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Questions tagged [set-theory]

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 relations, Boolean-valued models, embeddings, orders, relations, transfinite recursion, set theory as a foundation of mathematics, the philosophy of set theory.

5
votes
1answer
53 views

Injection into a proper class and choice without regularity

In $\sf ZF$, we have that the axiom of choice is equivalent to: For all sets $X$, and for all proper classes $Y$, $X$ inject into $Y$ and For all sets $X$, and for all proper classes $Y$, $Y$ ...
5
votes
1answer
260 views

Function on the set of limit countable ordinals

Let $\Lambda$ be the set of all countable limit ordinals. Does there exist an injective function $f:\Lambda\to\omega_1$ with the properties: $\forall \lambda\in\Lambda:~f(\lambda)<\lambda$ $\...
-3
votes
0answers
247 views

Some questions regarding “Elementary Embeddings and Correctness”

Consider the following, from Eskew's (and Friedman's, presumably, though Dr. Eskew's name is under the title) page-long abstract, "Elementary Embeddings and Correctness": Kunen's inconsistency ...
2
votes
1answer
82 views

Non-existence of countable base of arbitrary ultrafilter

$\scr{B}$ is the base of a nonprincipal ultrafilter $\scr{U}$ on $\omega$ if 1. $\forall U,V\in\mathscr{B}~\exists T\in\mathscr{B}:~T\subset U\cap V$, 2. $\forall X\in\mathscr{U}~\exists U\in\mathscr{...
0
votes
2answers
92 views

Existence of $\alpha$-sequence of infinitesimal sequences with $\alpha>\omega_1$

We are in ZFC & CH. Given family $Y=\{y_\alpha\}_{\alpha<\omega_1}$ of infinitesimal $\omega$-sequences (i.e. $\lim_{n\to\infty}y_{\alpha n}=0$) of rational numbers with the property: $\forall\...
-2
votes
0answers
112 views

Cardinals from ordinals in ZFC [on hold]

Did someone define the cardinal numbers as the equivalence classes from the relation of bijection on the class On of all ordinal numbers ? Gérard Lang
11
votes
2answers
482 views

What is the consistency strength of weak Vopenka's principle?

Weak Vopěnka's principle says that the opposite of the category of ordinals cannot be fully embedded in any locally presentable category. Recall that one form of Vopěnka's principle says that the ...
21
votes
4answers
5k views

What would be some major consequences of the inconsistency of ZFC?

Update (21st April, 2019). Removed the reference / initial trigger behind my question (please see comment thread below for the reasons). Am retaining, of course, the actual question, noted both in the ...
6
votes
0answers
136 views

Complexity of the set of closed subsets of an analytic set

Let $X$ be a compact Polish space and $K(X)$ the hyperspace of closed subspaces of $X$ with the Vietoris/Hausdorff metric topology. Question: If $A$ is an analytic subset of $X$, what is the ...
3
votes
1answer
135 views

K-analytic spaces whose any compact subset is countable

A regular topological space $X$ is called $\bullet$ analytic if $X$ is a continuous image of a Polish space; $\bullet$ $K$-analytic if $X$ is the image of a Polish space $P$ under an upper ...
14
votes
0answers
416 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? Some motivation: If $\delta$ is a Woodin cardinal, then it remains ...
7
votes
0answers
127 views

Making the precaliber number bigger than all Knaster numbers

Write $\mathfrak m_k$ for the Martin's axiom number for $k$-Knaster, i.e., for the smallest size of a family of dense subsets of some $k$-Knaster poset for which there is no generic filter. (A poset ...
5
votes
1answer
216 views

Intersection of iterated powerset in NFU

I am interested in the existence of the set $\bigcap_{n \in \mathbb{N}} \mathcal{P}^n(x)$ for any given set $x$, in the context of NFU (New Foundations with Urelements). It seems to me that the ...
2
votes
2answers
456 views

Who first discovered the concept corresponding to the symbol of class comprehension?

Who first discovered the concept corresponding to the symbol of class comprehension {x/𝛗}used today in set theory ?
12
votes
1answer
385 views

End-extending cardinals

Let us say a cardinal $\kappa$ end-extending if there is a function $F : V_\kappa^{<\omega} \to V_\kappa$ such that: (a) If $M \subseteq V_\kappa$ is closed under $F$, then $M \prec V_\kappa$. (b) ...
14
votes
1answer
542 views

Do we know the consistency strength of the Singular Cardinal Hypothesis failing on an uncountable cofinality?

Suppose that $\kappa$ is a strong limit cardinal. The singular cardinal hypothesis states $2^\kappa=\kappa^+$. We know that the failure of SCH requires large cardinals, and in fact is equiconsistent ...
45
votes
4answers
4k views

The origin of sets?

