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 theory, definability, infinite combinatorics, transfinite hierarchies; etc. More elementary questions should use the "elementary-set-theory" tag instead.

Filter by
Sorted by
Tagged with
-1
votes
0answers
15 views

Is this form of comprehension axiom consistent [closed]

Is there a problem by using this axiom: If something exists that wouldn't be A a and has the attribute c ,there is a set that contains just (all things that have the property c and wouldn't be A ,and ...
0
votes
1answer
39 views

Without the ZF axiom of regularity can any infinite sets be constructed?

Update with Direct Question Based on Asaf's comments, here is a related question: Prove that the mapping $n \mapsto n \cup \{n\}$ on the set $\Bbb N$ is injective without the axiom of foundation. ...
0
votes
1answer
28 views

Internet Site With Large Cardinal Concepts

I can't remember what was the name of an Internet site, which has a lot of ordinal, cardinal and large cardinal concepts, divided into 3 "staircases", or so I remember. Do you know what is ...
1
vote
1answer
83 views

Bourbaki's definition of function

I saw this definition and I got confused by it: "Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E d a variable element y of F is called a ...
0
votes
0answers
18 views

Is there an identity for $z$ if $z=2^z$ outside the Lambert-W function?

If $z=2^z$, this implies $z=-\frac{W_n(-\ln2)}{\ln2} \forall n \in \mathbb{Z}$ Taking into account the Lambert-W function having many special values surrounding the natural log, are there any unique ...
1
vote
0answers
13 views

Is Fodor's Lemma necessary for the $\omega_1$ train station puzzle?

A common homework problem with Fodor's lemma is to imagine a train network with stations $\langle S_\alpha\mid \alpha<\omega_1+1\rangle$. The train starts at $S_0$ and stops at each station. At ...
-2
votes
2answers
23 views

Countability of the set [closed]

Let $f$ be differentiable function from $\mathbb{R}$ to $\mathbb{R}$. Consider the set $$A_y=\{x \in \mathbb{R} : f(x)=y \}$$ I want to know whether $A_y$ is countable for each $y\in \mathbb{R}$. I ...
0
votes
0answers
28 views

What is a binary relation and transitive closure?

-Self studying 2Yr Undergraduate Discrete Math -Background: Calculus 3, probability 1, statistics 1, linear algebra 1, ODEs 1, stochastics 1 Hi, I'm studying relations and want to confirm I ...
0
votes
0answers
35 views

Find exercises infinite sets

Where can I find exercises with infinite sets (to check if they are countable or not)? I have searched the web and didn't find anything if you have a link i would appreciate it if you post it.
2
votes
0answers
58 views

How strong is “ZFC is 'very consistent'”?

Suppose we add a predicate "$x$ is very consistent" to $\textsf{ZFC}$, and we add the following axioms: $\textsf{ZFC}$ is very consistent If some theory $T$ is very consistent, then it is ...
0
votes
0answers
24 views

Could you guys correct my Peano's axiom induction principle translation in predicate logic?

I'm reading "classic set theory for guided independent study", and i'm studying the construction of natural numbers using peano's axioms, one of the axioms, called induction axiom, states: &...
2
votes
1answer
43 views

Find ordinals $\alpha,\beta$ such that $n^{\alpha}=\alpha$ and $\omega_1^{\beta}=\beta$

Find ordinals $\alpha$ such that (a) $n^{\alpha}=\alpha\; $ (b) $\omega_1^{\alpha}=\alpha$ On (a) I could verify that all ordinals of the form $\omega, \omega^{\omega},\omega^{\omega^\omega},\cdots$ ...
2
votes
1answer
38 views

Prove that $\xi+\omega= \omega \cdot \xi \ $ iff there is $\zeta$ satisfying $\ \xi=\omega^{\omega}\zeta+1$

I could prove the $\leftarrow$ implication. But assuming $\xi+\omega=\omega\cdot\xi \ $ I couldn't prove that $\xi$ is of the form $\xi=\omega^{\omega}\zeta+1$ for some ordinal $\zeta$. My attemptive ...
0
votes
1answer
27 views

The succersor operation is a injective function in the class of all the sets

Definition If $x$ is a set then the successor $S(x)$ of $x$ is the set $x\cup\{x\}$. Statement If $x$ and $y$ are sets then $S(x)=S(y)$ if and only if $x=y$. Cleraly if $x=y$ then $S(x)=S(y)$ but ...
2
votes
1answer
26 views

How many maximum binary pairs are possible in a Poset?

Answer : $n(n+1)/2$ Maximum binary pairs is possible iff the poset is a toset. A toset is reflexive so I don't have control over self loops like $(1,1),(2,2).$ They have to be there. My approach to ...
0
votes
1answer
71 views

How to prove that Von Neumann's set of natural numbers only contains 0 and it's successors?

