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21
votes
4answers
992 views

Why should we care about “higher infinities” outside of set theory?

Let's say you are a prospective mathematician with some addled ideas about cardinality. If you assumed that the natural numbers were finite, you'd quickly vanish in a puff of logic. :) If you ...
4
votes
2answers
129 views

Natural examples of $\bf\Sigma^0_3$ equivalence relations

I have been reading about Borel equivalence relations and I have noticed that while $\bf\Sigma^0_3$ equivalence relations are mentioned, there is a conspicuous absence of natural examples (other than ...
3
votes
1answer
124 views

Injective subset function

Let $X$ be a non-empty set and let $F: X \to {\cal P}(X)$ be a function with the following property: for $A \subseteq X$ we have $|A| \leq |\bigcup F(A)|$. Does this imply that there is an ...
2
votes
0answers
142 views

On generic forcing conditions

Let $P$ be a forcing poset, and $Q \in V^P$ a forcing poset in $V^P$. Let $M \prec H(\lambda)$ ($\lambda$ sufficiently large) countable with $P,Q \in M$. What I want to know is if then the following ...
5
votes
1answer
161 views

Consistency of the nonrigidity of $P(\omega_1)/NS$

Is it consistent with ZFC that there exists an automorphism of $P(\omega_1)/\mathrm{NS}_{\omega_1}$ which is not the identity?
0
votes
0answers
96 views

Defining Global Choice in terms of strong limit cardinals over $ZF$

In his answer to user33038's mathoverflow question "What axioms are stronger than the Axiom of choice?", Prof. Hamkins writes: "What's more, the axiom of choice is equivalent over $ZF$ to the ...
4
votes
1answer
135 views

Large Cardinal Principles that Imply $\Sigma_3^1$-Generic Absoluteness

It is known that (light-face) $\Sigma_3^1$ generic absoluteness is consistent with $\mathsf{ZFC}$: Friedman and Bagaria showed that it holds in the $\text{Coll}(\omega, < \kappa)$ extension of $V$ ...
2
votes
0answers
112 views

RCS iteration such that the RCS limit is semi-proper

For a countable ordinal $\eta$ and an ordinal $\gamma$ let $\langle P_\alpha, \dot{Q}_\alpha \colon \alpha < \gamma \rangle$ be an RCS iteration with RCS limit $P_\gamma$, such that ...
3
votes
2answers
176 views

Hedetniemi's conjecture for graphs with countable chromatic number

Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph_0$, but $\chi(G\times H) < \aleph_0$?
5
votes
1answer
179 views

A question regarding strong cardinals and measure sequence

Let $E$ be a $(\kappa, \lambda)$-extender such that $j: V\to M\simeq Ult(V,E)$ is the corresponding elementary embedding with critical point $\kappa$, $M\supset V_{\kappa+2}$, $M^\kappa\subset M$. Let ...
6
votes
0answers
146 views

rigidity of $\mathcal P(\omega_1) / NS$ under MA

In Woodin's book, Lemma 5.100 asserts that if $MA_{\omega_1}$ holds and there is an $\omega_2$-saturated ideal $I$ on $\omega_1$, then $\mathcal P(\omega_1)/I$ is a rigid boolean algebra, meaning it ...
2
votes
0answers
106 views

What other axioms for set theory can be written in the form: “If mathematical structures $X$ and $Y$ are equipotent, then they're isomorphic”?

The "injective continuum function hypothesis" (ICF) is the following statement. ICF (Version 0). For all cardinal numbers $\kappa$ and $\nu$, we have $2^\kappa = 2^\nu \rightarrow \kappa = \nu.$ ...
8
votes
1answer
369 views

Does there exist an uncountable partition of a Polish space so that the union of any collection of blocks is Borel?

