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9
votes
2answers
401 views

Ordering of large cardinals by cardinality

Let Type A and Type B be two types of large cardinals from, say, Cantor's Attic (http://cantorsattic.info/Upper_attic) Now assuming that ZFC + Type A + Type B is consistent (ie, both Type A and Type ...
10
votes
0answers
240 views

When does $HOD^{V[G]} \subseteq V$?

Assume that $\mathbb{P}\in HOD$ is non-trivial. It is well-known that if $\mathbb{P}$ satisfies some homogeneity properties, then $HOD^{V[G]} \subseteq V$, where $G$ is $\mathbb{P}$-generic over $V$. ...
9
votes
1answer
307 views

Pontryagin dual of the surreal numbers?

Has any work been done on the Pontryagin dual of the surreal numbers (suitably topologized)? I have not been able to find anything and am not sure if this is still unknown. Alternatively, has this ...
5
votes
1answer
157 views

almost disjoint ladder system on $\omega_2$

Suppose $\langle s_\alpha : \alpha \in \omega_2 \cap \mathrm{cof}(\omega_1) \rangle$ is a sequence such that each $s_\alpha$ is an increasing cofinal map from $\omega_1$ to $\alpha$. Is it possible ...
18
votes
2answers
555 views

History of set-class distinction

I have two questions concerning the history of set theory, both related to the distinction between the notion of a set and the notion of a class: Who was the first mathematician to make this ...
2
votes
1answer
220 views

Wholeness Axiom and Ultimate L

From what I understand: The Wholeness Axiom(s) is/are the "ultimate axioms of infinity", bordering on inconsistency with ZFC. Ultimate L (Completion of ZFC) attempts to extend the orderly world of ...
6
votes
1answer
152 views

Different ways of making $HOD$ far from $V$

There are different criteria for building a model $V$ of $ZFC$ which is far from its $HOD$, for example: $(A)$ Cardinality criteria: For this in a joint work with James Cummings and Sy Friedman, we ...
9
votes
2answers
768 views

Non null Turing antichain

This interesting question resulted from a query of Mushfeq: In ZFC, can we find a non null set of pairwise Turing incomparable reals?
8
votes
2answers
374 views

Some “axiom of choice” and “dependent choice” issues

I am probably about to ask some fairly basic questions, and yet I have found it quite hard to find the answers to these. If I understand correctly, mathematicians tend to be quite happy working with ...
21
votes
0answers
544 views

Relative null-ness

Here, "measure" always means Lebesgue measure on $\mathbb{R}$. This question is partly motivated by my answer ...
6
votes
0answers
187 views

Remote points in $\beta X$

It is known that in general convergence by sequences is not enough to account for all points in $\beta X \setminus X$, where $\beta X$ refers to the Stone-Cech compactification of a topological space ...
16
votes
1answer
491 views

Can we find CH in the analytical hierarchy?

Recently I was talking to my friend and I have mentioned to him that it was proven that CH is not provably (over ZFC) equivalent to any statement in second-order arithmetic. However, today I found out ...
15
votes
1answer
420 views

A problem of Keisler and Tarski

The following question dates back to Keisler and Tarski: From accessible to inaccessible cardinals, Fund. Math. 53, 1964 and also perhaps Mazur: On continuous mappings of Cartesian products, Fund. ...
7
votes
1answer
173 views

Forcing Extension of Countable Linearly Iterable Structures

Let $V$ satisfy there exists a measurable cardinal. Let $\kappa$ be a measurable cardinal and $U$ be the normal measure on $\kappa$ witnessing this. Let $\mathbb{P}$ be a forcing of size less than ...
5
votes
0answers
66 views

Is there an analytic $\mathrm{P}$- ideal on $\omega$ which is not $\Sigma^0_2$ and not $\Pi^0_3$-complete?

Soleski proved that for any analytic $\mathrm{P}$-ideal on $\omega$ is $\Pi^0_3$. The usually example , such as the density zero ideal $Z_0$ is $\Pi^0_3$-complete, $I_{\frac{1}{n}}$ is ...
4
votes
1answer
178 views

Sets not containing the vertices of unit triangles (Question posed by Erdős)

Following this post, I have been thinking about the problem posed by Erdős, Does there exist a constant $c > 0$ such that every subset $A$ of the plane of area more than $c$ contains the ...
8
votes
2answers
355 views

Large cardinal consistency strength and size

My understanding is that large cardinals are ordered by "consistency strength", but how does this correlate with their size (cardinality)? More specifically, are there any systematic results on the ...
20
votes
1answer
2k views

How far wrong could the Continuum Hypothesis be?

