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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31 views

Justification for ordinal arithmetic at limit ordinals

Wikipedia tells me that: $$\alpha + \lambda := \bigcup_{\beta < \lambda} \left ( \alpha + \beta \right ) $$ for a limit ordinal $\lambda$. Multiplication and exponentiation, as Wikipedia says, ...
2
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0answers
53 views

Supplementing FOL with the Härtig quantifier

"The Härtig quantifier captures a certain fairly large fragment of second-order logic and contributes to understanding higher-order logic." (p. 1154 ...
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0answers
32 views

Condition for an Ultrafilter to be Ramsey.

I read in the english Ultrafilter article on Wikipedia that one can prove that a non-principal ultrafilter $D$ on $\omega$ is a Ramsey ultrafilter if and only if for every coloration $c:[\omega]^2 ...
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1answer
48 views

Indiscernibles and colorations.

Let $L$ be a language, and $(X,\leq)$ be a total order contained in an $L$-structure $\frak{A}$. Now if we denote by $[X]^n$ the set of $n$-sized sequences in $X$ and consider a set $\Gamma$ of ...
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1answer
47 views

If $|A|<|B|$ does $B$ surject onto $\aleph(A)$?

After reading Proving existence of a surjection $2^{\aleph_0} \to \aleph_1$ without AC I became curious if there is a generalization to arbitrary cardinals. That is, if $\frak m<n$, does it follow ...
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1answer
39 views

P-names are comprehensible in the ground model

If my understanding of $\mathbb{P}$-names $\tau$ is correct, they must be comprehensible by someone living in the ground model $\mathcal{M}$. However, the value $\tau[G]$ of $\tau$ need not be ...
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1answer
45 views

How to show countability of $\omega^\omega$ or $\epsilon_0$ in ZF?

I know that with choice, the countable union of countable sets is countable, making $\omega^\omega$ and $\epsilon_0$ both countable. Can we show this without choice? E.g. in the case that $\omega_1$ ...
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1answer
123 views

Must $\mathfrak{sd} = \mathfrak{d}$?

This should be fun. Let $\mathfrak{sd}$ be the least cardinal $\kappa$ such that there exists $\langle s_i : i < \kappa \rangle$ satisfying the following. (1) Each $s_i: \omega^{\omega} \to ...
4
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1answer
86 views

How to prove that $\sf CH$ implies $2^{\aleph_0}=\aleph_1$

Of course, most of you will, upon reading the title, exclaim "But isn't that the definition of the continuum hypothesis?" So I need to be a little more careful about the exact definitions. Let ...
3
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0answers
48 views

Does “$(\exists f:A\twoheadrightarrow B)\implies(\exists f:B\hookrightarrow A)$” implies the axiom of choice? [duplicate]

Let $P$ denotes the property that if there exists a surjection from set $A$ to set $B$, then there exists an injection from $B$ to $A$. It's apparent that $P$ can be proved in ZFC. My question is that ...
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0answers
35 views

Intersection with the empty set in Principia Mathematica

In Whitehead and Russell's Principia Mathematica, "arithmetical class-sum" is defined as I have tried my best to decipher the dated notation in the book. It seems that the sum is defined as the ...
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0answers
126 views

ZF and weakly inaccessible cardinals

This question should probably have been asked 3 years ago (perhaps it has and was removed for some reason?) In 2011, Alexander Kiselev claimed to have proved in ZF that there are no weakly ...
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1answer
31 views

Question about $\text{non}(\mathsf{nonstat}_\gamma)$

A lemma in Kunen's (2011) set theory states that if $\gamma$ is a limit ordinal with $\kappa=\text{cf}(\gamma)>\omega$, then $\text{add}(\mathsf{nonstat}_\gamma)$ $=$ ...
4
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1answer
44 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 ...
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5answers
1k views

Are there any nontrivial examples of contradictions arising in non-foundational or applied math due to naive set theory?

I understand that naive set theory, whose axioms are extensionality and unrestricted comprehension, is inconsistent, due to paradoxes like Russell, Curry, Cantor, and Burali-Forti. But these all ...
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2answers
66 views

Set Theory : Unusual proof of Well Founded Closure in T Forster Logic, Induction and Sets [closed]

I am reading the Lemma 82 proof in T Forster Logic, Induction and Sets, but I can't understand the working in the red circled text below.
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2answers
153 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 ...
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2answers
62 views

The number of subsets of cardinality less than $\kappa$ of a cardinal $\kappa$ is $\kappa$

Let $L=\bigcup_{\alpha \in Ord} L_\alpha$ be Godel's constructible universe and thus $L \models GCH$. Let $\kappa$ be an infinite cardinal and $S:=\{A \subseteq \kappa : \#A < \kappa \}$. Is it ...
3
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2answers
289 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
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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 & = \{\} ...
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2answers
88 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 ...
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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 ...
6
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1answer
150 views

Dominating strategically $\omega_1$ reals

For a given $\kappa > \omega$, define the game $d(\kappa)$ that runs for $\omega$ stages as follows: At stage $n$, player I chooses a sequence of elements of $\omega$, $g_n$ of length $\kappa$, and ...
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1answer
108 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 ...
2
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1answer
74 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 ...
5
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1answer
74 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) ...
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2answers
67 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 ...
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1answer
333 views

Another way of extending the Banach-Tarski paradox?

This question is kinda a follow-up on Extending Banach-Tarski paradox? On a sphere, we can do all kinds of translations. We will, as usual, look to the translations that are a string of two ...
3
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1answer
103 views

Extending Banach-Tarski paradox?

I've learned the Banach-Tarski paradox as following: The points on the sphere (but not the fixed points) are drawn as a square grid, form each point there are three new directions plus the direction ...
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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 ...
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2answers
99 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, ...
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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 ...
1
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1answer
74 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 ...
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2answers
85 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 ...
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0answers
54 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 ...
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0answers
69 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 ...
3
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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 ...
6
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1answer
93 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 ...
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3answers
270 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 ...
5
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2answers
53 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 ...
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3answers
145 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 ...
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0answers
104 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 ...
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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 ...
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2answers
56 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 ...
4
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2answers
111 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 ...
4
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2answers
112 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 ...
6
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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: ...
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1answer
39 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
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1answer
88 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 ...
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1answer
60 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 $WF$ $\subset$ ...