I have been delving into ZFC set theory on my own for some time, and I am now learning about the possible definitions of $\mathbb N$ in this theory. I am lead to believe (and inclined to agree) that ...
-3
votes
1answer
23 views

Need help with partial and total orderings

Given $Y \subset X$ and $y\in Y$ and partial ordering $(X, \le)$ and total ordering $(Y,\le)$, i.e. $Y$ is a chain in $X$. Is it true in general that $\forall z\in X: [z\lt y \implies z\in Y]$? If ...
2
votes
1answer
45 views

Well-founded trees of any order

Suppose $T$ is a well-founded tree on $\mathbb{N}$, that is, a set of finite sequences of $\mathbb{N}$ closed under taking initial segments. Well-founded means that there is no infinite sequence $(x_n)...
1
vote
3answers
54 views

Enumerating open sets around elements of an uncountable set in topology - how do we justify it?

I was trying to prove for myself a basic result in point-set toplogy, namely that "Any compact subset of a Hausdorff space is also closed". To be precise we're taking compact set here to ...
0
votes
1answer
78 views

Need a reference for this result

I need a reference where I can find the proof of this Let $E$ be an infinite set and $G$ the set of maps from $E$ to $\mathbb{Z}$ with finite support, then there is a bijection between $E$ and $G$ ...
0
votes
0answers
30 views

Forcing: to find strictly stornger or weakly stronger condition?

In the $(*)$ below, I would like to understand whether "all we have to do is find $q>p$ such that ..." or as I would say $q\geq p$ such that... I have two questions about this: does this ...
0
votes
0answers
31 views

Can ordinal-stage replacement prove full replacement?

Working in first order logic with identity and membership: Define: $ f \text{ is ranking } \iff f \text{ is a function } \land \forall \alpha [ f(\alpha)= \bigcup (\{PP(\beta): \beta < \alpha \})]...
0
votes
2answers
50 views

Formally Introducing the Intersection Symbol into ZFC Set Theory

I am currently reading Lectures in Logic and Set Theory: Volume 2, Set Theory by Tourlakis. In the book, he formally introduces the power set notation, $\mathcal{P}(A)$, as well as union, $\bigcup A$, ...
0
votes
0answers
40 views

Minor difference in two proofs of Zorn's Lemma

I am working on a formal proof of Zorn's Lemma. I am modelling it on two informal proofs that do not require the use of ordinals. They differ slightly on, among other things, the definition of an ...
4
votes
1answer
59 views

Diagonal arguments for uncountable lists?

The diagonal argument is a general proof strategy that is used in many proofs in mathematics. I want to consider the following two examples: There is no enumeration of the real numbers. Because if ...
1
vote
1answer
111 views

Is a countable infinite union of $\Sigma_1$ sets is $\Sigma_1$?

I’m reading Kunen’s book Foundations of mathematics. My question is whether a countable union of $\Sigma_1$ sets in $HF$ is also $\Sigma_1$ or not. I wonder if we can think $\Sigma_1$ sets as open ...
4
votes
2answers
126 views

Intersection of Zermelo-Fraenkel universes containing all ordinals

I am reading my first Set Theory book (Set Theory and the Continuum Problem, Smullyan, 2010) and I find the subject pretty interesting. I am not a professional mathematician and can only study maths ...
2
votes
2answers
51 views

Separation properties of a topological space vs. characteristics of the continuum

Suppose that a set $X$ has a topology $\mathcal{T}$. Then $$\mathcal{T}\ \text{is T}_1\Rightarrow|\mathcal{T}|\geq|X|.$$ I'm curious about implications in the opposite direction, possibly assuming the ...
0
votes
1answer
112 views

Maths without the axiom of choice [duplicate]

The axiom of choice (AC) states that for any non-empty collection of non-empty sets, there exists a choice function defined on the collection. On the surface this seems very intuitive, and is easily ...
1
vote
1answer
48 views

Is there an intuitive way of justifying why the square of an infinite cardinal is itself?

By no means I am an expert in this subject, but I do have some knowledge of ZFC. While there are many proofs which are difficult to recollect, I feel like I have enough knowledge that if I am given a ...
1
vote
1answer
33 views

Subsets of $\omega_\alpha$ and $L_{\omega_{\alpha+1}}$

Under V=L, is it true that all subsets of $\omega_\alpha$ are contained in $L_{\omega_{\alpha+1}}$? If so, what would be the salient reason as to why its true? Reason for asking question: It seems to ...
0
votes
0answers
34 views

Graph coloring: general question

Assume we have a graph $ G=\langle V,E\rangle$ and there exists a coloring $ f\colon V \to A$ for a set $ A $ such that $ |A| = \alpha $. Is it true that for any set $ B $ such that $ |B|=\alpha $ ...
-5
votes
0answers
42 views

Gödel theorem, ZFC and consistency [closed]

Can it be shown that the theory of ZFC sets is consistent? Can we find a model for this theory? If yes then i think that we must do that under some assumptions? Is that correct thinking?
1
vote
1answer
54 views

Build an increasing $\omega^{\omega}$-sequence in real set

I'm trying to build an increasing $\omega^{\omega}$-sequence on the real set but I'm not sure about it. Here is my approach. Step $0$. consider the following $\omega$-sequence $\langle\overbrace{\;0,\;...
2
votes
1answer
70 views

Does this theme about extending ZF always work in both directions?