I hear it's consistent with ZFC to have $$ 2^{\aleph_0} = \aleph_n $$ for any $n = 1, 2, 3, \dots $. How much worse can it get? More precisely: are there models of ZFC with $2^{\aleph_0} \gt ...
0
votes
1answer
194 views

A question regarding models of $ZF+I_0$ [Revised]

In his answer to user42090's mathoverflow question"Minimal Generalized Contnuum Hypothesis & Axiom of Choice", Prof. Hamkins writes: "...one can build the analogue of the symmetric models for ...
6
votes
2answers
404 views

“Lebesgue-measurable” cardinals and real-closed fields

I understand the motivation behind measurable cardinals is to ask the question: "is there any set large enough to admit a non-trivial measure on all of its subsets?" Hence, it's also worthwhile to ...
12
votes
2answers
396 views

tree properties on $\omega_1$ and $\omega_2$

Are the following mutually consistent (relative to large cardinals)? (1) There are no $\omega_2$-Aronszajn trees. (2) There is an $\omega_1$-Kurepa tree. In the models I know of the tree property ...
2
votes
1answer
174 views

tree property at $\aleph_2$ and $\aleph_4$

It is claimed that if there are two weakly compact cardinals, then there is a generic extension in which $\aleph_2$ and $\aleph_4$ have the tree property. Assuming one knows Mitchell's forcing, what ...
6
votes
0answers
241 views

The least admissible above a dominating real

Let $\mathbb{P}$ be the usual forcing which adds a dominating real: conditions in $\mathbb{P}$ are pairs $(p, f)$ with $p:\omega\rightarrow\omega$ finite partial and $f:\omega\rightarrow\omega$ total, ...
11
votes
1answer
239 views

Partition relation at successor cardinal

Is it consistent that there are regular cardinals $\kappa < \lambda$, such that $\lambda$ is a successor cardinal and for every coloring $d\colon[\lambda]^2\to\kappa$ there is some ...
3
votes
1answer
132 views

Does totally proper forcing imply countable distributivity?

For a suitable model $M$ for $Q$ and a condition $q \in Q$ we say that $q$ is $(M,Q)$-generic if whenever $r \leqslant q$, $D \in M$ dense, $D \subset Q$, $r$ is compatible with an element of $D \cap ...
0
votes
2answers
310 views

A question regarding the relation between Freiling's Axiom of Symmetry and real-valued measurable cardnals

A major argument against Freiling's Axiom of Symmetry is the following (this from the wikipedia article of the same name): "The naive probabalistic notion used by Freiling tacitly assumes that there ...
8
votes
0answers
217 views

On an unpublished result of Magidor

In 1970th, Magidor proved the following important results: (1) Assuming the existence of a supercompact cardinal, it is consistent that $\aleph_\omega$ is strong limit and ...
26
votes
2answers
1k views

What is the fewest number of points you must delete from $\mathbb{R}^3$ to make it not simply connected?

This question concerns a set-theoretic aspect that I found interesting in the recent question asked by user Nick R., namely, Is $\mathbb{R}^3\setminus\mathbb{Q}^3$ simply connected? He had asked ...
2
votes
1answer
179 views

Notation: $Sigma$ and $Pi$ of intersections

In Jech - Set Theory, the proof of Theorem 31.7, I came along some notations I wish to understand correctly. For a countable elementary substructure $M \prec H_\lambda$ and $A \in M$ and a generic ...
28
votes
5answers
2k 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
250 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 ...
1
vote
1answer
169 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 ...
3
votes
1answer
276 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 ...
6
votes
1answer
185 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?
2
votes
0answers
118 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 ...
5
votes
1answer
182 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
129 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 ...
2
votes
2answers
204 views

Hedetniemi's conjecture for graphs with countable chromatic number

Are there graphs $G, H$ such that $\chi(G) = \chi(H) = \aleph_0$, but $\chi(G\times H) < \aleph_0$?
6
votes
2answers
245 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
164 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
120 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
486 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
302 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
238 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
348 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 ...
14
votes
3answers
463 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 ...
3
votes
0answers
151 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
197 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 ...
11
votes
2answers
412 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!
0
votes
1answer
116 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 ...