Is it consistent that there exists a partition $P$ of the real number line $\mathbb{R}$ such that $|P|>\aleph_{0}$ but where $\bigcup R$ is Borel whenever $R\subseteq P$? If ...
4
votes
1answer
276 views

“set of all irreducible representations of a group”, set-theoretic issues [closed]

I am working on a problem related to representations of the Weil group of a local field $\mathcal{W}_F$. In many articles one introduces the set $\hat{\mathcal{W}}_F$ of all equivalence classes of ...
1
vote
0answers
191 views

Some questions regarding Shelah's revised Generalized Continuum Hypothesis [closed]

It is well known that $\mathsf{ZF}+\mathsf{GCH}\vdash\mathsf{AC}$ (which means that the Kunen inconsistency can be proven in $\mathsf{ZF}+\mathsf{GCH}$). Consider now Shelah's revised Generalized ...
8
votes
2answers
292 views

Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?

I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try. Thank you in advance. Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that ...
15
votes
3answers
426 views

Does every set $X$ have a topology for which the only continuous self-surjection is the identity map?

This question is a special case of Dominic van der Zypen's question Reconstructing relations with the image relation of a topology, as discussed in the comments, particularly the comment of Eric ...
2
votes
0answers
114 views

How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,...,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if ...
6
votes
1answer
185 views

Reverse of a termspace forcing fact

Suppose $\kappa$ is an inaccessible cardinal. Consider the termspace forcing for adding a Cohen subset of $\kappa$ after $\mathbb P= Col(\omega,<\kappa)$. Members are Levy names for countable ...
10
votes
1answer
299 views

Connected but no path connected components

Is there a Borel subset of plane which is connected but whose only path connected components are singletons? I know that a Bernstein set is a non Borel example of such a set. Thanks!
3
votes
1answer
111 views

Cardinality of pairwise non-isomorphic complete lattices on an infinite cardinal $\kappa$

Suppose $\kappa$ is an infinite cardinal. Let $\cal S$ be a collection of pairwise non-isomorphic complete lattices on the ground set $\kappa$. What cardinality can $\cal S$ have at most? Is the ...
12
votes
1answer
580 views

Does “cardinal arithmetic is well-defined” imply axiom of choice?

Let me quickly explain what I mean with my question. Let $(\kappa_i)_{i\in I}$ be a collection of cardinal numbers, indexed by elements of some set $I$. We can try to define $\sum\limits_{i\in ...
7
votes
1answer
176 views

Consistency strength of being strong cardinal and indestructible under collapses

What is the consistency strength of the following statement: $\kappa$ is a strong cardinals and it is indestructible under $Col(\kappa, <\theta),$ where $\theta> \kappa$ is some fixed ...
10
votes
1answer
248 views

Does stationary reflection imply Mahloness?

Suppose $\kappa$ is strongly inaccessible and every stationary subset of $\kappa$ reflects. Must $\kappa$ be Mahlo? Remarks: It is possible for every stationary subset of $\kappa$ to reflect, but ...
4
votes
1answer
161 views

Can we always add sets without collapsing cardinals or adding [very] bounded sets?

Given a model of $\sf ZFC$, and an infinite ordinal $\alpha$. Can we prove that there is always a cardinal $\kappa$, and a forcing $\Bbb P$, such that: $\Bbb P$ does not add sets of rank ...
6
votes
2answers
404 views

Collapsing the cardinals between two singular cardinals

Question 1: Is it consistent that there is a forcing notion collapsing $\aleph_{\omega\cdot 2}$ to $\aleph_\omega$ without collapsing $\aleph_\omega$ or $\aleph_{\omega\cdot 2 + 1}$? If the ...
-1
votes
1answer
223 views

Class forcings and elementary embeddings

In the Hamkins-Kirmayer-Perlmutter paper "Generalizations Of The Kunen Inconsistency", they prove the following theorem: "Theorem 7: In any set forcing extension $V[G]$, there is no nontrivial ...
5
votes
1answer
245 views

Transcendental distance sets

Define a set $S \subset \mathbb{R}^d$ as a transcendental distance set if the distance between any pair of distinct points of $S$ is transcendental. For example, $S = \{ k \, \pi \;\mid\; k=1,2,\ldots ...
6
votes
0answers
173 views

$\alpha$-minimal degrees for singular $\alpha$

An important question in $\alpha$-recursion theory is whether there is a minimal $\alpha$-degree at $\alpha=\aleph_\omega.$ Question 1. Who first introduced the above question, and where can I find ...
11
votes
1answer
519 views

A question of Erdős

In the following paper (pages 122-23), Erdős asks if there is a constant $c > 0$ such that every subset $A$ of plane of area more than $c$ contains the vertices of a triangle of unit area. Is this ...
9
votes
1answer
552 views

Are there discontinuities in the large cardinal hierarchy?