The history of set theory from Cantor to modern times is well documented. However, the origin of the idea of sets is not so clear. A few years ago, I taught a set theory course and I did some digging ...
216
votes
31answers
30k views

What are some reasonable-sounding statements that are independent of ZFC?

Every now and then, somebody will tell me about a question. When I start thinking about it, they say, "actually, it's undecidable in ZFC." For example, suppose A is an abelian group such that every ...
-3
votes
0answers
335 views

Can we get rid of the primitive symbol $V$ in Ackermann's set theory without increment in consistency strength?

EDIT: Ackermann had presented his theory with a new primitive added to the language of set theory that is the "set-hood" primitive one place predicate symbol $\mathcal M$ or in common equivalent ...
61
votes
12answers
74k views

What practical applications does set theory have?

I am a non-mathematician. I'm reading up on set theory. It's fascinating, but I wonder if it's found any 'real-world' applications yet. For instance, in high school when we were learning the ...
3
votes
1answer
124 views

Radin forcing preserving large cardinals

I'm wondering if there are any known result for the maximum large cardinal strength which can be preserved by Radin forcing? For instance, with any large cardinal hypothesis in the ground model, can ...
9
votes
2answers
881 views

Terminology about trees

In set theory, a tree is usually defined as a partial order such that the set of elements below any given one is well-ordered. I am interested in the class of partial orders $P$ such that for every $...
46
votes
8answers
5k views

Why should we believe in the axiom of regularity?

Today I started reading Maddy's Believing the axioms. As I knew beforehand, it includes some discussion of ZFC axioms. However, I really hoped for a more extensive discussion of axiom of foundation/...
13
votes
2answers
600 views

Set-theoretical foundations of Mathematics with only bounded quantifiers

It seems that outside of researchers in Mathematical Logic, mathematicians use almost exclusively bounded quantifiers instead of unbounded quantifiers. In fact, I haven't observed any other practice ...
3
votes
0answers
208 views

Is there a standard meaning for this notation for ordinals in set theory?

In the paper ``Berkeley Cardinals and the Structure of $L(V_{\delta+1})$", by Raffaella Cutolo, a use is made of the notation $\alpha < < \beta$ for a pair of ordinals $\alpha, \beta$, without ...
117
votes
11answers
20k views

Solutions to the Continuum Hypothesis

Related MO questions: What is the general opinion on the Generalized Continuum Hypothesis? ; Completion of ZFC ; Complete resolutions of GCH; How far wrong could the Continuum Hypothesis be?; When was ...
-2
votes
0answers
105 views

Can every first order theory have a finitely axiomatizable conservative extension in this sense?

Note: This is an edit of the previous question. By $T_2$ conservatively extends $T_1$ if and only if: There exists a function $F$ such that $T_2$ extends $T_1$ through $F$; and for every function $G$...
9
votes
4answers
1k views

Very Large Cardinal Axioms and Continuum Hypothesis

Are very large cardinal axioms like $I_0$, $I_1$, $I_2$ consistent with $CH$ and $GCH$?
70
votes
19answers
8k views

Injectivity implies surjectivity

In some circumstances, an injective (one-to-one) map is automatically surjective (onto). For example, Set theory An injective map between two finite sets with the same cardinality is surjective. ...
5
votes
1answer
205 views

Radin generics from iterated ultrapowers

Let $u$ be a measure sequence derived from some embedding $j:V\rightarrow M$. I heard that from iterating $j$ is possible to define a generic filter for some Radin forcing $\mathbb{R}_w$. Specifically,...
75
votes
20answers
10k views

Proofs of the uncountability of the reals.

Recently, I learnt in my analysis class the proof of the uncountability of the reals via the Nested Interval Theorem. At first, I was excited to see a variant proof (as it did not use the diagonal ...
7
votes
3answers
534 views

Characterizing “bounded” distributivity in terms of dense open sets

Prikry forcing is a well-known example of a forcing which adds an $\omega$-sequence to some cardinal $\kappa$, but does not add bounded subsets to $\kappa$. There are other examples of forcings which ...
2
votes
1answer
176 views

Proving that family of sets has non-empty intersection

Let's say I have an object which can be viewed as family of sets $\mathfrak{S} \subseteq 2^S$, and I want to prove that its intersection is non-empty. What is known already: $S$ is set of measurable ...
5
votes
1answer
230 views

Number of ultrafilters in an extender

Assume GCH. Suppose $j: V\to M$ is an elementary embedding such that $\mathrm{crit}(j)=\kappa$, ${}^\kappa M \subset M$, $M\supset V_{\kappa+2}$. We can assume $j$ is defined from some extender of ...
8
votes
1answer
395 views

Stationarity and Fodor's lemma for a (nice) poset?

