# Tagged Questions

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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### Does the set of all ordinals strictly dominated by a given set exist in ZF?

How do I prove that $$\{\alpha\in\mathsf{On}\,|\,\alpha\prec A\}\in V,$$ assuming $A\in V$? I know that if AC is assumed, this set is equal to $\mbox{card}(A):=\mu_\alpha(\alpha\approx A)$, and ...
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### Every finite partially ordered set has a maximum length chain.

Every finite partially ordered set, $(A, \leq)$, has a maximum length chain. A chain is a sequence of distinct elements $a_1 \leq a_2 \leq .......\leq a_n$ with relation $"\leq"$ where $a_i \in A$ ...
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### $\varepsilon$-numbers

An ordinal $\xi$ is an $\varepsilon$-number (where does this name come from?) if $\omega^\xi = \xi$. Is the set of all countable $\varepsilon$-numbers closed in $\omega_1$? In other words, does an ...
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### If every nonempty subset of a set $S$ has a least and greatest element, is $S$ finite?

Some sets are well-ordered; all of their nonempty subsets have least elements. You can also have sets where all of their nonempty subsets have greatest elements. Some sets have both of these ...
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### A question about a two player game and axiom of choice

Suppose two players 1 and 2 play the following game: Player 1 starts by playing the set of reals $\mathbb{R}$. Player 2 plays an uncountable subset $Y_1$ of $\mathbb{R}$. Then player 1 plays an ...
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### Question regarding well-ordering theorem

The well-ordering theorem states that every set can be well-ordered. That is, there is a total order on the set in which every non-empty subset has a least element. I was wondering whether it's also ...
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### Showing there is only one isomorphism between well ordered sets using transfinite induction

I need to show specifically using transfinite induction that given two well-ordered sets $\left(A,<_{1}\right)$ and $\left(B,<_{2}\right)$ there is only one isomorphism between them. To do ...
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### Two questions regarding Ordinal Numbers.

I'm trying to prove that there is an uncountable ordinal all which members are countable ordinal. This is fairly easy if I can state that the class of all countable ordinals is a set and then take the ...
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### Question about Hausdorff Maximal principle and antichain

I have this prob and still haven't figured about Let (A, $\leq$) be a partially ordered set. A subset B of A is called an antichain if and only if for any two distinct elements x and y in B, ...
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### Clarification of a remark of J. Steel on the independence of Goldbach from ZFC

On page 424 of the following paper: S. Feferman, Harvey M. Friedman, P. Maddy and John R. Steel, Does Mathematics Need New Axioms?'' The Bulletin of Symbolic Logic, Vol. 6, No. 4 (Dec., 2000), pp. ...
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### How does Fraenkel's urelement proof show choice is independent of ZF?

I understand the actual proof Fraenkel gives but I can't see how it proves choice independent of the full ZF because he works in a very restricted universe. Can anyone show how to connect one to the ...
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### Axiom of choice for infinite strictly descending chains of subsets.

Let $X$ be an infinite set. Then there are infinite stictly decreasing chains of subsets of X. Does this fact require axiom of choice? If yes, what is the choice of function?
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### Interpretation of impure set theory within pure set theory?

I recently came across this paper, where the author (in section 3.1) gives an interpretation of ZFU within ZF. The author of the paper goes on to show what he calls the "synonymy" (i.e., that ZF and ...
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### infinite sequence of sets $\{X_i\}$ that for each $i$, $X_i\in X_{i-1}$

I need to show that the following infinite sequence $\{X_i\}$ doesn't exist: for all $i$, $X_i \in X_{i-1}$ I really don't know where to start. The only thing in my mind is the axiom : for every non ...
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### Axiom of choice in set theory

Just as the title stated, what is the main point of axiom of choice? I do not quite understand what is written in the axiom. The axiom that I know is: Given any collection of non-empty sets, there ...
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### pronunciation of “Kunen” [closed]

How should the last name of set theorist Ken Kunen be pronounced? I have heard both koo-nen and kyoo-nen from other people. Probably the best answer would be to say how he himself pronounces it, but ...
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### Problem understanding definition of ordering on stationary sets

In the book "Set Theory" of Thomas Jech, it is defined for any two stationary subsets $S,T$ of a regular uncountable cardinal $\kappa$, $S<T$ if and only if $S\cap\alpha$ is stationary in $\alpha$ ...
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### Infitive distributive law in boolean valued models

I'm posting the problem 2.14 and 2.15 of the book "Set theory" of J.L. Bell. These problem are proposed after the forcing relation chapter and I'm new in this kind of stuff, so I have some little ...
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### Paradox: Any set theory without universe set is not a model of itself

Because a model of a first order theory is not allowed to use a proper class as its domain, we can't use the universe of the set theory from the "meta-level" directly as a model for a first order ...
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### $\omega$ is a set

I'm trying to show from the $ZFC$ axioms that $\omega$ is a set and I have something that I think could be a proof but I'm not sure about one of its parts. In particular, I want to use the axiom that ...
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### Differences between pure and impure set theory?

What are some differences between pure and impure set theory? For example, this paper references the result that ZFC with urelements is categorical if you assume that the urelements form a set. ZFC, ...
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### For any two sets $A,B$ , $|A|\leq|B|$ or $|B|\leq|A|$

Let $A,B$ be any two sets. I really think that the statement $|A|\leq|B|$ or $|B|\leq|A|$ is true. Formally: $$\forall A\forall B[\,|A|\leq|B| \lor\ |B|\leq|A|\,]$$ If this statement is true, ...
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### Non-measurable subset of $\omega_1$

Consider $\omega_1$ equipped with the order topology. Then Borel subsets of $\omega_1$ are precisely those which contain a closed and unbounded set or the complement contains such a set. There must be ...
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### Is Choice an assumption or determined by category?

Is the axiom of choice an assumption, that one may "freely" choose (eg, ZFC) or discard (eg, ZF, ZF+AD), or is it determined by the nature of the categories being considered? The latter view is ...
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### Limiting of Power Sets

This is, I believe, a relatively simple set-theoretical question. I am ,however, not sure of the answer. If we take a set, say $A$, and if we call the power set of $A$, $P_{1}(A)$, and we define ...
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### What is the cardinality of the countable ordinals?

Every countable ordinal $\alpha$ can be written uniquely in Cantor canonical form as a finite arithmetical expression, say $C(\alpha)$. We thus have the 1-1 correspondence between the countable ...
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### What does it mean that $f$ is unbounded modulo $D$?

Given an ultrafilter on $\omega$ and a function $f:\omega\longrightarrow\omega$, what does it mean that $f$ is unbounded modulo $D$? Thanks
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### Problem with definition of Rudin-Keisler equivalence

I'm trying to do exercise 7.11 of Jech's "Set Theory": If $D$ and $E$ are ultrafilters on $\omega$, then $D\leq E$ and $E\leq D$ implies that $D\equiv E$, where $\leq$ is the Rudin-Keisler ordering, ...
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### Assuming $(GCH)$, strongly inaccessible and weakly inaccessible coincide

My book says "... If $GCH$ holds, then the notions of strongly inaccessible and weakly inaccessible cardinals coincide, ..." In $ZFC$ I can prove this. But the paragraph from which I have excerpted ...