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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Regarding Thomas Jech's demonstration of Zorn's lemma via induction

Thomas Jech, such as many other mathematicians, demonstrates $AC \rightarrow ZL$ via transfinite induction. He says: Proof. We construct (using a choice function for nonempty sets of P), a chain ...
4
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2answers
93 views

Proof that $\omega_1$ is countable (where is the flaw?)

I have discovered a "proof" of the fact, that $\omega_1$ can be countable, contradicting its definition. It mustn't assume axiom of choice, so I guess there are some parts of this construction which ...
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2answers
43 views

Problem understanding the Axiom of Foundation

I am just beginning to learn the ZF axioms of set theory, and I am having trouble understanding the Axiom of Foundation. What exactly does it mean that "every non-empty set x contains a member y such ...
3
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1answer
45 views

Is it true that for all infinite cardinal numbers $\nu,$ the set $\{\kappa \mid \kappa^\nu > \kappa\}$ is unbounded?

Bind all lowercase greek letters to cardinal numbers. Question. Is it true that for all infinite cardinal numbers $\nu,$ we have that $\{\kappa \mid \kappa^\nu > \kappa\}$ is unbounded? For ...
3
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1answer
49 views

Unique ultrafilter on $\omega$

We know that from axiom of choice (or just BPIT) we can deduce ultrafilter lemma, which states that every filter can be extended to an ultrafilter. From this lemma we can derive existence of at least ...
6
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1answer
119 views

Is it possible to prove the axiom of infinity from the real number axioms?

What I mean by the title is whether, given that there is a class $\Bbb R$ and operations $+,\cdot,<$ that satisfy the ordered field axioms and the least upper bound axiom, you can prove the ...
2
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1answer
83 views

Can the consistency of ZF be proved in MK?

When adding choice to Zermelo–Fraenkel (ZF) set theory, one can go further than ZFC, by first adding classes, then adding global choice, and then proving that the resulting von Neumann–Bernays–Gödel ...
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1answer
30 views

A question about Choice Functions.

Assume the axioms of ZFC. Suppose that X is an infinite set of infinite (and pairwise disjoint) sets, none of which has a cardinal number greater than that of X. Is the cardinal number of the set of ...
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2answers
129 views

What is the value of $\beth_{\omega_1}^{\aleph_0}$?

It is well known that $\beth_\omega^{\aleph_0} = \beth_{\omega+1}$. This follows since for strong limit $\kappa$, we have $\kappa^\kappa = \kappa^{\mathrm{cf}(\kappa)}.$ Question. To the extent that ...
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1answer
75 views

Theorem of Galvin, Mycielski and Solovay

I don't know is this the right place to ask this question, but can someone tell me where I can find the proof of the theorem of Galvin, Mycielski and Solovay. Theorem that says that a subset $X$ of ...
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4answers
73 views

How do I tell whether axiom of choice is used or not?

I am having a hard time understanding the Axiom of Choice(AC). Say I have an index set $A$ , and a collection of indexed sets {${V_\alpha}$}, where $\alpha$ is a member of $A$. Then, does the ...
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1answer
43 views

Is it true that if $\kappa < \kappa^\nu$, then ${\mathrm{cf}(\kappa)} \leq \nu$?

Let $\kappa$ denote an infinite cardinal number. Then we know the following. $$\kappa<\kappa^{\mathrm{cf}(\kappa)}$$ Question. Is it true that if $$\kappa < \kappa^\nu,$$ then ...
5
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1answer
87 views

Cardinality of all $\mathbf{\Sigma}^0_\alpha$-sets over Baire space without full choice

It is well-known that the set of all open (or closed) sets on Baire space has cardinality of the continuum. In context of choice, we can prove that the set of all $\mathbf{\Sigma}^0_\alpha$-sets over ...
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0answers
20 views

Intermediate Extensions and Coding Sets by Ordinals [duplicate]

Lemma 15.43 in Jech's "Set Theory" states that if $V \subseteq M \subseteq V[G]$ where $G \subseteq \mathbb{P}$ is some $V$-generic filter and $M$ is a transitive models of ZFC, then $M = V[D \cap G]$ ...
3
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1answer
64 views

What are the fixed points of cardinal exponentiation?

