forcing, large cardinals, descriptive set theory, infinite combinatorics, cardinal characteristics, forcing axioms, ultrapowers, measures, reflection, pcf theory, models of set theory, axioms of set theory, independence, axiom of choice, continuum hypothesis, determinacy, Borel equivalence ...

learn more… | top users | synonyms

2
votes
0answers
48 views

Optimal tiling for a collection of partitions

I'm interested in a possible generalization of Tiling relation on the set of partitions (the question has only been partially answered). Let $x$ be an infinite set and let $\text{Part}(x)$ be the ...
4
votes
1answer
97 views

Tiling relation on the set of partitions

Let $x\neq \emptyset$ be a set and let $\text{Part}(x)$ be the collection of all partitions of $x$. We need the following notation. Let $P \in \text{Part}(x)$ and $t\subseteq x$. We set $$P_{[t]} = ...
2
votes
1answer
112 views

Closed sets in ordinal spaces

I'm studying the ordinal space $[0,\kappa[$ where $\kappa\neq \omega$ is a cardinal of countable cofinality and I want to know why there are in $[0,\kappa[$ two disjoint closed sets of cardinality ...
2
votes
1answer
248 views

Axiomatic ZFC Set Theory [closed]

Can the Axiom Schema of Comprehension be omitted from ZFC since it is implied by the Axiom Schema of Replacement?
1
vote
1answer
159 views

A question regarding sets of Vitali's type in models of $ZF+GCH$ where $L$$\neq$$V$

Consider sets of Vitali's type in models of $ZF+GCH$ where $L$ $\neq$$V$. Are there sets of Vitali's type in both $L$ and $V-L$? If so, is there any way one can distinguish the constructible sets of ...
6
votes
2answers
621 views

A question on rank-to-rank embeddings

Consider a non-trivial elementary embedding $j:V_\lambda\to V_\lambda$ and, for each $A\subset V_\lambda$, set $j(A)=\bigcup_{\delta<\lambda}j(A\cap V_\delta)$. In Implications between strong ...
3
votes
3answers
323 views

The Set-Theoretic Multiverse and Joint Embeddings

I am curious whether or not the following axiom is independent of Hamkins's axioms for the Set-Theoretic Multiverse. Hamkins's axioms can be found here on pages 1-2 and here on pages 24-26. Consider ...
4
votes
5answers
860 views

A generalized diagonal?

A simple question. Let $ f:X\to Y $ be a function and let $ E_f:=\{(x, y): f (x)=f (y)\}\subset X\times X $. What is the name of the set $ E(f) $? It would be nice to have some reference also. It ...
1
vote
2answers
267 views

What is the consistency strength of a standard model of ZF versus a transitive model?

A standard model of ZF need not be transitive, of course, and Joel David Hamkins' answer to Large cardinal axioms and Grothendieck universes gives Tarski sets as an interesting example. I should ...
1
vote
2answers
225 views

Is there a pairing function from countable ordinals to $\mathbb N$? [closed]

It is well-known that there is a computable pairing function $<\ >:\mathbb N^2\to \mathbb N$. Let $X$ be some reasonable class of countable ordinals ($\omega_1^{CK}$, $\epsilon_0$, ...
5
votes
0answers
249 views

Cardinal characteristics without choice

(I'm taking my definition of a cardinal characteristic from Blass' excellent article http://www.math.lsa.umich.edu/~ablass/need.pdf, which cites Vojtas/Fremlin/Miller; theirs is more general, but I'm ...
2
votes
2answers
215 views

Is certain topology-related set a distributive lattice?

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 ($n$ ...
11
votes
2answers
675 views

Where is the Erdős–Rado theorem stated in Erdős and Rado's Bull AMS paper?

This may be inappropriate for MO, but here goes: if I have understood the statement of the Erdős–Rado theorem correctly, then it contains as a special case the following result: if $\mu$ is ...
4
votes
2answers
243 views

Is it consistent that $\frak{d} < 2^{\aleph_0}$?

Let $\omega^\omega$ denote the set of all functions $f:\omega\to\omega$. We write $f <^* g$ if there is $N\in\omega$ such that $f(n) < g(n)$ for all $n>N$. A set $D\subseteq \omega^\omega$ is ...
12
votes
4answers
1k views

Sierpinski's construction of a non-measurable set

In the early 20th century there was a lot of fuss over the axiom of choice implying that there are Lebesgue non-measurable sets of reals. In his book about The Axiom of Choice, Gregory Moore points to ...
2
votes
3answers
811 views

Where do Set Theory and Number Theory meet together?

