle \mathbb{P}_{\alpha}\rangle _{\alpha \leq \kappa},\langle \mathbb{\dot{Q}}_{\alpha}\rangle _{\alpha ...

**13**

votes

**1**answer

662 views

### Finitely generated group with $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups?

Let $X_G$ be the number of normal subgroups of a group $G$. Are there examples of finitely generated groups $G$ where it is consistent to have $\aleph_0<X_G<2^{\aleph_0}$ normal subgroups? Also ...

**9**

votes

**1**answer

369 views

### Just a little absoluteness might be cheaper?

Absoluteness is a wonderful thing, but expensive consistency-strength wise. My question is, when can we get large amounts of absoluteness in specific situations for much cheaper?
Specifically, fix a ...

**10**

votes

**0**answers

163 views

### Generic $\mathbf{\Sigma}_3^1$-absoluteness for class forcings

In the paper "Generic Absoluteness" by Bagaria and Friedman (http://www.logic.univie.ac.at/~sdf/papers/bagfried.pdf) it is shown that in ZFC generic $\mathbf{\Sigma_3^1}$-absoluteness is false for ...

**7**

votes

**2**answers

274 views

### Topological tameness beyond the Gandy-Harrington topology

The Gandy-Harrington topology on $\omega^\omega$ is the topology generated by all lightface $\Sigma^1_1$ sets; that is, all sets which are continuous-in-the-usual-sense images of $\omega^\omega$.
...

**25**

votes

**5**answers

1k views

### Forcing as a replacement of induction and diagonal arguments

Let me give some examples motivating the question.
The use of forcing instead of induction: For this consider Cantor's theorem:
Theorem 1. Any two countable dense linear orders $I, J$ without end ...

**9**

votes

**1**answer

270 views

### Saturation of null ideal

In ZFC, can we find more than continuum many non null sets of reals whose pairwise intersections are null?

**11**

votes

**1**answer

297 views

### Non meager rectangle

Suppose $G \subseteq \mathbb{R}^2$ is dense $G_\delta$. Must there (in ZFC) exist non meager sets of reals $A, B$ such that $A \times B \subseteq G$?

**5**

votes

**1**answer

295 views

### Details for Woodin's forcing argument for a saturated ideal from the Levy collapse

Theorem 2.65 in Woodin's book shows that a saturated ideal on $\omega_1$ exists after Levy-collapsing a Woodin cardinal $\delta$ to $\omega_2$. I am confused about the part of the argument where he ...

**2**

votes

**0**answers

52 views

### Strongly minimal covering subsets of $\text{Ind}(G)$

Let $G=(V,E)$ be any undirected, simple graph. Let $\text{Ind}(G)$ be the set of independent subsets of $V(G)$. We say that $K\subseteq \text{Ind}(G)$ is a cover (by independent subsets) if $\bigcup K ...

**4**

votes

**1**answer

196 views

### Measure algebra of a total extension of Lebesgue measure

Solovay shows that the existence of a measurable cardinal is equiconsistent with the existence of a countably additive extension of Lebesgue measure that is defined on all sets of real numbers. Given ...

**-1**

votes

**0**answers

27 views

### Set theory identity, can't derive it [migrated]

I see the following step in a textbook, and I can't follow it,
$A\cup(B\cup(C-D)-E)$
$=$
$(A\cup(B-E))\cup(C-(D\cup E)$
My progress from the LHS get stuck like this,
$$A\cup(B\cup(C-D)-E)$$
...

**8**

votes

**1**answer

183 views

### Is the Jordan decomposition of a self-adjoint functional constructive?

Let $A$ be an abstract C*-algebra, and let $\varphi\colon A \rightarrow \mathbb C$ be a bounded linear function. Assuming the axiom of choice, there exist unique positive bounded linear functions ...

**9**

votes

**1**answer

439 views

### Coding a model of $0^\sharp$ from a $\Pi^1_1$ Gale-Stewart game

As a preface to this question, this is my first time asking on Math overflow, and this seemed like the sort of question that would be acceptable here. However, I apologize if it is not.
A method for ...

**9**

votes

**0**answers

177 views

### The Chang model after collapsing an inaccessible limit of Woodins

If $\kappa$ is an inaccessible cardinal and $G \subset \operatorname{Col}(\omega,\mathord{<}\kappa)$ is a $V$-generic filter, then in $V[G]$ the Chang model $L(\text{Ord}^\omega)$ satisfies "every ...

**0**

votes

**0**answers

114 views

### Finitely additive measure over integers [duplicate]

We know that, with Axiom of Choice (AC), it can be shown that there exists a finitely additive uniform distribution defined for all subsets of the integers (see, e.g., Hrbacek and Jech 1999, Ch. 11).
...

**18**

votes

**3**answers

2k views

### Who needs Replacement anyway?

The set theory ETCS famously comes without the Replacement axiom schema (or an equivalent) that is part of ZFC. One (to me, not apparently useful) set that one cannot build in ETCS is $\coprod_{n\in ...

