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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6 views

Complex Borel Measure Decomposes as Discrete + Continuous Measures?

I am trying to prove that any complex Borel measure $\mu$ on a measure space $(X, \mathfrak{M})$ decomposes into the sum of a discrete measure $\lambda$ and a continuous measure $\nu$, $\mu = \lambda ...
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0answers
14 views

Connection beween closure property in forcing and preservation of $H_{\kappa}$

I would like to know about the relation between closure property of forcing notions and preservation of hierarchy of hereditary small sets, $\langle H_\lambda \mid \lambda\in \mathrm{Card}\rangle$, at ...
4
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2answers
138 views

Understanding proof of Hartogs’ Theorem on Set Theory

I'm trying to understand the proof of the Hartogs’ Theorem on page 100 of this book. My especific question is: If we have for each set $A$ $$\mathrm{WO}(A)=\{ (U,\leq_{U}) \, | \, U\subseteq A \, ...
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1answer
33 views

Condition to be a prewellordering.

I'm trying to do the problem 7.17 on th book Notes of Set Theory of Moschovakis. First I will define what is a prewellordering: A prewellordering on a set $A$ is any relation $(\lesssim) ⊆ A×A$ ...
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0answers
28 views

Infinite connected graph such that every vertex has finite degree

Let $G=(V,E)$ be an connected graph with $|V| \geq \aleph_0$ such that $\text{deg}(v)$ is finite for all $v\in V$. Does this imply that $|V|=\aleph_0$?
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2answers
41 views

Injections from all ordinals into a set $X$

We are working in $\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \alpha \to X$ for arbitrary ordinals $\alpha$. I am quite sure that, in fact, $A$ is a set, since if ...
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3answers
138 views

Does $A\times A\cong B\times B$ imply $A\cong B$?

This is similar to What does it take to divide by $2$? about $(A\sqcup A\cong B\sqcup B)\Rightarrow A\cong B$ which is valid in $\textsf{ZFC}$ by using cardinalities and also in $\textsf{ZF}$ by some ...
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1answer
52 views

Is this the real definition of an ordinal number?

I have to prove that if $\alpha$ is ordinal and $\beta \in \alpha$ then $\beta$ is oridinal. To do this I wanted to prove that for every $x\in\beta$, $x\in\alpha$. This seems like it should be true, ...
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0answers
90 views

Question about the foundation of mathematics [duplicate]

I have studied mathematical logic and set theory as an undergraduate. I studied mathematical logic (propositional and predicate logics) before set theory. When I studied mathematical logic, I was a ...
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1answer
86 views

In preparation forcing and large cardinal textbooks

Everybody in set theory refers to texts like Kunen, Jech and possibly Halbeisen's books as elementary references for forcing. Also Drake and Kanamori's books are well-known references for large ...
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2answers
67 views

Determining a charge through subsets

A charge is a finitely additive set function $c: \mathcal{P}(\mathbb{N}) \to [0, 1]$ such that $c(\mathbb{N}) = 1$ and $c(\{n\}) = 0$ for every $n \in N$. Here $\mathcal{P}(\mathbb{N})$ is the set of ...
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2answers
44 views

Prove that every totally ordered set has a well-ordered cofinal subset

I thought this was easy until I realized that for a set to be well-ordered, EACH of it's non-empty subsets must have a least element. Let $A$ be a non-empty totally ordered set. Let $x_0$ be some ...
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0answers
22 views

In an infinite dimensional real inner-product space , can any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis?

Let $V$ be an infinite dimensional real inner-product space , then is it true that any non-null orthogonal set of vectors in the space can be extended to an orthogonal basis ? Or at least is it true ...
2
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1answer
47 views

Trouble understanding elementary embedding proofs

Here are two pretty standard results about elementary embeddings that I don't understand. (1) Let $\pi:V \to M$ be a non-trivial elementary embedding with $M$ a transitive class. Let ...
5
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0answers
57 views

Iterated ultrapowers with arbitrary measures are well-founded

An iterated ultrapower of an inner model $M$ is a sequence $\langle M_\gamma:\gamma\leq\lambda\rangle$ such that $M_0=M$, $M_{\gamma+1}$ is a class of $M_{\gamma}$ using a measure in this model, and ...
3
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1answer
36 views