As all know, by absoluteness theorems in Set Theory, most of theorems in number theory are $ZFC$-provable if and only if they are consistent with $ZFC$, it's because of absoluteness of essence of ...
12
votes
3answers
690 views

Complete resolutions of GCH

Let's say that a "complete resolution of GCH" is a definable class function $F: \operatorname{Ord}\longrightarrow \operatorname{ Ord}$ such that $2^{\aleph_\alpha} = \aleph_{F(\alpha)}$ for all ...
9
votes
0answers
157 views

Preserving Jonsson cardinals

I am (still) interested in trying to characterize and describe forcings that preserve Jonsson cardinals. A cardinal $\kappa$ is a Jonsson cardinal if there is no Jonsson algebra on $\kappa$, i.e. ...
4
votes
1answer
182 views

Is there a “natural” bijection between models of $ZFC$ and $ZF\neg C$?

I'm quite confident that there is the same number of models of $ZFC$ and of $ZF\neg C$, which means that there would exist a bijection between the former and the latter, by definition. However, I was ...
3
votes
1answer
113 views

Classify set theories whose transitive models sharing the same sets of ordinals are equal

This question is a follow-up from my recent question, Classifying set theories whose standard models sharing the same ordinals are equal Let's say that a (recursively axiomatizable) set theory $T$ ...
3
votes
1answer
174 views

A question on models of set theory and Lebesgue measure

In a question and an answer at MO, Joel David Hamkins showed that (if ZFC is consistent) there are models of ZFC in which $V\neq HOD$ and every $\Sigma_2$-definable set has a definable member. Let ...
12
votes
2answers
272 views

Classifying set theories whose standard models sharing the same ordinals are equal

Let's say that a (recursively axiomatizable) set theory $T$ extending ZF is "ordinal-categorical" if, whenever $M$ and $N$ are standard models of $T$ sharing the same ordinals, one has $M = N$. For ...
11
votes
1answer
544 views

Is forcing computable?

By results similar to Tennenbaum's theorem we know that there exist no computable models of $ZF$. But suppose we are given, as a sort of oracle, access to some model of $ZF$ (e.g. we can make oracle ...
30
votes
1answer
773 views

Producing finite objects by forcing!

It is a trivial fact that forcing can not produce finite sets of ground model objects. However there are situations, where we can use forcing to prove the existence of finite objects with some ...
0
votes
1answer
212 views

A question regarding $ZFC^{-}$

Given $ZFC^{-}$, that is, ZFC-Powerset+Collection+Separation, is there a set of alternative axioms $X$ (other than the trivial one, namely, {Powerset}) that, when added to $ZFC^{-}$, allow one to ...
9
votes
1answer
217 views

Strongest large cardinal axiom compatible with $V = L$?

What is the strongest known natural large cardinal axiom compatible with $V = L$ (strongest in the sense that it implies all known "small" large cardinal axioms, where a large cardinal axiom is said ...
0
votes
1answer
121 views

Lowering from filters to ultrafilters for an infinitary relation

Let $U$ be a set. Let $N$ be a (possibly infinite) index set. Let $f$ be an $N$-ary relation on $U$ (that is $f$ is a set of functions $N\rightarrow U$). I denote $\mathcal{L}\in \upuparrows f ...
12
votes
1answer
665 views

Lots of combinatorial interpretations of Catalan numbers

During a lecture I gave on Catalan numbers, I pointed out that that it is possible to give a continuum number of combinatorial interpretations of these numbers. See the solution to (f$^5$) on page 54 ...
-2
votes
1answer
154 views

Expressing a value related to an infinitary relation through ultrafilters

Let $U$ be a set. I denote $\mathfrak{A}$ the lattice of filters on $U$ ordered reverse to set theoretic inclusion of filters. I denote $\bigvee$ and $\bigwedge$ correspondingly the supremum and ...
6
votes
1answer
129 views

Consistency of the collection axiom scheme compared to replacement

Set theory ZFC- is ZFC without power set, but with replacement. It does not imply the collection axiom scheme, as discussed in http://jdh.hamkins.org/what-is-the-theory-zfc-without-power-set/ Does ...
2
votes
0answers
66 views

A question on recursion and transfinite recursion in extensions of KP

Is the $\Sigma_{n}$-recursion supported by $\Sigma_{n}KP=KP+\Sigma_{n}$-separation + $\Sigma_{n}$-collection equivalent with $\Sigma_{n}$ transfinite recursion? If not, how do these notions differ?
2
votes
1answer
111 views

Interweaving two indexed families of filters

Conjecture Let $U$ be an (infinite) set. Let $f$ be an $N$-ary (where $N$ is an arbitrary index set) relation on $U$ (that is a set of functions $N \rightarrow U$). Let $\mathcal{L}_0$, ...
2
votes
0answers
80 views

An analogue of CH for proper classes

Working in NBG set theory, with AC but without Global Choice, we ask for two proper classes A and B such that A strictly injects in B and B strictly injects in P(A); so Question: In NBG set theory, is ...
7
votes
1answer
229 views

Are there known ways to posit definable global choice in ZF without positing V=L?

