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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2
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1answer
147 views

Assuming $(GCH)$, strongly inaccessible and weakly inaccessible coincide

My book says "... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..." In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...
1
vote
1answer
113 views

Explicit choice functions for finite sets in topological spaces

When dealing with finite nonempty sets of real or natural numbers it is always possible to define a explicit choice function, that choose one (arbitrary, but well defined) element out of that set: ...
1
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2answers
184 views

Defining strong limit cardinals in $ZF$

I do not understand the following passage/footnote in the book I am currently reading: An initial ordinal $\lambda$ is called a strong limit cardinal if $2^\kappa < \lambda$ for every $\kappa ...
1
vote
1answer
152 views

Proving Axiom Schema of Replacement holds in $H_\lambda$

I think I proved the following, can you tell me if my proof is correct? Exercise 23: Show that if $\lambda$ is a regular infinite cardinal then $\langle H_\lambda , \overline{\in} \rangle$ satisfies ...
1
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0answers
67 views

A sheaf of cumulative hierarchies

I've recently read a paper about sheaf forcing in which a sheaf of cumulative hierarchies was defined (defintion 5.3 on page 30). The same object is described in this English paper (defintion 3.1 on ...
3
votes
3answers
279 views

How to break power set for non-transitive models?

Apparently, $\varphi (x,y) : y = P(x)$ where $P$ denotes the power set is not absolute for transitive models. We call a formula $\varphi(v_1, \dots v_n)$ in $L_S$ absolute for a class $\mathbf X$ ...
0
votes
3answers
169 views

Extensionality of a relation and Axiom of Extensionality

What's the difference between the Axiom of Extensionality $(A1)$ and an extensional relation? The definitions are $(A1) \forall x,y ( x = y \leftrightarrow \forall z ( z \in x \leftrightarrow z \in ...
2
votes
3answers
231 views

Is there a bounded-quantifier formula in ZF without Foundation that defines the naturals?

Does there exist a first-order set-theoretic formula $\phi(x)$ using only bounded quantifiers such that, in ZF without assuming the Axiom of Foundation/Regularity, $\phi(x)$ is true if & only if $...
1
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4answers
174 views

A question about standard models

As I understand it a standard model is a model in which the relation is the $\in$ on the actual set of sets constituting the model. (i) Hence theories that aren't in the language of set $L_S$ ...
4
votes
2answers
333 views

A non-well-ordered set where principle of transfinite induction holds?

A theorem in my textbook says: Let $(A, < )$ be a totally ordered set. Set A has a least element and principle of transfinite induction holds in A if and only if A is well ordered. I understand ...
10
votes
4answers
581 views

How to introduce advanced set-theoretical objects to philosophy students?

First, I apologize if MSE is a bad fit for this question. I'm going to give a course as the last course of "elementary set theory" (the previous courses were not given by me). I planed to introduce ...
3
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0answers
185 views

A question about a passage in Just/Weese's Basic Set Theory

I have a question regarding the following passage in Just/Weese (p 192), $\mathbf{V}$ denoting the cumulative hierarchy: "But consider the following situation: Where in "($\beta$)" does the $\...
5
votes
1answer
214 views

Books on Axiom of Dependent Choices?

Are there any books about the axiom of dependent choice? There are books about the axiom of choice, e.g. Herrlich or Jech. But I can't seem to find any on the axiom of dependent choices. As far as I ...
2
votes
1answer
105 views

SDR for an infinite set of sets

Let $F$ be a set of nonempty sets. A set $R\subseteq\cup F$ is said to be a system of distinct representatives (SDR) of for every $B\in F$ there exists unique $x\in R$ such that $x\in B$ and $\forall ...
7
votes
2answers
362 views

Relationships between AC, Ultrafilter Lemma/BPIT, Non-measurable sets

How is it possible to reconcile the following... In 1970, Solovay constructed Solovay's model, which shows that it is consistent with standard set theory, excluding uncountable choice, that all ...
6
votes
1answer
441 views

Bourbaki Proof of Zorn's Lemma in Lang's Algebra

Serge Lang in his book Algebra has a nice appendix on set theory at the end of the book. In particular, in paragraph 2, pp. 881-884 he provides a proof of Zorn's Lemma from "other properties of sets ...
6
votes
2answers
217 views

Prove that the statement implies the Axiom of Choice

Prove that the following statement implies the Axiom of Choice: Let $ C $ is a set (of sets) and $ B $ is a set such that for all $ c \in C $, there exists a $ b \in B $ such that $ b \not\in c $. ...
4
votes
3answers
377 views

Defining cardinals without choice

According to Wikipedia if we assume AC we define a cardinals number to be an ordinal that is not in bijection with any smaller ordinal. Without AC, one takes the cardinality of a set $X$ to be the ...
0
votes
1answer
88 views

What is a conservative/intersective function?

