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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3
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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 = \...
3
votes
2answers
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 ...
2
votes
1answer
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 ...
7
votes
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 ...
5
votes
2answers
273 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 $\aleph_0$ ...
13
votes
3answers
1k views

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 ...
7
votes
1answer
256 views

Is Zermelo set theory finitely axiomatizable?

I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?
2
votes
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 ...
2
votes
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 ...
4
votes
1answer
281 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} ...
12
votes
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 ...
7
votes
2answers
324 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 ...
4
votes
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 ...
6
votes
1answer
78 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?
4
votes
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 ...
8
votes
3answers
203 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 ...
8
votes
2answers
447 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 ...
4
votes
1answer
156 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 ...
6
votes
1answer
398 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.
1
vote
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<\...
1
vote
1answer
260 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, ...
3
votes
1answer
119 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 ...
3
votes
2answers
326 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(\...
3
votes
1answer
87 views

The product of the finite non-zero ordinals, and the product of the finite non-zero cardinals.

I am trying to study for a test I have on Set Theory, and after being given the practice test, I am having some big problems with simple things. One question I am having a lot of problems with is: ...
1
vote
0answers
75 views

Borel sets used in Model Theory

In Lemma 4.4.13 (page 161) of D.Marker- Model Theory book it is proved that $D(F,T)$ (the set of all possible F-diagrams of models of T) is a Borel set. Can someone explain to me the proof given a ...
9
votes
3answers
818 views

Curious facts about ordinal numbers

I have some notes about curious facts about ordinal numbers, for example that their addition is not commutative, multiplication is not distributive from the right hand side and that the exponent rule ...
2
votes
1answer
636 views

Countable infinity and the axiom of choice

Many of the states of affairs about infinite cardinalities and their size, depend on the axiom of choice. What would a comprehensive list be of the properties of the cardinality of the natural numbers ...
5
votes
1answer
348 views

Set theory like first-order theory of ordered pairs

The widely-accepted Kuratowski definition of an ordered pair is $(a,b):=\{\{a\},\{a,b\}\}$. This definition makes perfect sense in the context of first-order set theory. However, how would a first-...
3
votes
1answer
623 views

Ordinal non-commutative addition example

I know this is a newbie question, so please bare with me :) I'd like to prove within ZF axiomatic set theory that the addition of two ordinals is not commutative. In particular, I'd like to prove ...
3
votes
1answer
140 views

Question about cofinality of an ordinal

Let $\alpha$ an ordinal and $\langle\alpha_\xi\rangle$ a cofinal sequence of elements of $\alpha$. The length, $\gamma$, of this sequence is at least $\operatorname{cf}\alpha$ but can be equal to any ...
4
votes
1answer
492 views

Counterexamples to the continuum hypothesis

Assume the continuum hypothesis is false, and add that as an axiom to ZF set theory. How many cardinalities are between the rationals and the reals in this case? Only one? Infinitely many? Countably ...
1
vote
1answer
91 views

Equivalent definition of a closed set.

I try to prove the equivalence between "C is closed in an ordinal $\alpha$" and "each strictly increasing sequence of elements of C of length $< cf\alpha$ converge in C". For $\Rightarrow$, it's ...
7
votes
2answers
448 views

A question about the cardinality of the set of all the bijections from $M$ to itself

$M$ is a set.We denote the set of all the bijections from $M$ to itself by $K$.Is the cardinality of $K$ larger than the cardinality of $M$?
14
votes
1answer
668 views

Set of All Groups

In my undergraduate Group Theory class, while discussing the impossibility of equivalence relations on groups, my professor said that the set of all groups does not exist. Are there any group ...
6
votes
1answer
337 views

Set theorist as a physicist or physicist as a set theorist?

I'm majoring physics, but really interested in mathematics. I liked physics since it was really beautiful to have an analysis on a nature with mathematical tool. However, the more i study, the more ...
1
vote
1answer
108 views

How can a non-dense, well-ordered set like a long ray be uncountable?

How can a non-dense, well-ordered set like a long ray be uncountable? If it's a set of an uncountable number of [0,1) line segments laid end to end, shouldn't there be a bijective function between the ...
1
vote
2answers
213 views

Inhabited versus nonempty sets - disasters without excluded middle?

The attached screenshot, from Goldblatt's Topoi, shows at a glance the distinction between the constructive concept of inhabited set versus that of the classical nonempty set. These definitions ...
2
votes
2answers
81 views

Reaching elements with finite applications of the successor relation in well-ordered sets

Given an well-ordered class $(A,\leq)$ we want to proof that every element $a \in A$ can be reached by applying the successor relation on a the smallest element of $A$ or on an limes element of $A$ ...
1
vote
0answers
118 views

Questions about $\sigma$-algebra, algebra and topology

I know the definitions of $\sigma$-algebra, algebra and topology, but why countable/finite union, as in $\sigma$-algebra/algebra, and finite intersection, arbitrary union as in topology? What inspire ...
2
votes
2answers
360 views

Hall's theorem vs Axiom of Choice?

From Wikipedia Let $S$ be a family of finite sets, where the family may contain an infinite number of sets and the individual sets may be repeated multiple times. A transversal for $S$ is ...
9
votes
4answers
294 views

Hyperreal field extension

In non-standard analysis, assuming the continuum hypothesis, the field of hyperreals $\mathbb{R}^*$ is a field extension of $\mathbb{R}$. What can you say about this field extension? Is it ...
0
votes
2answers
118 views

Injective function from set of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$

I'm looking for an injective function from the set $A$ of all functions $f: \mathbb{R} \to \mathbb{R}$ to $\mathcal{P}(\mathbb{R})$. Any hints? I think the other direction is easy: An injective ...
7
votes
1answer
608 views

Can an infinite cardinal number be a sum of two smaller cardinal number?

Let $\kappa$ be an infinite cardinal number. My question is whether there are $\lambda$ and $\mu$ such that both $<\kappa$ but $\lambda+\mu=\kappa$? If AC holds, then the answer is definitely NO....
2
votes
1answer
174 views

Is there a “more powerful” form of set theory that would enable this?

I was wondering about this. In Conway's "On Numbers and Games", he discusses the "surreal numbers", and in one point mentions that they are full of "gaps". That the surreal number line is riddled with ...
6
votes
1answer
228 views

Exercise on partially ordered sets from Kunen's *Set Theory*

This problem is Exercise (F4) from Chapter VII of Kunen's text. Apparently, it is a result of Tarski. It goes as follows: Let $ \mathbb{P} $ be a partial order. Define $ \text{c.c.}(\mathbb{P}) $ ...
4
votes
1answer
84 views

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular?

A cardinal $\kappa$ such that $2^{\lambda}<\kappa$ for all $\lambda<\kappa$ is regular? I would appreciate very much an answer
2
votes
1answer
90 views

A Simpler Characterization of Inductive Definitions?

While reading appendix A of John Harrison's "Handbook of Practical Logic and Automated Reasoning" a somewhat advanced theorem is appealed to as a prerequisite for characterizing when an inductive ...
3
votes
1answer
120 views

An infinite cardinal agrees with all its well-orders on sets of full size.

Suppose $\kappa$ is an infinite von Neumann cardinal (well ordered by $\in$), and take ${<}$ a well-order on $\kappa$. Does there necessarily exists a subset $X\subset\kappa$ of full size (in ...