I can't find the mistake in this argument (Cofinality and König's theorem)

I have some trouble explaining this apparent contradiction: we know that given $k>\aleph_0$ an infinite cardinal, $\mu=cof(k)$ is the minimum cardinal such that $k=\sum_{i\in \mu} k_i$ where ...
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2answers
65 views

Counterexample to the Hausdorff Maximal Principle

The Hausdorff Maximal Principle states: Every partially ordered set $\left(X,\leqslant\right)$ has a linearly ordered subset $\left(E,\leqslant\right)$ such that no subset of $X$ that properly ...
3
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2answers
353 views

Prove that a statement or its negation follows from ZFC

There are several problems which have been shown to be unprovable in ZFC. Has there ever been a case of the opposite? That is, has it ever been proven for some statement $\varphi$ that $\text{ZFC} ...
4
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2answers
70 views

How does Ulam's argument about large cardinals work?

I am looking for either a reference, a proof, or a suitable proof sketch that can explain Ulam's original argument about measure theory and measurable cardinals. Here is the result I am looking for: ...
6
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0answers
127 views

What are disasters with Axiom of Determinacy?

It is well-known that Axiom of Choice has several consequences which might be viewed as counter-intuitive or undesirable. For example, existence of non-measurable sets or Banach-Tarski Paradox. H. ...
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4answers
236 views

Uses of ordinals

Are there any interesting theorems outside of set theory that use ordinals in their proofs? The only example I know of is Goodstein's theorem, and I haven't been able to find anything else. In other ...
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1answer
53 views

What set theory or foundation of mathematics is most commonly used by applied mathematicians in the private sector? [closed]

In terser words, what set theory or mathematical foundation applied is the most economically productive outside of academia?
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1answer
40 views

Are objects built by a generic filter which is not in the ground model necessarily out of the ground model?

Let $G$ be a $\mathbb{P}$-generic filter over a ground model $M$ of ZFC and $G\notin M$. Are all objects built by this generic filter necessarily out of the ground model $M$? Particularly is limit of ...
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1answer
44 views

About Set Theory Axioms [duplicate]

The axiom of Replacement Scheme implies separate axiom. I can not show this lemma. Does someone have any idea about it?
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1answer
49 views

If there monomorphisms $f : A \rightarrow B$ and $g : B \rightarrow A$ then there is an isomorphism $h : A \rightarrow B$

Consider the following set theoretical result of Schröder-Bernstein-Cantor: Let $A$, $B$ be sets. Assume we have injections $f : A \rightarrow B$ and $g : B \rightarrow A$. Then there exists a ...
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2answers
76 views

Difference between a set and a class

I don't understand the difference between a set and a class. The definition which I studied is: A set $A$ is a class such that there exists a class $B$ such that $A \in B$. But isn't it always true ...
6
votes
1answer
68 views

Order type of real analytic monotonic functions ordered by eventual domination

Let $\mathcal F$ be the set of all functions $\mathbb R^+\to\mathbb R$ that are: real analytic on $\mathbb R^+$, monotonic on $\mathbb R^+$, and having derivatives of any order that are also ...
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1answer
64 views

Questions of Hechler forcing

Shows that Hechler forcing adds Cohen real. A suggestion please. Can you tell me reference Hechler forcing.
6
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2answers
64 views

Can you equip every vector space with a Hilbert space structure?

Suppose that we have a vector space $X$ over the field $\mathbb F \in \{ \mathbb R, \mathbb C \}$. Question: Does there exist a Hilbert space $\widehat X$ over $\mathbb F$ such that $\widehat X$, ...
1
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1answer
53 views

Notation about cardinals

Does a cardinal $\mathcal{k}$ such that $2^\mathcal{k}=k^+$ have any special name? I never encountered any name for this property, but I think it is possible they have one.
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0answers
45 views

How to show that the class of singletons, and the class of all ordered pairs are proper classes?