I can't find any information on what a conservative or intersective function is. ...
12
votes
1answer
658 views

Strength of the statement “$\mathbb R$ has a Hamel basis over $\mathbb Q$”

I would like to know if there are "interesting" equivalences to the statement "$\mathbb R$ has a Hamel basis over $\mathbb Q$". I am not interested in more general statements, like "every vector space ...
2
votes
1answer
229 views

Bijection Definition Clarification

I was reading A Walk Through Combinatorics: An Introduction to Enumeration And Graph Theory (Miklós Bóna) and came across a definition for bijection: Let $X$ and $Y$ be two finite sets, and let $f ...
3
votes
2answers
178 views

Axiom of Choice and Ascending Chain Condition

Matsumura, in his "Commutative Ring Theory" p. 14 proves that "A partially ordered set $\Gamma$ satisfies the ascending chain condition $\Leftrightarrow$ every nonempty subset of $\Gamma$ has a ...
1
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2answers
249 views

Axiom of Choice - Naive Counterexample [duplicate]

Possible Duplicate: Axiom of choice question I know there is a lot of discussion on the axiom of choice and, in fact, I attended once a lecture on it, but I still cannot understand the ...
2
votes
1answer
150 views

Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
3
votes
1answer
91 views

If $X=\bigcup_{n\in{\mathbb{N}}}\kappa^n$, is it provable from $ZF$ that $|X|=\kappa$?

My question is the following, if $\kappa$ is an aleph and $F$ is the set of all finite sequences in $\kappa$, then the fact that $|F|=\kappa$ is provable from $ZF$?. This can be proven from $ZF$ for ...
8
votes
1answer
280 views

If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$?

Greets This is from exercise 3.4 of Thomas Jech's "Set Theory", stated: "Show that $\Gamma(\alpha\times{\alpha})\leq{\omega^{\alpha}}$. Thus $\Gamma(\alpha\times{\alpha})=\alpha$ for all ...
22
votes
1answer
603 views

Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
2
votes
1answer
250 views

Does every infinite well-ordered set have an initial segment isomorphic to $\mathbb N$?

This is an exercise question from Chapter 2 of A Course in Galois Theory by D.J.H. Garling: Suppose that $(A,\leq)$ is an infinite well-ordered set. Show that there is a unique element $a$ such that $...
5
votes
0answers
105 views

Cardinality of $\mathbb R\setminus\mathbb Q$ without AC [duplicate]

Possible Duplicate: Does $k+\aleph_0=\mathfrak{c}$ imply $k=\mathfrak{c}$ without the Axiom of Choice? Showing that $\mathbb{R}$ and $\mathbb{R}\backslash\mathbb{Q}$ are equinumerous using Cantor-...
1
vote
1answer
195 views

Problem 1.30 (The Maximum Principle is equivalent to the axiom of choice) (J.Bell, Boolean-valued Models and Independence Proofs, 3rd edition)

Problem 1.30 (The Maximum Principle is equivalent to the axiom of choice) (i) Let $\{a_i : i ∈ I\} ⊆ B$ satisfy $\bigvee_{i∈I} a_i = 1$. A partition of unity $\{b_i : i ∈ I\}$ in B is called a ...
0
votes
0answers
77 views

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition)

Problem 1.45 (Boolean-valued ordinals) (J.Bell, boolean-valued models and indipendence proof, 3rd edition) (i) Show that, for any formula φ(x), $ [[ ∃α .φ(α)]] = \bigvee_α [[φ( \hat{α})]]$ and $[[∀α....
8
votes
2answers
377 views

Does negation of Axiom of Choice imply symmetry?

It seems that every construction of a model in which the Axiom of Choice fails involves some kind of symmetry. Is there an example of a construction of a model where AC fails but no argument involving ...
4
votes
3answers
209 views

What's the thorny issue on: “If all $S\in \ell $ are nonempty, does it follow that $\prod_{S\in \ell} S$ is nonempty? when $\ell$ is infinite?”

I'm reading Paolo Aluffi's ALGEBRA, Chapter 0. Here he proposes that there's a thorny issue: What is this thorny issue?
4
votes
2answers
262 views

How to construct $\{\{\{…\}\}\}$ in ZF without axiom of foundation

I used to think naively the construction is straightforward, which is, if we add one layer innermost each time, then we could have one that corresponds to $\omega$ in Neumann's representation, which ...
3
votes
1answer
340 views

What does AFA , “Every graph has a unique decoration” mean?

