# 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.

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### 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 ...
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### 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. ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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.
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### 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 ...
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### 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: &...
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### 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 $\omega, \omega^{\omega},\omega^{\omega^\omega},\cdots$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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, $\bigcup A$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $|B|=\alpha$ ...
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### 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?
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