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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7
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1answer
115 views

Existence of a bijection without AC

Let $A$ and $B$ be subsets of a set $X$ and let $f:A\rightarrow X\setminus A$ and $g:B\rightarrow X\setminus B$ be bijections. Is it possible to show without AC that there is a bijection $h:A\...
3
votes
3answers
375 views

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$)

If a metric space has the limit point property, is it separable? (ZF + AC$_\omega$) I'm struggling with this problem for a week. I'm talking about this in Metric space. Here's the part of argument ...
6
votes
2answers
167 views

Sum of cardinals without AC

Let $A$ and $B$ be infinite sets. To show $|A\cup B|=\max\{|A|,|B|\}$ we need AC. Now let us assume $|A|<|B|$. Can we show $|A\cup B|=|B|$ without AC?
0
votes
1answer
351 views

Is cardinality a total order? Is AC necessary? [duplicate]

Possible Duplicate: Is the class of cardinals totally ordered? Intuitively, it seems like for any sets $A,B$ either $\lvert A\rvert\leq \lvert B\rvert$ or $\lvert B \rvert \leq \lvert A\rvert$....
10
votes
1answer
323 views

Shelah Cardinals and Modern Set Theory

Do Shelah cardinals play an essential role in any modern set theory results or was the concept basically made obsolete by Woodin cardinals?
2
votes
1answer
1k views

Cardinality of cartesian square

Given an infinite set $A$ - does the cardinality of $A$ equal to the cardinality of $A^2$?
1
vote
1answer
108 views

(ZF) Dedekind infinite + Limit Point Compact ⇒ Separable

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF) Yesterday, i posted this question and got an answer that 'Limit Point Compact⇒Separable' is unprovable ...
4
votes
1answer
123 views

Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$

This is an exercise from Kunen's book. Show that for $n, m \in \omega$, the ordinal and cardinal exponentiations $n^m$ are equal. What I've tried: I want to prove by using induction on $m$. ...
5
votes
2answers
416 views

If every infinite subset has a limit point in a metric space $X$, then $X$ is separable (in ZF)

I can prove this in ZFC, but don't know how to prove this in ZF. Following is the argument of this in ZFC. Fix $0<r\in \mathbb{R}$ and $x_0\in X$. Let $A_i = \{x\in X\mid d(x,x_j)\ge r,\, j<i\}$...
2
votes
1answer
280 views

Play a slot machine for uncountable number of times

There is a slot machine. You insert one coin, it destroy the coin and return countable number of coins. Then you can pick any one of the returned coin and put in the slot machine again. Formally, ...
4
votes
2answers
231 views

Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an epsilon number

Firstly, I give the definition of the epsilon number: $\alpha$ is called an epsilon number iff $\omega^\alpha=\alpha$. Show that if $\kappa$ is an uncountable cardinal, then $\kappa$ is an ...
5
votes
2answers
1k views

some elementary questions about cardinality

I know little about set theory and while reading some Algebra proof I had difficulty on some details. So my questions are : If $X$ is an infinite set and $Y$ is the set of all finite subsets of $X$, ...
5
votes
2answers
333 views

What interpretations exists for the singleton of the singleton of the empty set $\{\{\varnothing\}\}$ and its singleton $\{\{\{\varnothing\}\}\}$?

What I know I know from Peano's axioms that the empty set is equivalent to the natural number $0$ and that the singleton of the empty set is equivalent to the natural number $1$. ( http://www....
5
votes
2answers
323 views

Equivalent characterizations of ordinals of the form $\omega^\delta$

Let $\alpha$ be a limit ordinal. Show the following are equivalent: $\forall \beta, \gamma<\alpha (\beta+\gamma<\alpha)$ $\forall \beta<\alpha(\beta+\alpha=\alpha)$ $\forall X\subset \alpha(...
2
votes
1answer
253 views

How do I prove the existence of infinite union in ZFC?

Given an infinite set of sets A - how can I prove in ZFC that the union of all the elements of A exists?
1
vote
1answer
127 views

How to show the two inequalities?

This is an exercise from Kunen's book. Show that $\alpha < \beta$ implies that $\gamma+\alpha<\gamma+\beta$ and $\alpha+\gamma\le\beta+\gamma$. Given an example to show that the "$\le$" cannot ...
4
votes
1answer
517 views

Is a countable product of compact intervals in $\mathbf R$ compact (without using the AC)?