Whenever $\kappa$ is an infinite cardinal number, write $\mathrm{cl}_\kappa$ for the unique function $\mathrm{Card} \rightarrow \mathrm{Card}$ given by $\mathrm{cl}_\kappa(\nu) = \nu^\kappa.$ It ...
4
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1answer
102 views

A question regarding the status of CH in the Gitik model

Consider models of ZF+"Every uncountable cardinal is singular" (eg. Moti Gitik: "All uncountable cardinals can be singular", Israel journal of Mathematics, 35(1-2): 61-88, 1980). How should CH be ...
0
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1answer
44 views

Proof of Hartogs' Lemma Q2

I'm reading "Sets, Models and Proofs" by I. Moerdijk and J. van Oosten to have some basic understanding of set theory. The last step in "Hartogs' Lemma" proof is a headache: "This is impossible ...
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0answers
58 views

Krull's theorem and AC

Lately I have been trying to prove axiom of choice based on krull theorem. The theorem states that for every ring with a unit $R$ that is not a field, there is a maximal ideal. I know it is equivalent ...
4
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2answers
66 views

Proof of Hartogs' Lemma

I'm reading "Sets, Models and Proofs" by I. Moerdijk and J. van Oosten to have some basic understanding of set theory. There's one step in "Hartogs' Lemma" proof that I don't see how to it works: I ...
4
votes
2answers
186 views

Why doesn't “$V$ is a model of $ZF$” imply consistency of ZF?

One can prove that $V$, Von Neumann's universe, satisfies all of $ZF$ axioms, so it's a model of $ZF$. But I can't see why this doesn't imply the consistency of that theorem. I'm aware that ...
5
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1answer
193 views

Asymmetric roles that are symmetric in every instance

This is similar to something else I posted, but this time we'll pretend we've never heard of infinite sets or infinite series. \begin{align} & \sin(\alpha+\beta+\gamma+\delta+\varepsilon+\zeta) ...
3
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1answer
108 views

Book recommendation in Foundational Mathematics

I have been navigating in this "foundational world" of mathematics for a while now ( but certainly not long enough and not deep enough ) and have read a bit about many different topics : set theory, ...
2
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1answer
50 views

Smallest Transitive Model Containing Certain Sets

Let $V$ denote the underlying universe. Let $M \in V$ be a transitive set or class such that $M \models \text{ZF}$. Let $x \in V$ be some set. One often sees the notation for $M[x]$ and $M(x)$. If ...
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0answers
43 views

How to define Set Theory and First Order Logic without circular reference? [duplicate]

Axiomatic set theory is basically a first order logic theory, so I believe we must certainly begin with a treatment of first order logic. However, to define a first order logic vocabulary we need to ...
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0answers
129 views

“Dual cardinality” in the graphs $(V_\alpha,\in_\alpha)$

04-25-2014 Enriched with more details Let define the graphs $(V_\alpha,\in_\alpha)$ in $ZFC$ $V_\alpha$ can be finite too $\in_\alpha \subseteq V_\alpha \times V_\alpha$ and ...
3
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1answer
84 views

Proving König's lemma (technical problems)

the aim of my exercise is to give a proof of the König's lemma. So, let $\kappa, \lambda$, be cardinals such that $cf(\kappa)\leq \lambda$. My professor's suggested us to prove that there exists a ...
7
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1answer
86 views

Maximal ordinal embeddable into poset of functions under eventual Domination

Consider the set of functions from $\mathbb{N}\to\mathbb{N}$. We can impose a partial order on this set by saying that $f>g$ if $f(n)>g(n)$ for all sufficiently large $n$. By a diagonalization ...
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1answer
67 views

Tarski's fixed point theorem

My professor stated the following "For every normal function $f \colon ON \to ON$ and for every ordinal $\beta$ there exist an ordinal $\gamma \ge \beta$ such that $f(\gamma)=\gamma$". (normal ...
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3answers
138 views

ZFC Axioms to be extended?

Sorry if this is going to be a really loaded question. I was told several times that for virtually all theorems/corollaries/propositions of mathematics (except those cases not compatible with ZFC ...
5
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1answer
84 views

Do we need choice to prove that $|\mathbb{N} \times A| = |A|$ for all infinite sets $A$?

I can't think of any way to prove it without choice.
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1answer
77 views

Simplify this expression of ordinal numbers: $(ω+1) ω²$ [closed]

Let $ω$ be the ordinal of the natural numbers. Simplify $(ω+1) ω²$
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1answer
41 views

How can we show that $\omega_1$ is a regular cardinal?

A cardinal $\kappa$ is regular if and only if there is no $\lambda<\kappa$ for which there is a function $f:\lambda\rightarrow\kappa$ with range cofinal in $\kappa$. How can we see in ZFC that ...
12
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2answers
252 views

Non-constructive axiom of infinity

Looking at the ZFC axioms (in the Wikipedia version), the axiom of infinity stands out because it contains an extremely specific construction. This seems rather unelegant to me, therefore I've thought ...
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1answer
64 views

Examples for intensional set theories.