I need a global choice function defined by a formula in (a fragment of) ZF. There is no harm in assuming V=L for my purposes. But I wonder if there are any familiar alternative ways to get this? ...
10
votes
2answers
315 views

Semiproper but not proper

Assume V=L. Is there a semi-proper notion of forcing that is not proper? Namba forcing isn't semi-proper in L, and Prikry forcing isn't even available there.
9
votes
0answers
168 views

countable OD sets in the Solovay model

It is known that the following is true in the Solovay model (SM) for ZFC: any countable OD (ordinal-definable) set $X$ of reals necessarily consists of OD elements. What about countable OD sets of ...
10
votes
3answers
198 views

First-order definable bijection between $P(On)$ (or $No$) and $V$? (Is this equivalent to $V = HOD$?)

It is known that locally one can ``code'' any set in the von Neumann universe $V$ by a set of ordinals. But can one do this globally? In other words, is there a first-order definable bijection ...
3
votes
1answer
139 views

CCC Forcing and $\omega_1$ conditions

I have a question about the proof of the Lemma 7.2 in the paper I. Juhász, P. Koszmider and L. Soukup, A first countable, initially $\omega_{1}$-compact but non-compact space, Topology and its ...
11
votes
1answer
346 views

What are the current views on consistency of Reinhardt cardinals without AC?

It's well known that Reinhardt cardinals are inconsistent, provided that we have access to axiom of choice, but, as far as I know, we are clueless about this when we don't assume choice. For me, the ...
5
votes
2answers
416 views

When do we have a bijection between a proper class A and its power set class P(A)?

We work in the set theory NBG with the axiom of (local choice but without global (class) choice. For every class A P(A) is the class of all sets x included in the class A. We know that P(A) is a set ...
3
votes
1answer
132 views

Formal systems needed to formalize relative independence results

We know that Con(ZF) implies Con(ZFC+GCH), Con(ZF+neg(AC)) and Con(ZFC+neg(CH)). But what are some weak theories in which these relative independence results are provable? In particular, are they ...
3
votes
0answers
351 views

Can there be ordinals larger than those contained in Ord, and if so, can they be used to extend the constructible universe $L$?

Can there be ordinals larger than those contained in Ord, the class of all ordinals,and if so, can these ordinals be used to extend the constructible universe $L$? In a simplified form, my question ...
16
votes
2answers
1k views

Is non-existence of the hyperreals consistent with ZF?

I know that it is possible to construct the hyperreal number system in ZFC by using the axiom of choice to obtain a non-principal ultrafilter. Would the non-existence of a set of hyperreals be ...
2
votes
1answer
137 views

A question regarding forcing extensions

Can one, for an infinite set A in ZFC, use forcing to add so many generic subsets of A as to make the collection of all subsets of A a proper class? Consider now a model $M$ of ZFC and use $Add( , ...
1
vote
2answers
134 views

A Question Regarding Defining Generic Extensions of ZF and ZFC in Morse-Kelly Set Theory

It is known that Morse-Kelly (MK) set theory forms a metatheory for ZFC. For example: MK proves Con(ZFC). In fact, Joel David Hamkins claims in his blog post "Kelly-Morse set theory implies Con(ZFC) ...
2
votes
2answers
127 views

Order in bijective-equivalent collections of proper classes in set-theory

We work in the set theory NBG (with local choice, but not global choice), because if there is global choice, every proper class is well-ordered, so that every proper class is bijective with the class ...
5
votes
1answer
252 views

Injection of the proper class of ordinals in every proper class

Is it possible to prove in the set theory NBG (with local choice but without global choice) that the proper class of ordinals injects in every proper class ?
2
votes
1answer
150 views

Injection of every proper class in the ordinal class

Is it possible in the set theory NBG (with local choice but without global choice) that every proper class injects in the proper class of ordinals ?
3
votes
0answers
105 views

Cores of infinite graphs

Let $\kappa$ be a cardinal and let $\textrm{Grph}(\kappa)$ be the set of graphs $G = (V,E)$ such that $V \subseteq \kappa$ and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We ...
7
votes
0answers
258 views

Two questions about universally measurable sets

I have two questions about universally measurable sets: (1) Is there a universally measurable set of reals which does not have the Baire property? (2) Is there a universally measurable set of reals ...