Let $\{I_n=[a_n,b_n]\}_{n\in\mathbf N}$ be a countable collection of closed, bounded intervals in $\mathbf R$. Is the infinite Cartesian product $$\prod_{n=1}^\infty I_n$$ compact without using the ...
6
votes
1answer
1k views

Proving that the set of algebraic numbers is countable without AC

A complex number $z$ is said to be algebraic if there is a finite collection of integers $\{a_i\}_{i\in n+1}$, not all zero, such that $a_0z^n + … + a_n = 0$. Then can I prove the set of all ...
3
votes
1answer
710 views

What does the continuum hypothesis imply?

Are there any fundamental/interesting results that are a consequence of assuming the continuum hypothesis as an additional axiom? I'm sorry if this question was already asked. I'm also sorry if there ...
3
votes
3answers
117 views

Does the Cartesian product get smaller if I use fewer sets?

My introduction into Axiom of Choice has been kind of confusing (Zorn's lemma) for the start, so it took me some time to realize it's nothing but to say The non-empty product of non-empty sets is non-...
1
vote
1answer
87 views

Name for a set which has an order?

As we all know, a set is a collection of elements which have no particular order and no multiplicity. So what do you call a construct which does store its elements in a specific order? What is the ...
5
votes
1answer
490 views

Definition of Cantor Set without AC

You can see the original text that i thought AC is used here; From Walter Rudin: Principles of Mathematical Analysis, 3rd ed., ISBN 0-07-054235-X, p.41-42. 2.44 The Cantor set The set which we ...
23
votes
1answer
922 views

Universally measurable sets of $\mathbb{R}^2$

$$\text{Is }{{\cal B}(\mathbb{R}^2})^u={{\cal B}(\mathbb{R}})^u\times {{\cal B}(\mathbb{R}})^u\,?\tag1$$ Is the $\sigma$-algebra of universally measurable sets on $\mathbb{R}^2$ equal to the product ...
2
votes
2answers
62 views

Cocountable fibers

Let $C$ be an uncountable set. Can we construct a set $A \subseteq C^2$ such that it has a cocountable number of cocountable horizontal fibers, and a cocountable number of countable vertical fibers?
3
votes
2answers
325 views

True Statement in ZF is true statement in ZFC?

How do we know an $ \aleph_1 $ exists at all? Here, you can see that $\aleph_2≦2^{\aleph_0}$, which is a contradiction to CH in ZFC. So if 'true statements in ZF is true in ZFC' is true, CH must be ...
5
votes
1answer
251 views

Stronger than ZF, weaker than ZFC

Can you please name axiom system that is strictly weaker than ZFC and strictly stronger than ZF? (such as DC, AC$_\omega$) I searched for it but i could only find these two. If there are statements ...
3
votes
1answer
140 views

Which infinite cardinals can be defined using partition relations?

Several types of infinite cardinals are easily defined in terms of partition relations. For instance, if $\kappa > \omega$ then $\kappa$ is weakly compact if $\kappa$ satisfies $\kappa\to(\kappa)...
2
votes
1answer
123 views

An inequality in a proof of Kunen's Inconsistency

This may be a silly question, but I keep coming back to it. Let $j:V\prec M$ be a non-trivial elementary embedding with $M$ a transitive class and $\kappa$ the critical point of $j$. Define the ...
1
vote
0answers
98 views

Infinitesimals and infinite elements among the transseries

In the quest for extensions of $\mathbb{R}$ and $\mathbb{C}$ that contains infinitesimals, infinities (and even more exotic beasts like $\omega - 1$ and $\sqrt{\omega}$) I came across the theory of ...
2
votes
1answer
142 views

A characteristic of intersection with cartesian product

Fix some binary relation $f$. Does there necessarily exist a set $C$ such that $(x\times x)\cap f\ne \varnothing \Leftrightarrow x\cap C\ne \varnothing$ for all sets $x$?
10
votes
1answer
536 views

Is the axiom of choice needed to show that $a^2=a$?

A comment on this answer states that choice is needed for the statement that $a^2=a$ for all infinite cardinals $a$. In Thomas Jech's Set Theory (3rd edition), his theorem 3.5 proves this statement ...
10
votes
3answers
500 views

An order type $\tau$ equal to its power $\tau^n, n>2$

In this question we are concerned only with linear (aka total) order types. By a cardinality of an order type we understand a cardinality of an instance of this type, which obviously does not depend ...
0
votes
1answer
197 views

$<$ on a preorder is a strict partial order

Definition: Suppose $X$ is a preorder. Define $x < y$ as $x \le y$ and $y \not\le x$ for each $x, y \in X$. Question: Show that this gives a strict partial order on $X$.
10
votes
1answer
396 views

Linearly ordered sets “somewhat similar” to $\mathbb{Q}$

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{\eta} + \pmb{1}$ - order type of $\mathbb{Q}\cap(0, 1]$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. Let's say that a linear ...
16
votes
2answers
374 views

Is $(\pmb{1} + \pmb{\eta})\cdot\pmb{\omega_1} = \pmb{1} + \pmb{\eta}\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
10
votes
1answer
191 views

Is $\pmb{\eta}\cdot\pmb{\omega_1} = (\pmb{\eta} + \pmb{1})\cdot\pmb{\omega_1}$?