The notion of a stationary set is peculiar in that it applies to subsets of certain very particular posets -- ordinals or powersets. At least to a non-set-theorist, the situation seems to beg for the ...
0
votes
3answers
837 views

Sets = structured sets without structure

Motivation There is presumably no single and widely accepted formal definition of structured sets = sets plus structure based on sets as primitive objects, but several approaches are around. See e.g. ...
3
votes
0answers
167 views

A consistent way to “decide” independent statements [closed]

Let us fix some mathematical theory T=T(0), such as ZFC. The aim is to develop algorithm A, which takes any statement S independent of T(0) as an input, and outputs axiom a(S) such that T(1)=T(0)+a(S) ...
2
votes
0answers
50 views

Cardinal characteristics of the ideal of $\sigma$-continuity of the Pawlikowski function

A function $f:X\to Y$ between topological spaces is called $\sigma$-continuous if there exists a countable cover $\mathcal C$ of $X$ such that for every $C\in\mathcal C$ the restriction $f{\...
9
votes
2answers
340 views

Small uncountable cardinals related to $\sigma$-continuity

A function $f:X\to Y$ is defined to be $\sigma$-continuous (resp. $\bar \sigma$-continuous) if there exists a countable (closed) cover $\mathcal C$ of $X$ such that the restriction $f{\restriction}C$ ...
7
votes
1answer
412 views

Cardinal characteristics of amorphous sets

In a universe where the continuum hypothesis ($CH$) fails we can ask about combinatorial cardinal characteristics of the continuum, but in a universe where $CH$ is true no such cardinals exist so this ...
1
vote
1answer
98 views

Some kind of idempotence of dense filter

In discussion of following questions question1, question2, question3 became clear (see definitions in question3 ) that for the Frechet filter $\mathcal{N}$ we have $\mathcal{N}\nsim\mathcal{N}\otimes\...
0
votes
0answers
95 views

How slowly can it takes for the Fibonacci terms in a partially permutative self-distributive algebra to stabilize?

A self-distributive algebra is a structure $(X,*)$ where $*$ is a binary operation that satisfies the identity $x*(y*z)=(x*y)*(x*z)$. Let $L,T:X^{2}\rightarrow X^{2}$ be the operations where $L(x,y)=(...
1
vote
0answers
126 views

Is the variety of ternary self-distributive algebras generated by its finite members?

A ternary self-distributive algebra is an algebra $(X,t)$ that satisfies the identity $$t(u,v,t(x,y,z))=t(t(u,v,x),t(u,v,y),t(u,v,z)).$$ Is the variety of ternary self-distributive algebras generated ...
2
votes
1answer
74 views

The trace of the filter on a big subset

Let $\scr{F}$ be free filter ($\cap\scr{F}=\emptyset$) on a countable set $X$ and $B\in\scr{F}$. We define the trace of $\scr{F}$ on $B$ as follows $\mathscr{F}_B=\{Y\cap B:~Y\in\scr{F}\}$. $\scr{F}$ ...
21
votes
1answer
702 views

Does “every” first-order theory have a finitely axiomatizable conservative extension?

I originally asked this question on math.stackexchange.com here. There's a famous theorem (due to Montague) that states that if $\sf ZFC$ is consistent then it cannot be finitely axiomatized. ...
1
vote
0answers
86 views

Two small uncountable cardinals related to Q-sets

A subset $A$ of the real line is called a Q-set if any subset of of $A$ is of type $F_\sigma$ in $A$. Let $\mathfrak q_0$ be the smallest cardinality of a subset $X\subset\mathbb R$ which is not a Q-...
5
votes
2answers
213 views

Ordered union of Borel sets

Let $\mathfrak{A}$ be an uncountable collection of Borel sets in $\mathbb{R}^d$ such that for any $A,B\in\mathfrak{A}$, either $A\subset B$ or $A\supset B$. Then is it necessarily true that the union $...
14
votes
0answers
380 views

A meager subgroup of the real line, which cannot be covered by countably many closed subsets of measure zero?

Is there a ZFC-example of a subgroup $H$ of the real line $\mathbb R$ such $H$ is meager, has zero Lebesgue measure, but cannot be covered by countably many closed subsets of measure zero in $\mathbb ...
0
votes
0answers
59 views

Existance of bijective function which maps tensor product of subsets of a selective ultrafilter into the ultrafilter

In the answer on this question Andreas Blass had shown that for any selective ultrafilter $\scr{U}$ on $\omega$ and for any free subfilter $\scr{F}\subset{U}$ doesn't exist bijection $\varphi:\omega^2\...
10
votes
0answers
382 views

Sunflower / $\Delta$-system lemma in a more general poset?

The sunflower lemma (or $\Delta$-system lemma) may be viewed as a statement about the poset $P_\omega(\omega_1)$, and the generalized sunflower lemma may be viewed as a statement about the poset $P_\...