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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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 ...
2
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0answers
28 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 ...
2
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1answer
46 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 ...
4
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0answers
45 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 ...
4
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1answer
133 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
44 views

Are there any large cardinals that are not ordinals? [on hold]

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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1answer
45 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 $X=\mathbb N$. ...
2
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1answer
35 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 ...
2
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1answer
27 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 ...
13
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4answers
700 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 ...
3
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1answer
49 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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35 views

set theoretic uni, continuum hypothesis and independency [on hold]

Explain, in terms of set theoretic universes, what it means for a statement, like the continuum hypothesis to be independent from some set theoretic axioms, for example those wich are denoted $ZFC$. ...
7
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1answer
93 views

Members of $\Sigma^1_4$ sets

I'm pretty sure this is an easy descriptive set theory question that I'm just blanking on. Is it consistent with large cardinals - say, with a measurable - that every (nonempty) $\Sigma^1_4$ class ...
2
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1answer
35 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 ...
7
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2answers
163 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. ...
4
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1answer
133 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
50 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 ...
1
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3answers
41 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, ...
5
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0answers
62 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 ...
4
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1answer
75 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 ...
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0answers
44 views

Stronger Konig's Lemma

Assume T is "the" infinite binary tree: the one generated by branching, starting at the root, indexed by $\left\{0,1\right\}$. Konig's Lemma (binary case) states that: "There is an infinite (and ...
6
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1answer
119 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, ...
3
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1answer
32 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 ...
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2answers
47 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
46 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 $M$ ...
2
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1answer
40 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 $P=Fn(\lambda,\kappa)$ ...
0
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2answers
47 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 ...
3
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1answer
88 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 ...
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votes
1answer
55 views

return of Skolem's Paradox?

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 ...
12
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5answers
492 views

When can ZFC be said to have been “born”?

The "History" section of the Wikipedia article on ZFC isn't particularly helpful. The only thing I have understood from it is that ZFC had appeared after 1922. In what book or paper was ZFC first ...
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0answers
30 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 ...
4
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2answers
56 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 ...
8
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3answers
2k views

What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
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2answers
307 views

Is the notion of a set always a primitive notion?

Is there an approach in set theory where we have a definition of a set, or do we always consider the notion of a set as being a primitive notion that cannot be defined in terms of previously defined ...
2
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1answer
42 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 ...
2
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1answer
28 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 ...
0
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1answer
93 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 ...
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0answers
39 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 ...
2
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1answer
33 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 ...
0
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2answers
30 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 ...
3
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4answers
149 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 ...
2
votes
3answers
73 views

Proving the class of countable ordinals is closed under ordinal exponentiation in ZF

I managed to prove that given the axiom of choice, the class (or is it a set?) of countable ordinals is closed under exponentiation, since the axiom of choice implies that the countable union of ...
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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??
2
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1answer
52 views

Every $\alpha < \kappa^+$ can be embedded in any interval of $\kappa^{<\omega}$.

Let $\kappa$ be a cardinal. I want to show that every ordinal $\alpha < \kappa^+$ can be embedded (as an order) in any interval of $\kappa^{<\omega}$, ordered lexicographically. This is exactly ...
3
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2answers
34 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 $\text{cof}(\omega_\alpha)=\text{cof}(\alpha)$. ...
1
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1answer
42 views

For every cardinal $\kappa$, $\kappa^+$ is regular

Again I'm struggling with a proof from this introduction to cardinals. Lemma 2.6. For every cardinal $\kappa$, $\kappa^+$ is regular. Proof. If not, then there would be a cofinal map ...
1
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1answer
33 views

Using the axiom of choice to choose bijections

I couldn't think of a better question title. I am trying to understand the proof of theorem 1.8 in this introduction to cardinals. Theorem 1.8. Let $\kappa\in CARD$. Let ...
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1answer
69 views

Is ZFC the Minimal Structural Theory that Models R [closed]

Is ZFC minimal or "simplest" in the sense that is it is the structural theory with the minimal number of sub-structures that can model R and its "classic" continuity and completeness properties (maybe ...
17
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4answers
5k views

Why do we need to learn Set Theory?

I was planning to write some article for the Mathematics magazine of our college and it occurred to me that it will be a good idea to write about the impact and importance of Set Theory. I plan ...
0
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1answer
95 views

Measure and set theory.

I have read that if we assume the continuum hypothesis then it can be proved or concluded tha there exist a set function μ that has the three following properties: μ(A) is defined for each set A of ...