Suppose that we have all 3 of: $ZF + I \text { equi-interpretable } ZF + \neg I$ $ZF + I \vdash \neg \theta$ $ZF + \neg I + \theta \vdash Con(ZF + I)$ Can we always find a statement $\pi$ such ...
4
votes
0answers
91 views

Proof with transfinite induction

I'm trying to prove the following statement: Suppose that for every $r\in\mathbb{R}$ we are given a finite set $A_r\subseteq\mathbb{R}$ and that for any finite set $D\subseteq\mathbb{R} $, there ...
1
vote
0answers
28 views

Localization forcing adds a random reals

$\mathbf{Localization\,\,forcing}$, $\mathbb{Loc}$ consists of all pair $\langle \sigma, F\rangle$ such that $\sigma\in([\omega]^{<\omega})^{<\omega}$ is a finite sequence with $|\sigma(n)|=n$ ...
0
votes
1answer
34 views

***For any*** $f: B \to A$ with $(B \ne \emptyset)$, can a function $h:A \to B$ be constructed in such a way that $fhf = f$?

I've been stuck on this exercise for a while now, any help would be greatly appreciated. Here's my try. I constructed $h$ as a left inverse of $f$. For any function $f: B \to A$ with $B \ne \emptyset$...
0
votes
1answer
64 views

split perfect set into countable many pairwise disjoint perfect sets

We know that each perfect set can be written as a continuum many pairwise disjoint many perfect set. This will rely on the well know theorem which says: Let $X$ be a nonempty perfect polish space. ...
0
votes
0answers
21 views

is there a difference between the original and the following modified axiom of extensionality? [duplicate]

The axiom of extensionality in set theory is $$\forall x\forall y[\forall z((z\in x)\iff(z\in y))\Rightarrow(x=y)]$$ Why did they decide to use the $\iff$ relation instead of the $\land$ relation? ...
8
votes
0answers
92 views

Is this statement equivalent to AC?

Let $Frege(X)$ be the set of all equivalence classes of subsets of $X$ under equivalence relation bijection. formally: $Frege(X)= \{\{Y \subseteq X: |Y|=|Z|\}: Z \subseteq X \}$ Where $||$ is ...
0
votes
1answer
40 views

Prove there isn't an increasing $\omega_1$ sequence on real set

Although I have read that it's quite easy to prove there isn't an $\omega_1$ increasing sequence on real set I spent a lot of time figuring out why it happens and finally I think I made it, but I'm ...
0
votes
0answers
33 views

AC in Countable Union of Countable Sets [duplicate]

If I understand correctly, AC is not needed in the case of a finite union of countable sets because the choice function can be built with recursion, using the case of 2 sets (which is straight-forward)...
1
vote
1answer
121 views

Justification of the assumption that a generic ultrafilter exists.

I do not understand why the assumption of existence of a genric ultrafilter $G$ is justified. For the transitive model $V=\{x:x=x\}$ of $ZFC$ and a complete Boolean algebra $B$ in $V$, can we prove ...
0
votes
0answers
69 views

Assume axiom of choice and prove the well ordering theorem

I tried to prove the well-ordering principle. Here's what I've done: let $ A $ be some arbitrary set. if $ A=\emptyset $ then $ A $ is an ordinal, and therefore $ A $ is well ordered. So, we can ...
1
vote
0answers
67 views

“Objects of a category $\mathcal{C}$ constitute a set in some universe”

The following is taken from Borceux Handbook of Categorical Algebra p.59: Proposition 2.71. Consider the category $\mathcal{C}$ such that, for every category $\mathcal{D}$ and functor $F:\mathcal{D}\...
0
votes
2answers
47 views

How to choose infinite number of different values from infinite set of infinite sets.

Let $ \aleph_{\alpha} $ be a cardinal and assume that $ \left\{ A_{\beta}:\beta<\aleph_{\alpha}\right\} $ is a set of sets, such that $ |A_{\beta}|=\aleph_{\alpha} $ for any $ \beta<\aleph_{\...
0
votes
1answer
55 views

How prove the general recursion theorem

Theorem Let be $A$ a set and let be $S:=\bigcup_{n\in\Bbb{N}}(A^n)$ and finally let be $g:S\rightarrow A$ a function. So there exist a unique function $f:\Bbb{N}\rightarrow A$ such that $$f(n)=g(f|_n)$...
-4
votes
1answer
57 views

Is Hausdorff maximal principle and Kuratowski lemma same things? [closed]

In the John's Kelley "General Topology" (pages 32-34) there is definition of both of them and its look like its just different ways of notation of one proposition. But also there is ...
2
votes
0answers
44 views

Series ending with the trivial group

I'm interested in the following seemingly simple question. When we define the composition series or derived series of solvable groups we state that they have to end with the trivial group. Does this ...

1
2 3 4 5
145