All stuff is from Page 1 - 6, Non-Well-Founded Sets, Peter Aczel, which can be found here. Here a GRAPH will consist of a set of NODES and a set of EDGES, each edge being an ordered pair $(n, n')$ ...
4
votes
2answers
175 views

Modal theorems valid in a set theory model

This is the question i would like to discuss, properly stated. Given a model $M$ for a collection of set theory axioms (ZFC, for example), list all basic modal formulas $\phi$ such that $M\Vdash \phi$...
2
votes
2answers
318 views

What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?

The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
1
vote
2answers
116 views

Infinite union of internal sets not internal

This is homework problem. I need to give an example of internal sets $A_n \subset \mathbb{R}^*$ for which the union $\bigcup _{n=i}^\infty A_n $ is not internal. Also, this whole internal set ...
2
votes
2answers
158 views

Why $\mathrm{rank}(x^y) < \alpha+\omega$, if $x$, $y$ have rank $\le$ $\alpha$?

This question is from Set Theory, Jech(2006), Page 70, 6.5. Rank function is defined as on Page 64: $V_0=\emptyset$, $V_{\alpha+1}=P(V_{\alpha})$, $V_{\alpha}=\bigcup_{\beta<\alpha}V_\beta$, ...
2
votes
2answers
440 views

Using Zorn's Lemma

Background: I am trying to use Zorn's lemma to show the existence of ultrafilters containing an arbitrary filter on a set $X$. My argument goes as follows: Let $\mathcal{F}_0$ be a filter on $X$. If $...
3
votes
1answer
44 views

Proof of every cofinal subclass of $\mathbf{ON}$ is proper

Can you please tell me if my proof of the following claim is correct? Thank you! Claim: Every cofinal subclass of $\mathbf{ON}$ is proper. Proof: Let $A \subset \mathbf{ON}$ be cofinal. Assume $A$ ...
1
vote
1answer
81 views

Operations and relations

To what extent do operations and relations overlap? Is there some more general structure that encompasses both of these things? Thanks
8
votes
2answers
1k views

Simple and intuitive example for Zorns Lemma

Do you know any example that demonstrated Zorn's Lemma simple and intuitive? I know some applications of it, but in these proof I find the application of Zorn Lemma not very intuitive.
20
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6answers
5k views

Textbooks on set theory

I want to do a survey of textbooks in set theory. Amazon returns 3582 books for the keywords "set theory". A small somewhat random selection with number of references in Google scholar is the ...
14
votes
2answers
500 views

What's the difference between saying that there is no cardinal between $\aleph_0$ and $\aleph_1$ as opposed to saying that…

... $\aleph_1$ is the immediate successor of $\aleph_0$? I was reading the wiki article on $\aleph_1$ where a distinction is made between the two. If there's isn't a cardinal between $\aleph_1$ and $\...
2
votes
1answer
57 views

Axiom of choice on function [duplicate]

Possible Duplicate: Using a choice function to find an inverse for $F\colon A\to P(B)$ Let $F:A \rightarrow \mathcal P (B)$ be arbitary functions which covers $B$. Use AC to show there is a ...
0
votes
1answer
51 views

How to Understand Collection Principle in the Form of First-order Predicate Calulus

On Page 65, Set Theory, Jech(2006), Collection Principle is formulated as follows: $\forall{X}\exists{Y}(\forall{u}\in{X})[\exists{v} \psi(u, v, p) \to(\exists{v\in{Y}}) \psi(u, v, p)]$ ($p$ is ...
3
votes
1answer
91 views

Ordinal arithmetic washing line

Let $\eta$ be the order type of $\mathbb{Q}$. I'm trying to calculate $(1+ \eta) \cdot (\eta + 1)$ and $(\eta + 1) \cdot (1+ \eta)$. So for the first one I'm thinking that you just do this $(1+\eta) \...
6
votes
2answers
149 views

Question about trees and generalizing the Principle of Dependent Choices.

One form of the Principle of Dependent Choices is that for any tree $T$ of height $\omega$ such that every node of $T$ has a successor, there is a branch of $T$ of length $\omega$. In this post, I ...
3
votes
2answers
222 views

Ordinal arithmetic

So I'm having trouble understanding ordinal arithmetic. So if you have $\omega = \bigcup \{ n | n\in\mathbb{N}\}$ How is this defined $\omega^2$ as in the notes I'm reading it has $\omega^2 = \...