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Questions tagged [set-theory]

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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24 views

Classes in Set Theory

Im having a problem with the first rule for class usage enunciated in a book on set theory. ("Basic Set Theory" by Azriel Levy) T Its the one where a predicate infers some other predicate. It is not ...
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1answer
33 views

How strong are weak choice principles?

For this purposes of this question, a weak choice principle $W$ is a statement for which the following three statements hold $ZFC$ proves $W$ $W$ is independent of $ZF$ $ZF+W$ does not prove $C$ ...
2
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1answer
34 views

The element relation

I am studying set theory and I have the following question: Are all the mathematical objects which we can quantify over, in essence, sets? I ask you this also because I wish to know if, when we say ...
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1answer
31 views

Proof of MM implies non-stationary ideal on $\aleph_1$ is $\aleph_2$ saturated.

I am trying to understand the proof of thm 37.16 of Jech on page 687. I don't understand the first 4 lines, why does that suffice to proof the theorem? I don't see how they are related. It sais the ...
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1answer
87 views

Proving Hausdorff maximality principle without Choice

I have heard that it is possible to prove a variant of the Hausdorff Maximality Principle without the axiom of choice. This is called "Hausdorff Maximality Principle for well-ordered partial orders" ...
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1answer
74 views

Cardinality of an uncountable union of countably infinite sets?

Let $I$ be an uncountable directed set and $T = \{E_{\alpha}:\alpha \in I\}$ denote a collection of Countably infinite sets with $E_{\alpha} \subset E_{\beta}$ whenever $\alpha \le \beta$. Also $T$ ...
2
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1answer
51 views

Class Absoluteness on $V$ with ZF-Models

There's a bit about models and absoluteness in particular that confuses me (I think my question is related to Noah Schweber's answer to Why cumulative hierarchy of Sets is not model of ZF). In Kunen's ...
2
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1answer
46 views

Choicelike consequence of compactness

Could someone please elaborate on the proof of Corollary 3.2.? Why does one need a total order *on $F$* if one is just interested in well-ordering each $F_x$ (in order to choose one element)? Also, a ...
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0answers
54 views

Set-theoretic worries about functor transformation from $\text{Id}$ to $V\mapsto V^*$

Upon reading the wikipedia article for natural transformations of functors, I stumbled across the section on the dual of a vector space. This is not a question about why the transformation from the ...
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2answers
24 views

Bijection between collection C and proper class PC makes C a proper class?

If there exists a bijection between a collection $C$ and a proper class $PC$, is $C$ necessarily a proper class as well? I've read and have been told by math professors that the answer is yes, but ...
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1answer
64 views

Ex.1.7 Jech “Set Theory”. (The shortest proof)

There is an exercise on page 14 in the Tomas Jech's "Set Theory": Every nonempty $X \subseteq \mathbb{N}$ has an $\in$-minimal element. [Pick $n \in X$ and look at $X \cap n$.] and there is a ...
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0answers
28 views

Injective module and transfinite induction

There is a proposition that: Each $R$-module $A$ can be embedded into an exact sequence $$0\to A\to Q\to N\to 0$$ where $Q$ is injective. The proof eventually requires transfinite induction. The idea ...
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2answers
49 views

Understanding the proof of $V=L \rightarrow \Diamond$

I am trying to understand the proof that $\Diamond$ holds in the constructible universe. I am following Kunen's Set Theory, where the proof is on pages 230-231 (the more recent, 2011 edition). I am ...
3
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1answer
50 views

ZF(C): Model or Inner Model

I am quite confused about models of ZF(C) set theory; in particular the wording that is used frequently. I have seen many cases of statements such as "assume $V$ satisfies ZFC" where $V$ is the class $...
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1answer
49 views

Could there be an $\omega_1^{CK}$th hyperoperation?

If addition is the first hyperoperation, multiplication is the second, and the $(\alpha+1)$th hyperoperation is repeated occurrences of the $\alpha$th one. Is it possible for a limit ordinal (for ...
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1answer
36 views
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1answer
107 views

Is $\omega + \omega^2 = \omega^2$ true?

Just a simple question. Is $\omega + \omega^2$ equal to $\omega^2$, I’ve just been thinking about it and if that’s false and it equals $\omega^2+\omega$ then we could define a set of countable ...
2
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2answers
85 views

Löwenheim-Skolem and proper class models of ZFC. [duplicate]

Let $N$ be a proper class model of ZFC and $x \subset N$ a set. Show that there is a set $y \in N$ such that $x \subset y$. If $x \subset N$, I think that by the downward's part of Löwenheim-Skolem, ...
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2answers
31 views

Is $F ↾ a$ a set?

