# 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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### Independent Transcendental Numbers

I've been thinking about numbers which have not yet been proven nor disproven to be transcendental, such as $e + \pi,\, \pi - e,\, \frac{\pi}{e},\, \gamma,\,\zeta(3),$ etc. Some of these numbers ...
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### Which rules of inference does Suppes use?

I'm reading Axiomatic Set Theory by Suppes, and I'm having a bit of trouble understanding which rules of inference (logical system) he is using, here's an example (capital letters are used for sets): ...
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### Simple question about stationary sets in transitive models of ZFC

Let $\kappa$ be a regular cardinal, and suppose $X\subseteq\kappa$ is stationary in $\kappa$. Furthermore, let $\mathcal{M}$ be a transitive class model of ZFC with $X\in\mathcal{M}$. I'm trying to ...
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### Least ordinal $\beta$ such that it is provable that $2^{\aleph_0} \leq \aleph_{\aleph_\beta}$

What is the east ordinal $\beta$ such that it is provable in $\mathsf{ZFC}$ that $2^{\aleph_0} \leq \aleph_{\aleph_\beta}$?
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### Need this transfinite construction reach a closed set in the absence of the Axiom of Choice? Or Hartogs' theorem?

In this question I'm primarily concerned with the workings of a set theory that lacks both Foundation and Choice. Given a set $X$ and a function $\sigma:\mathcal{P}(X)\to\mathcal{P}(X)$, we have a ...
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### Filters and antichains in preorders

In a preorder $\mathbb{P}$ (reflexive and transitive), what means to say that a subset $C\subset\mathbb{P}$ is not contained in any filter of $\mathbb{P}$? For example, it is obvious that if $C$ ...
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### How well-behaved can well-orders on $\mathbb{R}$ be?

Well-orders on $\mathbb{R}$ are interesting because they witness the "alephness" of the cardinality of the continuum, and they're therefore connected to the continuum problem. It seems reasonable, ...
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### Original proof of Zorn's Lemma

On the wiki page for Zorn's Lemma it says that this lemma was Proved by Kuratowski in 1922 and independently by Zorn in 1935 but then it says: Zorn's lemma is equivalent to the well-ordering ...
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### Ernst Zermelo's counterexample

According to the book Real Analysis by Royden page 6(=1+2+3): Given an equivalence relation on a set X, it is often necessary to choose a subset C of X which consists of exactly one member from ...
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### If two posets have same dense open sets, are they equivalent as notions of forcing?

Suppose that $\mathbb{P}_0=(P,\leq_0)$ and $\mathbb{P}_1=(P,\leq_1)$ are partial orderings (in the weak sense, i.e., reflexive and transitive relations) on the same underlying set $P$, and such that ...
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### Statements independent of ZF that quantify over the real numbers

(This question is a bit vague, because I probably haven't aquired all the logical tools needed to express it in a more concise way) I've seen a few examples of statements in set theory that can ...
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### What is “non-simple applied first-order functional calculus” (60's set theory)

Azriel Lévy says in his 1960 paper Axiom Schemata of Strong Infinity in Axiomatic Set Theory, that the $\sf{ZF}$ set theory is formalized with a finite number of axioms in "non-simple applied ...
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### Why does $\epsilon_0 = \omega^{\epsilon_0}$

I saw this on wikipedia and wasn't sure why. It seems obvious given the definition but how can we be sure? Also, what would be $\epsilon_0 \bullet \epsilon_0$? Would it be $\epsilon_0$? Thanks
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### Can epsilon induction be derived from the transitive closure induction?

I was wondering (and could not seem to prove or disprove) if epsilon-induction could be derived from the transitive closure operator for binary relations. The induction of the transitive closure ...
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### Bartoszyński's results on measure and category and their importance

I have seen this interesting paragraph on a talk page of the Wikipedia article about Polish mathematician Tomek Bartoszyński: Tomek's paper "Additivity of measure implies additivity of category, ...
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### In which order should I learn the foundations of mathematics? [closed]

I know from Wikipedia that those are the four pillars of the foundations of mathematics: Proof theory Aximatic Set theory Model Theory Recursion Theory and I want to learn all of them, the problem ...
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### Is there a set theory in which the reals are not a set but the natural numbers are?

Is there any known axiomatization of set theory in which the real numbers are not a set, but the natural numbers and other infinite sets do exist? Such a set theory would have an Axiom of Infinity, ...
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### Range of a P-name

I am working on a set theory problem from Kunen's Set Theory book, and it involves knowing $\text{ran}(\tau)$ where $\tau$ is a $\mathbb{P}$-name. The entire section loves to talk about the domain of ...
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### Applications of the Axiom of Regularity to non-set-theoretical Mathematics [duplicate]

In the beginning of my mathematics studies at university, we have learnt that nearly all of ordinary mathematics not dealing with proper classes can be formalized within ZFC, which is a famous ...
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### Is this an accurate layman's description of the Anti Foundation Axiom

I'm writing an article that covers as one of its topics hypersets/non-well founded sets. In order to do so I have to describe what the anti-foundation axiom (AFA) is my description is currently as ...
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### Show that $3\cdot |A|<9^{|A|}$ for every $A$

I am trying to prove this question which came up in my university's set theory exam last year. A few similar questions have been asked over the last few years and I cannot figure out the method to ...
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### Non-principal ultrafilters on a set [duplicate]

Let $X$ be a set. If $X$ is finite then all ultrafilters on $X$ are principal, i.e. have the form $\{A \subseteq X : x \in A\}$ for some $x\in X$. But now suppose $X$ is infinite, say $X=\mathbb N$. ...
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### Forcing and violation of the $GCH$ at $\aleph_\omega$

In page 295 of Kunen's Set theory the author asserts that if $M$ is a countable transitive model of the axiom of constructibility $V=L$ then no forcing extension of $M$ can satisfy the theory ...
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### Soft Question: Why does the Axiom of Choice lead to the weirdest constructions?

I hope this is not too off-topic / soft for math.stackexchange. My basic question is: why does the Axiom of Choice allow for some of the weirdest constructions in math? I'll make a list of the weird ...
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### Are there any large cardinals that are not ordinals? [closed]

In ZF, are there any useful large cardinal that cannot be well-ordered? I think that some of the partition cardinals are that way, since with AC, we cannot have $\kappa \to (\omega)^{\omega}$. Are ...
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### Axiom of Choice implies the Well-Ordering Principle

I am trying to understand the proof of this implication we were taught in my set theory module. I cannot seem to tie it together with the final line of the argument... We used this lemma: Given ...
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### Proving Dedekind Finiteness of Set of Functions.

I'm struggling with a question from a past paper for a set theory exam. Can't really see a way forward as the different types of finite make it tricky. The question is 'Let A be finite and B be ...
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### Countable Transitive model where $\exists A\subset \omega_1\;(L[A]\vDash\, \neg CH)$
It is well known that for every subset $A\subset \omega_1$ if $V=L[A]$ then $L[A]\vDash GCH$. In particular $L\vDash \exists A\subset \omega_1\,(L[A]\vDash\, GCH)$. Nonetheless, it is also consistent ...
What I really want is this: A sequence $P_0$, $P_1$,... such that each $P_n$ is a countable partition of $\omega_1$, $P_{n+1}$ is a refinement of $P_n$, and such that if $A_n\in P_n$ for all $n$ and ...