The normal set theory of todays mathematics (ZFC) is extensional, i.e. it has the axiom of extensionality $$\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)$$ Are ...
5
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1answer
274 views

How can I prove that there is no set containing itself without using axiom of foundation?

I've already found some similar questions in here (and other sites), but in most of the case, the use of axiom of foundation is required to complete the proof. Is there any way to prove $\not\exists ...
2
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1answer
41 views

Preserve Cardinals and Adding No Bounded Subsets

In Chapter 15 at the bottom of page 228 of $\textit{Set Theory}$ by Jech, he writes that if $\kappa$ is a cardinal in $V$ and if $\kappa$ has no new bounded subsets in $V[G]$, then $\kappa$ remains a ...
9
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1answer
106 views

The cardinality of the set of all linear order types over $\omega$ is $2^{\aleph_0}+\aleph_1$ in ZF+AD?

In ZFC, cardinality of set of linear orders over $\omega$ is $2^{\aleph_0}$. By the argument given by here, we can prove (without the choice) the number of linear orders over $\omega$ is at least ...
6
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1answer
63 views

Large cardinals and $V$

I am confused by something: $\mu$ is a large cardinal if $\lambda<\mu\Rightarrow 2^{\lambda}<\mu$ and any union of less than $\mu$ sets of size less than $\mu$ is less than $\mu$. On Wikipedia ...
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0answers
58 views

Quotient Forcing in Iterations

I am trying to understand a proof of a lemma used to prove a preservation theorem for $^\omega \omega$-bounding for countable support iterations. In that quotient forcing is used to get a certain ...
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1answer
61 views

How many ways can we totally order a set $X$ (but not necessarily the whole set $X$, perhaps just a subset) up to isomorphism?

Here's Attempt #2 at asking this question. Suppose $X$ is a set (not necessarily finite), and let $T$ denote the collection of all totally ordered sets $(A,\leq)$ such that $A \subseteq X.$ Now let ...
1
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1answer
27 views

Splitting Frechet filter into two proper filters

Let $\Omega$ be a Frechet filter (=cofinite filter) on an infinite set. Do there exist proper filters $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}\cap\mathcal{B} = \Omega$ and ...
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6answers
686 views

Question about the Continuum Hypothesis

The Continuum Hypothesis hypothesises There is no set whose cardinality is strictly between that of the integers and the real numbers. Clearly this is either true or false - there either exists ...
2
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1answer
43 views

Question concerning a statement about separability

So here is my question, Let $X$ be topological space. If every disjoint familiy of open sets is at most countable, then $X$ is separable. I am pretty sure that this is true but I still wanted to ask ...
3
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2answers
70 views

Finite-case symmetry leads to infinite-case asymmetry

Formulas for sines or cosines of sums superficially appear to have a certain symmetry, specifically it looks as if sine and cosine play something like symmetrical roles: $$ \begin{align} & ...
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1answer
60 views

Equivalence of Axiom of Regularity

So Axiom of regularity states: every non-empty set A contains an element that is disjoint from A I'm wondering if this is equivalent as any set is not a member of itself? If so, how do we prove it? ...
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2answers
69 views

In axiomatic set theory every set is a “collection” of “empty sets”?

Based on the answers of this question: How elements are defined in axiomatic set theory and this part of this book: (page 9) I will examine this reasoning in depth: Let's take a random example: ...
3
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1answer
86 views

Size of topological space depending on the size of local basis. (With elementary submodels)

Recall that the character of a topological space $\chi(X)$ is the minimum cardinal $\kappa$ such that every point in $X$ has a local basis of size $\kappa$. I need to prove that if $X$ is compact ...
8
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1answer
125 views

Can all theorems of $\sf ZFC$ about the natural numbers be proven in $\sf ZF$?

I know a proof of Hindman's theorem that uses ultrafilters on the natural numbers, and ultimately, the axiom of choice. But the theorem itself is essentially a combinatorial property of the natural ...
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1answer
61 views

Question about rank into rank cardinals

I do not understand why λ is smaller than $\kappa_0$ if λ is the supremum of a growing sequence that starts at $\kappa_0$ (see definition below). From cantor's attic ...
6
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1answer
98 views

Building normal filters around a stationary set

Recently I've been looking at connections between Laver functions on large cardinals and diamonds. While $\diamondsuit$-like principles tend to readily generalize to Laver function-like concepts, I've ...