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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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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2answers
100 views

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, ...
6
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2answers
85 views

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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1answer
63 views

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 ...
5
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1answer
66 views

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 $\...
5
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1answer
87 views

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 ...
4
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1answer
40 views

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 first-...
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1answer
71 views

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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23 views

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 ...
4
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0answers
57 views

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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3answers
160 views

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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3answers
812 views

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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1answer
29 views

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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0answers
26 views

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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0answers
42 views

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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1answer
53 views

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 ...
1
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1answer
52 views

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 $X=\mathbb N$. ...
5
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1answer
86 views

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 $$ZFC\,\...
5
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1answer
295 views

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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1answer
56 views

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 ...
2
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1answer
40 views

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 $F\...
2
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1answer
32 views

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 ...
3
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1answer
58 views

The well-ordering principle implies Zorn's Lemma

I have read and understood proofs for each implication between $AC$, $ZL$, $WO$ except this one. These proofs need about 10 lines each. Can someone share a neat, hopefully short, proof for $WO\implies ...
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4answers
744 views

Please Explain Kuratowski Definition of Ordered Pairs

I've seen this Kuratowski definition for ordered pairs, but can't fathom why it implies an order to $x$ and $y$ $(x,y):=\{\{x\}, \{x,y\}\}$ As I understand sets, $\{\{x\}, \{x,y\}\}$ is also $\{\{x,...
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1answer
38 views

If $A \subset \mathcal{N}^2$ is a $\mathbf{\Sigma}^0_\alpha$ set, then $\{x : (x,x) \in A\}$ is also $\mathbf{\Sigma}^0_\alpha$.

This is the boldface Borel hierarchy on Baire space. Jech states this with a "clearly". What am I missing that makes the statement completely obvious? I clearly have zero intuition for this material....
4
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1answer
141 views

$\vDash \varphi$ iff $\| \varphi \|_A =1$ for every boolean valued structure $A$

In the book Axiomatic Set Theory (Takeuti, G; Zaring, W.M. - 1973) the theorem 6.4 states that if $\varphi$ is a closed formula of a given language then it is satisfied in every boolean valued ...
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0answers
66 views

Enlarging the continuum with $\sigma$-centered forcing

How large can we force the continuum to be if we force with a $\sigma$-centered forcing notion? References to texts discussing the subject would be much appreciated. [A forcing notion $P$ is called ...
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3answers
53 views

Ordinals and Countability

I've been watching Prof Su's (Harvey Mudd College) incredible lecture on Ordinals and Transfinite Induction, and I have a few related questions. He starts by drawing and examining three sets of dots, ...
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0answers
89 views

Measure on $\omega$ defined in the generic extension by an atomless measure algebra is atomless

Work in Cantor space with standard probability measure $m$. Suppose we are given a sequence of measurable sets $\bar{A}=\langle A_n : n\in \omega\rangle$ and a non-principal ultrafilter $U$ and the ...
1
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1answer
52 views

Infinite Spectrum Problem

Let us work in a class theory like NBG. For a given first order sentence $\phi$ define $\infty\text{-spectrum}(\phi)$ to be the class of all cardinal numbers $\kappa$ for which there is a model $M$ ...
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2answers
62 views

I don't understand the axiom schema of separation.

I understand that the axiom schema of separation should assert the existence of a set $y$, subset of a set $z$, where $y=\{x\in z:\varphi \,x\}$ (with $\varphi$ some formula). In the book I'm ...
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1answer
129 views

How can mathematics work in wildly different set theories?

There are several set theories, e.g. ZFC and NF, which often have different axioms or are even outright contradictory. And yet most of other branches of mathematics, e.g. abstract algebra or topology, ...
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2answers
218 views

Kunen exercise III.8.21

Let $f: \omega_1\to \mathbb{R}$ be one-one. Let $g:[\omega_1]^2\to 2$ be such that for any $\alpha<\beta<\omega_1$, $g(\{\alpha, \beta\})$ is $0$ when $f(\alpha)<f(\beta)$, and $1$ otherwise. ...
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2answers
53 views

Existence of how many sets is asserted by the axiom of choice in this case?

