This tag is for set theory topics typically studied at the advanced undergraduate or graduate level. These include cofinality, axioms of ZFC, axiom of choice, forcing, set-theoretic independence, large cardinals, models of set theory, ultrafilters, ultrapowers, constructible universe, inner model ...

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9
votes
2answers
155 views

Can $T$, $T+A$, and $T+\neg A$ all have different consistency strengths?

Let $T$ be a consistent theory, and let $A$ be a statement in the same language. Consider the three theories $T$ $T+A$ $T+\neg A$ Is it possible for them to be pairwise distinct in consistency ...
4
votes
1answer
45 views

Is there a simple formula for the cardinality of $\{A\subseteq\kappa\mid |A|\leq\lambda\}$ when $\lambda\leq\kappa$?

If $\lambda\leq\kappa$ are infinite cardinals, how many subsets of $\kappa$ of size $\lambda$ are there? And of size $\leq\lambda$? Is there some sort of explicite formula for this? The internet isn't ...
7
votes
4answers
692 views

number of infinite sets with different cardinalities

I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple? Thanks.
2
votes
1answer
76 views

Is there a 'largest' second-order categorical axiomatization of set theory, extended from ZFC2

While it's possible to obtain categorical second-order axiomatizations of set theory by extending ZFC2 with additional axioms (see [1]), these axioms seem somewhat arbitrary (e.g. adding an axiom that ...
4
votes
2answers
52 views

Ordered tuples of proper classes

From time to time I encounter notation like this: A triple $\langle \mathbf{No}, \mathrm{<}, b \rangle$ is a surreal number system if and only if ... The confusing part is that a proper class ...
3
votes
0answers
130 views

Is it known if “Homotopy type theory” (HTT) can consistently model objects beyond V?

I read the free book on HTT but could not find an answer to this question. If that were the case HTT would be stronger than ZFC+LC (for any choice of the Large Cardinal axiom). That would be ...
3
votes
2answers
294 views

Why is it important, that mathematics can be formalized in set theory?

Why is it important, that mathematics can be formalized in set theory? As one can read in the thread Are there areas of mathematics that cannot be formalized in set theory? Today known mathematical ...
2
votes
0answers
38 views

Is every mathematical object representable by sets? [duplicate]

I know that most mathematical objects can be represented by a complex structure of sets. For example one can use von Neumann ordinals for representing natural numbers: $$\begin{align} 0 & = \{\} ...
3
votes
2answers
473 views

On the definition of weakly compact cardinals

I am reading in Jech's Set Theory the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book. I ...
5
votes
1answer
864 views

Explain Zermelo–Fraenkel set theory in layman terms

What does Zermelo–Fraenkel set theory mean? According to Wikipedia, Zermelo–Fraenkel set theory is a set theory that is proposed to overcome issues in naive set theory. I really appreciate if ...
4
votes
2answers
90 views

Equivalence between different forms of the Axiom of Infinity

In Zermelo-Frankel set theory, the Axiom of Infinity is often stated as "There exists a set $X$ such that $\emptyset \in X$ and such that if $y\in X$ then $S^{+}_1(y)\in X$", where we take the ...
2
votes
2answers
197 views

Measure on Baire space

Inspired by the first parenthetical sentence of Joel's answer to this question, I have the following question: is there any useful notion of measurability in the Baire space $\omega^\omega$? Some ...
1
vote
1answer
42 views

Limiting the set of “constructible” properties, and loosening comprehension axiom

My historical understanding (which may very well be wrong) is that initially there was naive comprehension for set construction, which required no superset. Russell's Paradox came along and blew that ...
15
votes
3answers
810 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
0
votes
2answers
100 views

Cardinality and Concrete Mathematics

First, let me distinguish Concrete Mathematics ( Mathematic branches that studies fixed structures ) from Abstract Mathematics ( branches that study classes of structures, such as algebra , topology, ...
9
votes
5answers
432 views

When can ZFC be said to have been “born”?

