# 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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### Axiom of Choice - Naive Counterexample [duplicate]

Possible Duplicate: Axiom of choice question I know there is a lot of discussion on the axiom of choice and, in fact, I attended once a lecture on it, but I still cannot understand the ...
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### Proving properties of enumeration of infinite cardinals

I am doing the following exercise from Just/Weese: where $F$ is defined as follows: (a) I'm not sure whether the following passes as "convince myself": Apparently $F$ is defined for all ...
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### If $X=\bigcup_{n\in{\mathbb{N}}}\kappa^n$, is it provable from $ZF$ that $|X|=\kappa$?

My question is the following, if $\kappa$ is an aleph and $F$ is the set of all finite sequences in $\kappa$, then the fact that $|F|=\kappa$ is provable from $ZF$?. This can be proven from $ZF$ for ...
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### If $\alpha$ is an indecomposable ordinal, why $\Gamma(\alpha\times{\alpha})=\alpha$?

Greets This is from exercise 3.4 of Thomas Jech's "Set Theory", stated: "Show that $\Gamma(\alpha\times{\alpha})\leq{\omega^{\alpha}}$. Thus $\Gamma(\alpha\times{\alpha})=\alpha$ for all ...
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### Model existence for infinitary logics

One of the problems of infinitary logic is that it is possible for compactness to fail in a spectacular way: for example, one can concoct an inconsistent set of axioms whose proper subsets are all ...
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### Does negation of Axiom of Choice imply symmetry?

It seems that every construction of a model in which the Axiom of Choice fails involves some kind of symmetry. Is there an example of a construction of a model where AC fails but no argument involving ...
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### What's the thorny issue on: “If all $S\in \ell$ are nonempty, does it follow that $\prod_{S\in \ell} S$ is nonempty? when $\ell$ is infinite?”

I'm reading Paolo Aluffi's ALGEBRA, Chapter 0. Here he proposes that there's a thorny issue: What is this thorny issue?
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### How to construct $\{\{\{…\}\}\}$ in ZF without axiom of foundation

I used to think naively the construction is straightforward, which is, if we add one layer innermost each time, then we could have one that corresponds to $\omega$ in Neumann's representation, which ...
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### What does AFA , “Every graph has a unique decoration” mean?

All stuff is from Page 1 - 6, Non-Well-Founded Sets, Peter Aczel, which can be found here. Here a GRAPH will consist of a set of NODES and a set of EDGES, each edge being an ordered pair $(n, n')$ ...
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### Modal theorems valid in a set theory model

This is the question i would like to discuss, properly stated. Given a model $M$ for a collection of set theory axioms (ZFC, for example), list all basic modal formulas $\phi$ such that $M\Vdash \phi$...
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### What does Russell mean when he defines the “Posterity… with respect to the immediate predecessor”?

The the Introduction to Mathematical Philosophy, Russell defines the "posterity" of a given number with respect to the relation "immediate predecessor" as all those terms that belong to every ...
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### Infinite union of internal sets not internal

This is homework problem. I need to give an example of internal sets $A_n \subset \mathbb{R}^*$ for which the union $\bigcup _{n=i}^\infty A_n$ is not internal. Also, this whole internal set ...
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### Why $\mathrm{rank}(x^y) < \alpha+\omega$, if $x$, $y$ have rank $\le$ $\alpha$?

This question is from Set Theory, Jech(2006), Page 70, 6.5. Rank function is defined as on Page 64: $V_0=\emptyset$, $V_{\alpha+1}=P(V_{\alpha})$, $V_{\alpha}=\bigcup_{\beta<\alpha}V_\beta$, ...
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### Axiom of choice on function [duplicate]

Possible Duplicate: Using a choice function to find an inverse for $F\colon A\to P(B)$ Let $F:A \rightarrow \mathcal P (B)$ be arbitary functions which covers $B$. Use AC to show there is a ...
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### How to Understand Collection Principle in the Form of First-order Predicate Calulus

On Page 65, Set Theory, Jech(2006), Collection Principle is formulated as follows: $\forall{X}\exists{Y}(\forall{u}\in{X})[\exists{v} \psi(u, v, p) \to(\exists{v\in{Y}}) \psi(u, v, p)]$ ($p$ is ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $\aleph_0$ ...
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### 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 ...
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### Is Zermelo set theory finitely axiomatizable?

I know that ZF is not finitely axiomatizable, but what about Z (i.e. ZF without Replacement)?
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### 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 ...