If $F$ is a class function and $a$ is a set, is $F ↾ a$ a set? Note: It appears in Transfinite induction
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1answer
57 views

Why are there continuum many nowhere dense subsets of $\mathbb R$?

I am able to see why the closed nowhere dense subsets of $\mathbb R$ are equinumerous with $\mathbb R$: Every closed nowhere dense subset of $\mathbb R$ is the boundary of an open set (namely its ...
0
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0answers
62 views

How does countable collapse work?

As I’ve started delving deeper into the world of infinite ordinals, I’ve started seeing large, recursive, countable ordinals expressed in the form: $\varepsilon_{\kappa+1}$ where kappa is a large ...
1
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1answer
43 views

How do I proceed to prove $\tau<\alpha\cdot\beta\iff \tau=\alpha\cdot\eta+\zeta$ for a unique $\eta<\beta,\zeta<\alpha$?

Let $\alpha,\beta,\gamma,\tau$ be ordinals. Then $\tau<\alpha\cdot\beta\iff \tau=\alpha\cdot\eta+\zeta$ for a unique $\eta<\beta,\zeta<\alpha$ $\tau<\alpha^\beta\alpha^\gamma\...
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0answers
84 views

Set theoretic issues in the definition of a site in Stacks Project

I've been learning about sites from the Stacks Project, which is generally very precise in its terminology, but I've found some of their conventions very confusing in this part. Their definition of a ...
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0answers
59 views

In NBG set theory, if I have a class with a bijection to a proper class, is that class necessarily a proper class?

In other words, is a bijection with a proper class a sufficient condition for a class to be a proper class? The question just arose as I was learning about ordinals and cardinals and the associated ...
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1answer
48 views

Can the function $S(\alpha)$ have a fixed point ordinal?

I feel like there is no fixed point of the successor function, as for any ordinal $\alpha$, the ordinal $\alpha \cup \{\alpha\}$ must be greater. Though maybe I’m wrong and some ordinals with ...
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0answers
60 views

Creating a 'sequence' of subsets using the subset axiom, then using induction to prove they are all equal

Setup: $(N,\sigma)$ satisfies Peano axioms, i.e. $N \equiv ω$. Using the subset axiom we define, for each $k \in N$, $F_k = \{x \in N \; | \; \phi_k(x) \}$. Next we define $D = \{d \in N \; | \; ...
3
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0answers
53 views

Prikry's Forcing using Rowbottom's theorem

Rowbottom's theorem says: if $\kappa$ is a measurable cardinal and $U$ is a normal, nonprincipal and $\kappa$-complete ultrafilter over $\kappa$, then for every $f : [\kappa]^{< \omega} \to \gamma$,...
5
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0answers
94 views

Why is the existence of large cardinals believed to be true?

I am in the middle of watching a video of a presentation given by W. Woodin about the continuum hypothesis. This is not really something I know about, but I am confused by one of the slides (at 22:23 ...
6
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1answer
84 views

Is the étale site a small category?

Consider the étale site $X_{ét}$ of a scheme $X$. As a category, this is the collection of all étale schemes over $X$. Now, is this a set (i.e., is the étale site a small category)? If $X=Spec\ k$, ...
0
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1answer
60 views

Models of extensionality and comprehension without union

I want to find a model that satisfy comprehension and extensionality axiom but that it does not satisfy union axiom. I think that $M:=\{a,b,c,d,e,g\}$ with $\in^M=\{(a,b),(a,c),(b,e),(c,d),(e,g),(d,g)\...
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4answers
2k views

Why can't we prove consistency of ZFC like we can for PA?

this might be a silly question, but I was wondering: PA cannot prove its consistency by the incompleteness theorems, but we can "step outside" and exhibit a model of it, namely $\mathbb{N}$, so we ...
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5answers
56 views

Showing $\mathbb{R}^2$ has a Hamel basis using Zorn's lemma?

To get some intuition for Zorn's lemma I want to use it explicitly in the proof of the following theorem in the case when $X = \mathbb{R}^2$: Every vector space $X \neq \{ 0\}$ has a Hamel basis. ...
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1answer
38 views

Preservation of cardinality under ordinal exponentiation.

In Kenneth Kunen's The Foundations of Mathematics, he presents the following exercise and hint. Exercise I.11.34 Ordinal arithmetic doesn't raise cardinality. That is, assume that $\alpha$, $\beta$ ...
2
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1answer
81 views

Book on set theory

I'm looking for a rigorous book on naive set theory that is heavy on proofs (with maybe some problem sets of proofs much like spivak's books for example) and points out what the flaws of naive set ...
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0answers
82 views

Is Epsilon induction a theorem schema of Zermelo set theory?