**11**

votes

**0**answers

186 views

### Adding a saturated ideal

Is it consistent that there is no $\omega_2$-saturated ideal on $\omega_1$, but one is introduced by an $\omega_2$-closed forcing?

**10**

votes

**0**answers

198 views

### c.c.c forcing notions and adding minimal generic reals

Is the following statement consistent:
``There is no non-trivial c.c.c forcing notion adding a minimal generic real''?
The question is related to Prikry's question: Is it consistent that any ...

**5**

votes

**0**answers

112 views

### Homogeneous $\omega$-monolithic compact space

Under CH, is the cardinality of every homogeneous $\omega$-monolithic compact space
$X$ not greater than $2^{\omega}$?

**11**

votes

**3**answers

418 views

### The size of Lindelof space

Question. Suppose that $X$ is a Lindelof space such
that every point of $X$ is a $G_{\delta}$-point. Then is it true that $|X| ≤ 2^{\omega}$?

**9**

votes

**1**answer

376 views

### Is $\clubsuit_{\omega_1}$ enough to get Suslin tree?

This is problem 15.3 in Arnie Miller's problem list:
(Juhasz) Suppose there exists $\langle A_{\alpha} : \alpha \in L \rangle$, where $L$ is the set of limit ordinals below $\omega_1$ and for each ...

**7**

votes

**1**answer

242 views

### For a ring $k$ and a set $X$, what are the $k$-algebra homomorphisms $k^X \to k$?

Let $k$ be a commutative ring. Feel free to assume it's a field.
Let $X$ be a set. This question is only interesting when $X$ is infinite.
Write $k^X$ for the $k$-algebra of functions $X \to k$, ...

**0**

votes

**1**answer

200 views

### Provability of unprovability

I have three questions (without any real background, this is just something I've been wondering about recently)
Can PA prove that PA can't prove nor disprove, say, Goodstein theorem (or any natural ...

**4**

votes

**1**answer

161 views

### Cohen algebra and $\mathcal P(\omega)/ \mathrm{fin}$

Let $C$ be the Cohen algebra, the boolean completion of the partial order of finite partial functions from $\omega$ to 2, ordered by reverse inclusion. Does there exist an ideal $I$ on $C$ such that ...

**2**

votes

**0**answers

197 views

### Is there a $\Sigma^0_3$-complete ideal on $\omega$?

In Kechris's book "Classical Descriputive set theory" Chapter 23 (Exercise 23.4), it claim that there is a $\Sigma^0_\xi$-complete ideal on $\omega$ for each $\xi\geq 3$.
There is a candidate ...

**4**

votes

**1**answer

193 views

### Does the minor graph of graphs on $\mathbb{N}$ have an uncountable independent set?

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. We write $V(G) := V$.
If $S, T$ are disjoint subsets of $V(G)$ we say that ...

**34**

votes

**3**answers

1k views

### Does an existence of large cardinals have implications in number theory or combinatorics?

Does an existence of large cardinals have implications in more down-to-earth fields like number theory, finite combinatorics, graph theory, Ramsey theory or computability theory? Are there any ...

**4**

votes

**2**answers

183 views

### Minimum cardinality of lower-bounding subset of $[\omega]^\omega$

Let $[\omega]^\omega$ denote the collection of all infinite subsets of $\omega$. Let us call $S\subseteq [\omega]^\omega$ lower-bounding if for all $a\in [\omega]^\omega$ there is $s\in S$ such that ...

**5**

votes

**1**answer

287 views

### Uncountably many countable graphs with no homomorphism between them

By a graph I mean a pair $G = (V, E)$ where $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\}: a\neq b \in V\}$. A graph homomorphism between graphs $G, H$ is a map $f:V(G)\to V(H)$ such ...

**5**

votes

**1**answer

474 views

### $\Sigma_1$ elementary substructure

This is a question I was given in the exam: show that if $\lambda, \kappa$ are uncountable cardinals then
$$(H_\lambda,\in) \prec_1 (H_\kappa,\in)$$
where $H_\lambda$ is the class of all sets $x$ such ...

**8**

votes

**1**answer

263 views

### Can there be a tree of height $\omega_2$ having all levels countable, with no cofinal branch?

For many years I had the idea that if a well-founded tree is both very tall and very narrow, then it must have a cofinal branch. For example, it is a fun exercise to show that any $\omega_1$-tree ...

**8**

votes

**5**answers

2k views

### Naturally occuring groups with cardinality greater than the reals.

In group theory, the single most important piece of information about a group is its cardinality, which is of course either finite, countably infinite, or uncountably infinite. Usually, however, ...

**7**

votes

**1**answer

187 views

### Introducing meets while preserving directed closure

A poset $\mathbb{P}$ is called well-met iff every pair of compatible conditions in $\mathbb{P}$ has a greatest lower bound.
Question: Suppose $\mathbb{P}$ is a separative partial order which is ...