$\pmb{\eta}$ - order type of $\mathbb{Q}$. $\pmb{1}$ - order type of a singleton set. $\pmb{\omega_0}$ - order type of $\mathbb{N}$. $\pmb{\omega_1}$ - order type of the first uncountable ordinal. ...
4
votes
2answers
161 views

Is every order isomorphism of sets induced by a bijection?

Let $f$ is an order isomorphism $\mathscr{P}A \rightarrow \mathscr{P}B$ (where $A$ and $B$ are some sets, the order is set-inclusion $\subseteq$). Is it true that it always exist a bijection $F: A \...
10
votes
1answer
369 views

Question about whether axiom of choice is needed in this proof

Do I need axiom of choice in this proof here? I think not: at each step we choose one element from a set $N - \langle g_1, \dots, g_k \rangle $. So while there is indeed a countable number of sets ...
2
votes
1answer
163 views

How does one go to prove two forcing extensions are equivalent?

I want to know how does someone usually go, that is, what is the canonical way to go proving that having two notions of forcing $P$ and $Q$, the respective generic extensions $P[G]$ and $Q[H]$ are ...
1
vote
1answer
291 views

Unprovable statements in ZF [duplicate]

Possible Duplicate: Advantage of accepting the axiom of choice Advantage of accepting non-measurable sets As you all know, Banach-Tarski paradox is solely a consequence of Axiom of Choice, ...
2
votes
1answer
103 views

$\kappa$-complete, $\lambda$-saturated ideal equivalence

More or less the follow-up to this question, I asked a few days ago. Now I need to show the equivalence to the seemingly weaker version of $S(\kappa,\lambda,\mathbb{I})$ - the version where there is ...
3
votes
2answers
190 views

Equivalent condition for a nonprincipal ultrafilter to be Ramsey

I came across this problem in my study of set theory and am not quite sure how to approach it. A solution or starting point would be welcome. Prove that if $\mathcal U$ is a nonprincipal ultrafilter ...
5
votes
3answers
405 views

pseudo numbers and surreal numbers

A surreal number $\{x_L\|x_R\} \in No$ is a number when for all $\xi\in x_L$ and all $\eta \in x_R$ we have $\eta > \xi$. All the things $\{x_L\|x_R\}$ which are not of that form are called "pseudo-...
3
votes
3answers
825 views

Advantage of accepting non-measurable sets

What would be the advantage of accepting non-measurable sets? I personally feel that non-measurable sets only exist because of infamous Banach-Tarski paradox...
3
votes
1answer
184 views

Additive inverse

Let $F$ be the set of $\alpha\subset \mathbb{Q}$ with following properties. (I) $\alpha ≠ \emptyset$ and $\alpha ≠ \mathbb{Q}$ (II) $p\in \alpha$ and $q<p$ ⇒ $q\in \alpha$ (Notice that it's ...
2
votes
1answer
254 views

Preorders, chains, cartesian products, and lexicographical order

Definitions: A preorder on a set $X$ is a binary relation $\leq$ on $X$ which is reflexive and transitive. A preordered set $(X, \leq)$ is a set equipped with a preorder.... Where confusion cannot ...
7
votes
2answers
657 views

Looking for a 'second course' in logic and set theory (forcing, large cardinals…)

I'm a recent graduate and will likely be out of the maths business for now - but there are a few things that I'd still really like to learn about - forcing and large cardinals being two of them. My ...
2
votes
1answer
151 views

$\kappa$-complete, $\lambda$-saturated ideal properties

Kunen, II.56. Having trouble proving the properties of the following: The definition: $S(\kappa,\lambda,\mathbb{I})$ is the statement that $\kappa > \omega$ and $\mathbb{I}$ is a $\kappa$-complete ...
4
votes
1answer
200 views

A maximal system of hyperreal numbers

Let $( \mathbb{R}^\mathbb{N}/\mathcal{U} )_{\mathcal{U}\in\beta\mathbb{N}}$ be the set of all the hyperreal number systems, does there exist a set $\mathbb{X}%$ and embeddings $i_{\mathcal{U}}: \...
4
votes
2answers
165 views

$<$ in a preorder

The author of the book I am studying defines $<$ for a poset as If $x, y \in X$, where $X$ is a poset, then we shall write $x < y$ to mean that $x \le y$ and $x \ne y$. From this, I can ...