# 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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### Is there a constructible flat pairing function?

In ZFC set theory, is there a Skolem function f such that ZFC can prove f is a flat pairing function? And if so, can someone explicitly give me a formula?
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### Can every ccc forcing of size $2^{\aleph_0}$ be $\sigma$-linked?

It is well known that $MA_{\aleph_1}$ implies that every ccc forcing of size $< 2^{\aleph_0}$ is $\sigma$-linked (in fact - a countable union of filters). On the other hand, a separative forcing ...
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### Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?

Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that is well-founded. This means that for every non-empty set $a$ there is a set $b\in a$ such that $cRb\implies c\notin a$. Here ...
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### Ramsey Combinatorics and linear order

Prove that the following are equivalent for an infinite cardinal $\kappa$. (1) $\kappa \to (\kappa)^2_2$ (2) In any linearly ordered set of cardinality $\kappa$ there is either a well-ordered or a ...
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### Closed and unbounded set on a specific set or ordinals.

I'm having problems on showing the following: Let $\kappa$ be a strongly inaccesible cardinal. Show that the set of all ordinals $\alpha <\kappa$ such that $V_\alpha \models ZFC$ contains a club. ...
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### A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
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### First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
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### Are there collections of sets that are neither a set nor a definable proper class?

Working in ZFC, are there collections of sets that are neither a set nor a definable proper class? I mean if some collection of sets is not a set can we necessarily conclude that it is a definable ...
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### What conditions must be checked for that $c$ is Cohen over $V$.

$\textbf{Hechler forcing}$ Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ ...
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### References for Introductory Model Theory focusing on applications other than algebra

I would appreciate suggestions for references (books, lecture notes, articles etc...) on Model Theory (at an introductory level) that don't focus mostly on algebra when giving examples and/or applying ...
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### ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence ...
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### Proving that $\alpha\approx|\alpha|$ is not constructive

$\sf ZF$ tells us that for every ordinal $\alpha$ there is a bijection $f:\alpha\to|\alpha|$, more or less by the definition of $|\alpha|$, the cardinal of $\alpha$. What I would like to show is that ...
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### The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that "it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone." And in Jech ...
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### Explanation of a “sentence” of Halmos's Naive Set Theory

While reading Halmos's Naive Set Theory I found that he has remarked in a place, If $\mathcal{C}$ be a collection of subsets of a set $E$ (that is, $\mathcal{C}$ is a subcollection of ...
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Russell's paradox prevents us from allowing any expression of the form $\{x \mid P(x)\}$ from being a set. His observation shook up the field of set theory, prompting formal axiomatizations of the ...
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### Singularity of small cardinals under AD

It appears to be quite a folklore result that under AD we know which small well-orderable cardinals (for my purposes I mean below $\omega_\omega$) are regular, namely only $\omega,\omega_1,\omega_2$. ...
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### Why is the axiom of pairing needed in Von Neumann-Godel-Bernays Set Theory?

Why do we need the axiom of pairing in Von Neumann-Godel-Bernays Set Theory? Doesn't the following prove it? Suppose $A$ and $B$ are sets. Using the axiom of class comprehension, we can form a ...
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### Regarding chains and antichains in a partially ordered infinite space [duplicate]

I've been given this as an exercise. If P is a partially ordered infinite space, there exists an infinite subset S of P that is either chain or antichain. This exercise was given in the Axiom ...
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### What is wrong with this “proof” that there is no $\omega$th inaccessible cardinal?

"Theorem": There is no $\omega$th inaccessible cardinal. "Proof": Assume ZFC. Let $\kappa_n$ be the $n$-th inaccessible cardinal; since $V_\kappa$ is a model of ZFC for inaccessible $\kappa$, ...
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### Do we know that we can't define a well-ordering of the reals?

Folklore has it that it is impossible to define a well-ordering of the reals explicitly. There exist pointwise definable models of ZFC where every set is definable without parameters: it is the ...
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### Ordinal arithmetic $(\omega+1) \cdot \omega$ and $\omega \cdot (\omega +1)$

Here is where I am so far: $(\omega+1) \cdot \omega = \sup\{(\omega +1) \cdot n, n \in \omega\} = \omega^2$ and $\omega \cdot (\omega +1) = \omega \cdot \omega + \omega = \omega^2 + \omega$ Hence ...
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### Cardinality of a set of subsets with given max cardinality [duplicate]

Let $S$ be a set of infinite cardinality $\kappa_1$. What is the cardinality $k$ of the set of subsets of $S$ with cardinality $\kappa\le\kappa_0<\kappa_1$? I understand that if $\kappa_0$ is ...
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### Exercise on forcing

I got this homework in my forcing class: Let $G\subseteq P$ be generic over M. Show that there is a cardinal of M, $\lambda$ such for every set of ordinals $X\in M\left[G\right]$ there is a set ...
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### Non WellFounded Set theories and Russell's Paradox

I am very puzzled by set theories which reject the axiom of regularity. If we reject the axiom of regularity, and allow a distinction to be drawn between wellfounded and non-wellfounded sets/classes, ...
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### Ordinal exponentiation, is $3^\mu = \mu$?