The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I understood from it is that ZFC appeared after 1922. In what book or paper was ZFC first explicitly ...
7
votes
1answer
112 views

Diamond at singulars

I'm sure this is a silly question. Suppose $\lambda$ is a singular cardinal of uncountable cofinality. Then surely $\diamondsuit_\lambda$ must fail. But why? Just in case, let me specify that by ...
5
votes
1answer
76 views

Find a dense embedding from specific forcing poset to any countable forcing poset

I tried to prove this in the Kunen's set theory: Let $P$ be a countable non-atomic partial order. Show that there is a dense embedding from $T = \{p\in\operatorname{Fn}(\omega,\omega) ...
6
votes
1answer
96 views

Definability in $L(\omega_1)$

I'm trying to solve problem II.6.35 in Kunen's book, which asks to prove that if $V=L$, then the set $B$ of $\beta<\omega_1$ such that $L(\beta)\models ZF-P$ and every element of $L(\beta)$ is ...
2
votes
2answers
68 views

Are categorical second-order axiomatizations of set theory inconsistent due to the axiom of replacement

Second-order ZFC is nearly categorical, except that it does not determine the 'height' of the cumulative hierarchy (intuitively speaking). However, additional axioms can be added to second-order ZFC ...
0
votes
2answers
48 views

Definition of $\mathbb{P}$-name with index number

I've just started studying forcing. Currently, I am struggling to understand what is a $\mathbb{P}$-name in the first chapter in the Shelah's book page 6 ...
0
votes
1answer
107 views

Breaking the AC barrier using Kelley-Morse set theory

In her blog post Variants of Kelley-Morse set theory, Prof. Gitman proves that every model of the the common version of Kelley-Morse set theory ($\mathsf{KM}$) is a model of the Wikipedia version of ...
1
vote
1answer
76 views

In P. Cohen's models (or others) may we have $\neg\mathsf{AC}+\mathsf{CH}$? May we have $\neg \mathsf{AC} + \neg \mathsf{CH}$?

I know we have the consistency of $\mathsf{ZF} + \mathsf{AC} + \mathsf{GCH}$, $\mathsf{ZF} + \neg \mathsf{AC}$, and $\mathsf{ZF} + \mathsf{AC} + \neg \mathsf{GCH}$. What about $\mathsf{ZF} + \neg ...
1
vote
0answers
48 views

Choice of a skeleton

Suppose we are in presence of a strong enough axiom of choice (e.g., choice for conglomerates). I know that any category has a skeleton, but I would like to know if I can choose a skeleton which ...
5
votes
2answers
90 views

What questions are independent from the axiom of constructibility?

Wikipedia gives a list of statements true in L which would be true also for set theory if the axiom of constructibility (V=L) holds. However I wonder about the converse: Are there any important open ...
3
votes
0answers
184 views

Can Egoroff's Theorem be Strengthened for A Sequence of Smooth Functions?

I have posted this question on MO, and I'll raise the question here in a more concrete way. To simplify the situaton, we assume the measure space we are dealing with right now is a closed interval $I$ ...
6
votes
2answers
73 views

There is no sequence $X_n$ such that $\forall n(\mathscr P(X_{n+1})\preceq X_n)$.

I'm working on the following exercise from Kunen: Define, in ZF without the axiom of regularity, $\aleph(X)=\{\alpha: > \exists f \in \, ^\alpha X(f \text{ is } 1-1\}$. Show: ...
2
votes
1answer
94 views

Justification for the Axiom of Union

Is the Axiom of Union included in $\sf ZFC $ because one cannot construct the union of two sets using the Axiom Schema of Specification?
1
vote
0answers
73 views

First order logic and first order set theory

(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order ...
1
vote
0answers
55 views

First order logic and Second order logic: a question regarding domains

(1) I understood that in the language of first order logic quantification cannot be over sets (there are no predicate variables that can be bound by quantifiers), whereas the language of second-order ...
3
votes
1answer
79 views

Exercise 23.10 of Jech's book

I would need a hint for Exercise 23.10 of Jech's Set Theory (third edition), which states: If $\kappa$ is a regular cardinal, then there exists a strongly almost disjoint family ...
63
votes
7answers
13k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
10
votes
3answers
273 views

Does every cover have an irredundant subcover?

While composing an answer for this question, I got troubled by a technical point. I wanted to assert the existence of an irredundant subcover of a given open cover, but realized I'm not sure how to ...
4
votes
2answers
112 views

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined?

How should the relativized Kleene pointclass $\Sigma^1_1(A)$ be defined, when $A$ is a set of reals ($A \subset \omega^\omega$)? I assume that there is a standard definition, but I can't seem to find ...
1
vote
3answers
147 views

Isn't the axiom of determinacy inconsistent with ZF? What am I overlooking?

