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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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 ...
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1answer
151 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 ...
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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 ...
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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 ...
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1answer
605 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 ...
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1answer
251 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 $...
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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-...
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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 ...
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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....
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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 ...
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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?
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263 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 ...
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1answer
345 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 $(n, n')$ ...
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176 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 $M\Vdash \phi$...
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2answers
321 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 ...
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119 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 ...
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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: $V_0=\emptyset$, $V_{\alpha+1}=P(V_{\alpha})$, $V_{\alpha}=\bigcup_{\beta<\alpha}V_\beta$, ...
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2answers
446 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 $...
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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 $A$ ...
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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
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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.
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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 ...
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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 $\...
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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 ...
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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 ...
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1answer
93 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) \...
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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 ...
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234 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 = \...
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131 views

Questions about generalizations of the Principle of Dependent Choices

I've encountered two different generalizations of the Principle of Dependent Choices (DC) to initial ordinals $\kappa>\omega$. First, some notation. By $A\prec B$, I indicate that $A$ is injectable ...
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86 views

Proof that recursive construction is equivalent to dependent choice

I am trying to prove the following: In Zf, dependent choice (DC) and recursive construction (RCW) are equivalent where and (RCW): Can you tell me if my proof in one direction is correct and ...
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1answer
197 views

Why can countable recursive constructions not be done using countable choice?

Why can countable recursive constructions not be done using countable choice? For example, replace $Z$ by $\omega$ in the following theorem Why can't one prove it using countable choice? (The proof ...
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274 views

Trying understand a move in Cohen's proof of the independence of the continuum hypothesis

I've read a few different presentations of Cohen's proof. All of them (that I've seen) eventually make a move where a Cartesian product (call it CP) between the (M-form of) $\aleph_2$ and $\aleph_0$ ...
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Every Number is Describable?

Loosely, a number is describable if it can be unambiguously defined by a finite string over a finite alphabet. Numbers such as $\frac{1}{3}$, $\sum_{n=0}^\infty \frac{1}{n!}$, and "the ratio of ...
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1answer
258 views

Is Zermelo set theory finitely axiomatizable?

I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?
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1answer
76 views

A formula that is upwards absolute but not downwards absolute

I know that any existential formula is upwards absolute (an any universal formula is downwards absolute) but I was looking for an example of a formula that is upwards absolute but not downwards ...
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1answer
155 views

question on formulations of Generalized Continuum Hypothesis and Singular Cardinal Hypothesis

I hope this is not a silly question(well, not too silly, I hope). After all, a relevent question at a deeper level is already out there, even though it seems the solution is missing. Why not ...
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1answer
288 views

Adding Cohen reals one at a time

We know that if we start with a ctm $\mathbb{B}$ and force with the poset of finite functions from $\omega$ to $2$, we add a single Cohen real. We also know that if we force with the poset $\mathbb{P} ...
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4answers
1k views

What would happen if ZFC were found to be inconsistent?

If, one fine day, someone found a contradiction in ZFC (or even ZF), what implications would such an event have for mathematicians? Is there currently any backup axiomatic system on par with ZFC that ...
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325 views

What fragment of ZFC do we need to prove Zorn's lemma?

It is extremely well-known that Zorn's lemma is a theorem of ZFC. My interest is in a certain finitely-axiomatisable fragment of ZFC, sometimes called RZC (restricted Zermelo with choice) or ZBQC. The ...
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1answer
367 views

An exercise from Levy's Basic Set Theory (Exercise 3.17)

Exercise 3.17 on page 136 asks to prove the Milner-Rado paradox. Levy gives hints for two different proofs. I have no problem with the first one but I do not understand the ordering he mentions in ...
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1answer
79 views

Does $\beta \mathbb N$ embed into $\beta \mathbb N \setminus \mathbb N$?

Is there a clopen subset of $\beta \mathbb N \setminus \mathbb N$ homeomorphic to $\beta \mathbb N$? If so, is there any plausible description of any such a subspace?
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1answer
184 views

Always win without a winning strategy

On Page 141, Axiom of Choice, Herrlich(2006) Show that if in a game of the form $G(1, X_1, Y_1, A)$, the first player has no winning strategy, then the second player can always win, even ...
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205 views

Cardinality of the Union is less than the cardinality of the Cartesian product

Suppose I have two collections of sets, $\{A_i\}_{i \in I}$ and $\{B_i\}_{i \in I}$. It is given in the problem that $$\mathrm{card}(A_i) < \mathrm{card}(B_i) \text{ for all }i \in I $$ I want to ...
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453 views

Is there any generalization of the hyperarithmetical hierarchy using the analytical hierarchy to formulas belonging to third-order logic and above?

As I understand, hyperarithmetical sets are defined according to the analytical hierarchy, that is, second-order-logic formulas. There is a generalization of hyperarithmetic theory named α-recursion ...
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1answer
158 views

A countale partially ordered set that has an uncountable number of maximal chains

I'm looking for a countable set S with a partial order < that has an uncoubtable number of maximal chains. I had many ideas but non of then is correct (for example- S= natural numbers, "<" is ...
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1answer
399 views

Using Zorn's lemma show that $\mathbb R^+$ is the disjoint union of two sets closed under addition.

Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
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1answer
279 views

How to show the following equality of ordinal exponentiation: $n^{\omega^\omega}=\omega^{\omega^\omega}$?

Let $n > 1$ be a positive integer. Prove that $n^{\omega^\omega}=\omega^{\omega^\omega}$ I was able to prove that, since $ω$ is a limit ordinal, and $n > 1$, $$n^\omega= \bigcup_{\beta<\...
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1answer
264 views

Decreasing sequence of ordinals must stabilize

If $x_1, x_2, x_3, \ldots$ are ordinals such that $x_1\ge x_2\ge x_3\ge \ldots$, then there exists a positive integer $n$ such that $x_n=x_{n+1}=x_{n+2}=\ldots$. I think that this statement is true, ...
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1answer
120 views

If the von Neumann rank $x$ is less than that of $y$, is $x\in y$?

If $x$ and $y$ are sets such that $\newcommand{\rank}{\operatorname{rank}}\rank x < \rank y$ , then $x\in y$? I understand that generally if $\rank x < \rank y$, then it should be true, but ...
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2answers
327 views

The cofinality of $\aleph_{\omega\cdot9+3}$

I am studying for a test and I was able to find the cofinality 3 of the 4 ones given, but am having a lot of trouble with the 4th. the 3 first ones are: $\newcommand{\cf}{\operatorname{cf}}\cf(\...