The proof that axiom of foundation implies Epsilon induction in $ZF$ that I know of relies on existence of Transitive closures for all sets, but Zermelo set theory doesn't prove those, so is it the ...
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1answer
38 views

minimum cardinal for a set of sets

Let A be a set of sets. Has A an element with minimum cardinality? For example, if we consider A a finite set, the A has an element (which is not necessarily unique) with minimum cardinal. With many ...
2
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0answers
36 views

About $L \vDash GCH$ [duplicate]

In the notes I'm following to understand the proof that $L \vDash GCH$, there is this crucial following passage: "Let $A \subseteq X$. There is some $\alpha$ such that $A ∈ L_\alpha$ , and it ...
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1answer
35 views

Can we have a cardinal exponent of any number?

Can we have a cardinal such as ${1.5}^{\aleph_0}$ in the way that we have a powerset as $2^{\aleph_0}$, or is $2^{\aleph_0}$ just the notation we use, rather than actual exponentiation. Or is ...
0
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1answer
33 views

About $\in$-Recursion and Mostowski Collapse

The principle of $\in$-Recursion is a consequence of the regularity axiom. The formulation I know is: $\in$-Recursion: Let $G: V \to V$ be a class function, defined everywhere. Then there is a ...
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3answers
2k views

Where is the mistake in this “proof” of the inconsistency of ZFC?

This is a "proof" that ZFC is inconsistent, but I haven't found the mistake yet. Let $\{\varphi_n \colon n <\omega\}$ be an enumeration of all formulas in $L_{\in}$ with exactly one free ...
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0answers
26 views

About transfinite recursion

Transfinite recursion: If $F: V \to V$ is a class function, then there is a unique $G : ON \to V$ such that $G(\alpha) = F(G \restriction \alpha)$ for all ordinals $\alpha$. How from this theorem (...
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3answers
1k views

Why do we use von Neumann ordinals and not Zermelo ordinals?

Why do we use von Neumann ordinals, $$ 0 = \emptyset $$ $$ n+1 = n \cup \{n\} $$ and not Zermelo ordinals? $$ 0 = \emptyset $$ $$ n+1 = \{ n \} $$
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1answer
36 views

Is ordinal multiplication commutative?

DISCLAIMER I am sure this is a duplicate yet I coudn’t find an answer I was looking for so I’m probably going to ask and then eventually flag it as a duplicate. Is every instance of ordinal ...
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1answer
47 views

Create a well ordering for functions “being 0 near limits”

Let $\gamma$ be an ordinal and let $A$ = {$\alpha_i$|$i$ < $\gamma$ }. Define $P$ to be $\Pi${$\alpha_i$|$i$$\in$ $\gamma$ }. We can also view $P$ as the set of all the functions $f$ pointing from $...
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1answer
33 views

Measurable cardinals admit homogeneous set

I'm trying to prove that if $\kappa$ is a measurable cardinal with a normal ultrafilter $U$, then for every $f : [\kappa]^{< \omega} \to \gamma$, where $\gamma < \kappa$, there exists $H \in U$ ...
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1answer
24 views

Function for determining the Arity of an $n$-tuple

If we define an ordered pair in the standard way (the Kuratowski definition): $$(x_1,x_2)\equiv\{\{x_1\},\{x_1,x_2\}\}$$ We can recursively define $n$-tuples as an ordered pair of an $(n-1)$-tuple ...
1
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1answer
84 views

Is $(ω^ω)^ω$ equal to $ω^{(ω^2)}$? [closed]

I was just wondering if exponentiation rules such as in the title apply to transfinite ordinals... So things like: $\omega^\omega \cdot \omega^\omega = \omega^{\omega2}$
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0answers
56 views

Is there a bijection between $\Bbb{R}$ and $\Bbb{R} / \Bbb{Q}$? [duplicate]

$\Bbb{R} / \Bbb{Q}$ is a quotient set of $\Bbb{R}$ with the following equivalence relation $\sim$ : $$r \sim s \Longleftrightarrow r-s \in \Bbb{Q}$$ Then is there a bijection between $\Bbb{R}$ and ...
0
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1answer
53 views

Is it possible to prove Regularity with Transfinite Induction only?

Let us assume that we have only statement of transfinite induction. (And maybe some other well-know axioms) My question: "Is it possible to derive from it a regularity axiom as a theorem?". Some of ...
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1answer
74 views

Statement that implies axiom of choice

Consider the following statement: for any set $E$ and $G\subseteq E\times E$, you can to get a function $f:A\rightarrow B$ where $A=dom G$, $B=ran G$ and $f\subseteq G$. I want to show that this ...