I'm sure there's something I'm missing here; probably a naive confusion of mathematics with metamathematics. Regardless, I've come up with what looks to me like a proof that (first-order) ZF+AD is ...
4
votes
2answers
113 views

A Ramsey-type result for families of subsets

Let $S$ be a set of cardinality $\aleph_1$. Consider the directed family $\mathcal{C}$ (here directed means directed with respect to the inclusion) of all countably infinite subsets of $S$. Suppose ...
5
votes
2answers
54 views

show that there is a $\mathbb P$-name $\sigma$ such that $M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$

I want to show that there is a $\mathbb P$-name $\sigma$ such that for every $G$ a generic filter we will have $$M[G]\vDash \exists x\phi (x) \iff M[G]\vDash \phi (\sigma [G])$$ while $\phi (x)$ is a ...
1
vote
0answers
105 views

How can nontrivial elementary embeddings of the universe to some inner model be surjective?

Consider $\kappa$ to the least measurable cardinal, or equivalently $\kappa$ is the critical point for an elementary embedding $j:V \rightarrow M$ from the universe $V$ to an inner model $M$ (critical ...
4
votes
0answers
50 views

Adding bounded quantifiers increases complexity in $L$

I'm reading Devlin's Constructability to learn about $L$. Following the proof that $L_\alpha$ has a $\Sigma_1$ skolem function for limit $\alpha>\omega$ (II.6.5), the author notes Notice that ...
1
vote
2answers
57 views

The class of all functions between classes (NBG)

Is it possible in NBG (von Neumann-Bernays-Gödel set theory) to construct the class of all functions $X \to Y$ between two (proper) classes $X,Y$? I guess that this does not work. In the special case ...
0
votes
1answer
46 views

Using ordinal arithmetic calculate the following ordinal numbers

(ω + 1) x ω (ω + 1) x 2 For Question #2, I can simplify to the point where I get (ω + (ω + 1)), but I'm not sure how to proceed from there
3
votes
1answer
96 views

Hilbert–Bernays provability conditions

Let "provability formula" ${\rm Prf}(x, y)$ written in the manner that provability operator $\square A$ defined as $\exists x\ {\rm Prf}(x, \overline A)$ satisfying Hilbert–Bernays axioms: If ZF ...
1
vote
1answer
64 views

Exercise (3) of Chapter III from Kunen's Set Theory: Intro to Independence Proofs

I'm a little stumped on the aforementioned question. It's statement is as follows: Let $M$ Be any class such that $\forall$x (x $\subset$ $M$ $\rightarrow$ x $\in$ $M$). Show that $WF$ $\subset$ ...
158
votes
0answers
5k views

Does the open mapping theorem imply the Baire category theorem?

A nice observation by C.E. Blair1, 2, 3 shows that the Baire category theorem for complete metric spaces is equivalent to the axiom of (countable) dependent choice. On the other hand, the three ...
1
vote
2answers
88 views

Every II-finite set is III-finite

I need some help proving that if a set $X$ is II-finite then it is III-finite, i.e. if every non-empty family of subsets of $X$ which is linearly ordered by inclusion has a maximal element under ...
3
votes
1answer
167 views

how many infinite cardinals are smaller than $\aleph$?

What can be said in general (if we do not assume the continuum hypothesis) about the cardinality of the set of all infinite cardinals smaller than $\aleph$?
2
votes
1answer
51 views

forcing over a dense set.

I want to show that given a countable transitive model $M$ a notion of forcing $\mathbb P$ and a generic filter $G$, then forcing over a dense set of $\mathbb P$, $D$ is just as forcing over $\mathbb ...
80
votes
16answers
8k views

Why did mathematicians take Russell's paradox seriously?

Though I've understood the logic behind's Russell's paradox for long enough, I have to admit I've never really understood why mathematicians and mathematical historians thought it so important. Most ...
7
votes
0answers
106 views

Order-preserving map of regressive functions on $\omega_1$

I posted the following question a year ago on MO. It did receive some attention, but the answer there remains incomplete (as discussed in some detail in its comments). Meanwhile I got the impression ...
5
votes
1answer
94 views

No Borel well-order of the reals?

I'm told there is no Borel well-order of the reals (in ZFC). I'm told, in fact, that this is because of Borel determinacy. However, this is usually a vague handwave of the form (a) take the usual ...