Applying the axiom of choice to $\{\{1,2\}, \{3,4\}, \{5,6\},\ldots\}$, does only one choice set necessarily exist, or all of the $2^{\aleph_0}$ I "could have" chosen? Or something in between? It ...
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2answers
84 views

return of Skolem's Paradox? [closed]

The Lowenheim-Skolem theorem means that if ZFC has a model, it has a countable model. However, I think we can show that there aren't any countable models. The overview is that all the elements of ...
2
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1answer
43 views

GCH is preserved when forcing with $Fn(\lambda,\kappa)$.

Given a countable transitive model $M$ where $GCH$ holds it is an exercise from Kunen's book to show that GCH also holds in $M[G]$ when $G$ is a $P-$generic filter over $M$, and $P=Fn(\lambda,\kappa)$ ...
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0answers
34 views

Cantor normal form to compute sums and products of ordinals: $\omega^{\beta} c+\omega^{\beta'} c' = \omega^{\beta'}c'$ if $\beta'>\beta$

From Wikipedia on Ordinal arithmetic: The Cantor normal form also allows us to compute sums and products of ordinals: to compute the sum, for example, one need merely know that $$\omega^{\...
3
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1answer
38 views

Is BPIT equivalent to some ordering principle?

Working in $\mathsf{ZF}$, is $\mathsf{BPIT}$ (Boolean Prime Ideal Theorem) equivalent to some statement of the form "every set can be ***ly ordered"? I know that $\mathsf{BPIT}$ implies that every set ...
4
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2answers
73 views

Is it provable in $ZFC$ that if $V_\kappa\vDash ZFC$, then $\kappa$ is strongly inacessible?

The other direction of this implication is pretty obvious, but I'm having a hard time seeing why this direction might be true. I suspect that it isn't, but part of my suspicion comes from my inability ...
2
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1answer
36 views

Is there an agreed upon convention for naming ZFC+Large Cardinal Axioms?

Is there an agreed upon convention in general for what to name ZFC+[Large Cardinal Axiom]? Or would one have to state explicitly which axiom was being added? To explain what I mean, note that anyone ...
2
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1answer
49 views

$\kappa$-closed forcing preserves stationary sets.

Let's take an uncountable cardinal $\kappa$ which is regular inside the ground model $M$ and $\mathbb{P}\in M$ a forcing notion which is $\kappa-$closed in $M$. I'm trying to prove that every ...
3
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1answer
92 views

Ultraproduct with no long descending sequence

I have a countably infinite well-ordered structure $M$ (over a countable language if it helps), and an uncountable regular cardinal $κ$, and I wanted to construct an elementarily equivalent structure ...
0
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1answer
100 views

Does the regularity of $\omega_{\alpha+1}$ need Axiom of Choice?

Many books indicate yes to this question. However, I found the only lemma they claim to use AC is the following statement: If $\{A_i\}_{i\in I}$ is a family of sets, then $|\bigcup_{i\in I}A_i|\leq|I|...
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0answers
44 views

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 ...
0
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1answer
47 views

Explicit indexing of countable ordinals by sequences of integers?

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 $...
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2answers
44 views

Question regarding countable ordinals

Feel free to suggest a better title. We're going to regard $0$ as a limit ordinal, as people sometimes do. Let $L$ be the set of countable limit ordinals. If $\alpha\in\omega_1$ there exist a unique ...
4
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1answer
82 views

Well-ordering of sets of cardinal numbers

Proposition For every cardinal number $m$ there is a definite next larger cardinal number. This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal ...
3
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4answers
172 views

Is $\omega-1$ finite?

I saw some videos and read some stuff about ordinals, and it came to me that $\omega-1$ should be finite. My logic is that $\omega$ is the smallest transfinite number, so $\omega-1$ should be finite.....
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1answer
29 views

Question about Hartogs' theorem proof [closed]

Is it possible to prove this theorem without the use the replacement axiom??
3
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2answers
38 views

“Proof” that $\text{cof}(\omega_\lambda)<\omega_\lambda$ if $\lambda$ is a nonzero limit ordinal

The following lemma is from this introduction to cardinals. Lemma 2.7. Let $\omega_\alpha$ be a limit cardinal. Then $\alpha$ is a limit ordinal and $\text{cof}(\omega_\alpha)=\text{cof}(\alpha)$. ...