Suppose that $\Phi(x,y)$ is a formula in the language of set theory so that for each natural number $n$, the axiom $\exists x\Phi(n,x)$ is a large cardinal axiom (for example consider $n$-huge ...
15
votes
0answers
374 views

What sort of cardinal number is the Löwenheim-Skolem number for second-order logic?

In their paper "On Löwenheim-Skolem-Tarski numbers for extensions of first order logic", Magidor and Väänänen make the following statement: "For second order logic, $LS(L^{2})$ ...
7
votes
1answer
278 views

A question related to Woodin's $HOD$ conjecture

Recall that an $\kappa$ is $\omega$-strongly measurable in $\text{HOD}$ if there exists $\lambda < \kappa$ such that $(2^{\lambda})^{\text{HOD}} < \kappa$ and such that there is no partition of ...
10
votes
2answers
374 views

Constructing an $\omega_1$-sequence of functions that almost extend all previous functions

I want to construct a sequence of functions $$f_\alpha: \alpha \rightarrow \omega,\ \alpha < \omega_1$$ such that for all $\alpha < \omega_1$ the following holds: $f_\alpha$ is injective. ...
10
votes
1answer
233 views

Posets preserving stationary subsets of $\omega_1$ and no new $\text{cof}(\omega)$ ordinals, but without countable covering property

What are some examples of posets $\mathbb{P}$ which have the following properties? It's OK if the definition uses large cardinals or some other hypothesis. $\mathbb{P}$ preserves stationary subsets ...
30
votes
2answers
1k views

Measurability and Axiom of choice

In some situations, you need to show Lebesgue-measurability of some function on $\mathbb{R}^n$ and the verification is kind of lengthy and annoying, and even more so because measurability is "obvious" ...
1
vote
0answers
133 views

A question regarding extendible cardinals and a result of M. Magidor

The following definitions and Theorems come from M. Magidor's paper "On the Role of Supercompact and Extendible Cardinals in Logic" (Israel J. Math., Vol. 10, 1971): "Definition: Logic is called ...
7
votes
1answer
592 views

Is every real a member of some CTM?

I read somewhere about the hyperuniverse of countable transitive models of ZFC (http://www1.maths.leeds.ac.uk/maloa/lecturenotes/RW3%20Munster/friedman.pdf). It states an assumption, namely that every ...
7
votes
1answer
399 views

Why is the set-theoretic principle $\diamondsuit$ called $\diamondsuit$?

A shallow answer would just point to theorem 6.2 in Jensen's 1972 paper "The fine structure of the constructible hierarchy", where Jensen introduces this property. Or was this symbol used already ...
4
votes
1answer
256 views

Plausibility argument for a measurable cardinal

The following question is not mathematically precise but perhaps of some philosophical interest. A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: ...
5
votes
1answer
172 views

References for Forcing with Side Conditions

I'm looking for some good references about Forcing with Side Conditions, including expository papers that explain the main ideas with some details in order to give me a fairly clear insight of those ...
5
votes
1answer
264 views

Diagonalizing against a non stationary set of functions

Suppose $\kappa$ is regular uncountable (assume $2^{< \kappa} = \kappa$ and $\kappa$ is weakly Mahlo if needed). Is the following true? For every sequence $\langle f_i: i \to 2 \mid i \in A ...
1
vote
1answer
200 views

Well-ordered reference

I'm wondering if there is a standard reference for the following straightforward facts about well-orderings. (They are quite easy to prove, I just need a handy/standard reference.) Let $(S,<)$ ...
10
votes
1answer
520 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
0answers
105 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 ...
2
votes
2answers
317 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}: ...
0
votes
0answers
149 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 ...
8
votes
0answers
262 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 ...
5
votes
1answer
157 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 $\bigcup {\cal C} = X$ and $X\notin {\cal C}$. If ${\frak U}$ and $\frak{W}$ are collections of covers of a ...
5
votes
0answers
147 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 ...