I'm revising for my set theory final, and I've been asked to find an ordinal $\mu > \omega$ with $2^\mu = \mu$, then to answer whether $3^\mu = \mu$. The ordinal I picked as $\mu$ was the union ...
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### “Every set is equinumerous to a well-founded set” - provable in $\sf ZF-Reg$?

What is the relationship of the following to other axioms of $\sf ZFC$? $\sf WB$: Every set $A$ is in bijection with a set well-founded by $\in$. Obviously, $\sf ZF$ implies $\sf WB$ (because ...
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### What are some simple example of “forcing” in set theory?

Can someone illustrate the idea of "forcing" in set theory through some simple examples? The article on forcing on wikipedia goes straight to axiom of choice and continuum hypothesis, I wonder if ...
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### Choosing a $x \in$ $X$, with $X$ an infinte set, $<$ a well-ordering on $X$, such that $x < x'$ for only finitly many $x' \in X$

Let $X$ be an (countable or not) infinte set, $<$ a well-ordering on this set. Lately I read a proof in which was explicity stated to choose an element $x \in X$ such that there are infintly many ...
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### A consequence of zero sharp (successor cardinals having countable cofinality)

So the existence of $0^\sharp$ in set theory is really the assertion on the existence of indiscernibles for the constructible universe $L$ that also "generate" $L$ (see ...
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### Is there a set theory that avoids Russel's paradox while still allowing one to define the set of all sets not containing themselves?

The main idea of Russel's paradox is that, in Naive Set Theory, if we define $R = \{x\ |\ x \not\in x \}$, then $R \in R \Leftrightarrow R \not \in R$. ZFC deals with this by making unrestricted set ...
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### Cardinality of a Grothendieck Universe

Let us work in $ZFC+U$ where $U$ is the existence of a Grothendieck Universe. Let $\mathrm{On}(U) = \mathrm{On}\cap U$ denote the set of ordinals in $U$. How can I show that the cardinality of $U$ is ...
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### stationary set,club,module theory,Auslander lemma,

Here http://www.ams.org/journals/tran/1990-322-02/S0002-9947-1990-0974514-8/S0002-9947-1990-0974514-8.pdf I do not understand the first two lines of the proof of lemma 9,on page 550:What and why is ...
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### Axiom of foundation allows sets consisting of a descending sequences plus some 'atom'?

I am reading a text which describes how the Axiom of Foundation prevents sets that are built from a descending sequence such $$X=\{x_0, x_1,\ldots\},\text{ with } x_1\in x_0, x_2\in x_1,\ldots$$ ...
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### Diamond and Suslin tree

I'm reading the proof (assuming $\Diamond$) of the existence of a Suslin tree in Nik Weaver's Forcing Mathematicians (Theorem 18.4 page 71) and I have difficulty seeing the use of $\Diamond$. Given a ...
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### What do the ZFC axioms look like in terms of subset?

I'm reasonably familiar with the ZFC axioms. I know they can be formalized in many different ways. However, I usually see them presented in terms of the elementary "propositional calculus primitives" ...
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### Splitting the Real Line

By definition a $\mathfrak c$-dense subset of $\mathbb{R}$ has $\mathfrak c$-sized intersection with every non-empty open set. Using transfinite recursion it is quite easy to prove that every ...
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### Is there a set $A$ such that power set of $A$ has a bijection with $\mathbb{N}$? [duplicate]

Has this statement any relation with continuum hypothesis ?
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### The Diamond Principle implies the Club Principle.

So the Diamond and the Club principles are both combinatorial principles in set theory. They are defined as follows (there are thinner definitions but I stick to this ones is $\omega_1$, as I am sure ...
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### Elementary submodels in stationary logic.

In the paper "Stationary Logic" by Barwise, Kaufmann and Makkai the authors prove that stationary Logic L(aa) has Löwenheim number $\aleph_1$, i.e. every satisfiable set of sentences has a model of ...
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### Splitting Stationary Sets

So by a theorem of Solovay we know that a stationary subset of a regular (uncountable) cardinal $\kappa$ can always be split into $\kappa$-many stationary subsets. Is the regularity assumption ...
Suppose $1\leq\xi<\omega_1$ is a countable ordinal and that $1\leq\zeta<\omega^\xi$. Is it always true that $\zeta+\omega^\xi=\omega^\xi$? If so, why?