I need to prove that the class of all one element sets is a proper class and also that the class of ordered pairs of the form $\langle x,y\rangle=\{\{x\},\{x,y\}\}$ is a proper class. I can assume ...
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0answers
63 views

what does “a wff f(x, y)” mean exactly? (context: transfinite recursion)

I'm currently working through Herbert B. Enderton's book "Elements of set theory". I have a question concerning notation in logic, of which I know the basics but in which I'm not that firmly grounded. ...
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0answers
94 views

Do an infinite set and its double have the same cardinality? [duplicate]

My question was inspired by this answer. Suppose $A$ is an infinite set. Does its double, $A\times\{0,1\}$, always have the same cardinality? In my head I quickly spotted a simple proof that the ...
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0answers
24 views

characterization of well-founded subclasses that are Zermelo-Fraenkel universes

Let $V$ be the universe. Elements of $V$ are called sets. A function is a relation $F$ such that for every $x$, there is at most one $y$ such that $(x,y)$ is in $F$. If $x$ is a set, $F''(x)$ is ...
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0answers
68 views

Background & Advice for a self-learner of Descriptive Set Theory

A rather straight to the point soft-question: What kind of background should have somebody who wants to study properly descriptive set theory? More specifically, how much analysis should she/he ...
2
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2answers
120 views

A question about $\aleph_1$-dense sets and the basis problem for uncountable linear orderings

I have a question which I have been unable to find a reference for and which I explain as follows: Recall that a set of reals $X$ is $\kappa$-dense if between any two real numbers there are exactly ...
3
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1answer
53 views

Is there a standard notation for building sets up form a given one?

In ZFC each set $S$ has a well-founded membership tree building $S$ up from the empty set $\emptyset$. You could attach the membership tree for any given set $A$ on each of the bottom nodes for the ...
4
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2answers
119 views

Set theoretic universe in consistency proofs

I am having difficulties understanding the relative consistency proof $Con(ZF)\rightarrow Con(ZFC)$. Most authors seem to assume at the outset the existence of some universe $V$ satisfying $ZF$ and ...
3
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1answer
70 views

General Distributive Law and Axiom of Choice

Where can I find the proof of the fact that general distributive law of union over intersection and intersection over union is equivalent to Axiom of Choice? The mathematical formulation of the ...
3
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0answers
29 views

Existence of formula $\phi$ satisfying $\phi^M\to \mathrm{ZF}^M$ for every transitive proper class $M$

I try to prove such exercise problem in Kunen: Let $M$ be a transitive proper class, then there is a finite conjunction $\phi$ of axioms of ZF, such that whenever $M$ is a transitive class which ...
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0answers
25 views

On relation between absoluteness and elementary substructures in ZF

Let $V$ be the universe of sets. $A$ a class in $V$. Definition: A formula $\phi(x_1,\ldots,x_n)$ with $x_1,\ldots,x_n$ free variables is absolute with respect to $A$ if for all $x_1,\ldots,x_n$ in ...
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4answers
305 views

Why aren't valid higher order logic sentences recursively enumerable in full semantics?

It's said (proven in some reduction to the Gödel–Rosser theorem?) that second order logic and higher fails to be complete for full semantics; that is there isn't any semi-algorithm for determining if ...
1
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1answer
59 views

Computing the rank funciton of the Well founded universe of sets

Definitions: $(1)$ We define the $V_\alpha$ function by transfinite recursion as: $V_0=\varnothing$; $V_{\alpha+1}=P(V_\alpha)$; Lim$(\lambda)\rightarrow ...
1
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1answer
69 views

Does $\aleph_0\cdot\kappa=\kappa$ for every $\kappa\ge\aleph_0$ hold in ZF?

It is easy to show that for any (Dedekind) infinite cardinal $\kappa$ we have $\aleph_0+\kappa=\kappa$. Definition of an infinite cardinal is a cardinal such that $\aleph_0\le\kappa$. (I believe ...
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2answers
163 views

How to prove that Gödel's Incompleteness Theorems apply to ZFC?

Let us denote Robinson Arithmetic as Q and Primitive Recursive Arithmetic as PRA. Let $T$ be a formal theory formulated in the language of arithmetic. According to this page on the Stanford ...
2
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1answer
75 views

Modern algebra and set theory: ZFC vs. NBG

This may be somewhat of a philosophical question and is probably nitpicking, but it is also one that has always bothered me a little: Is it not more natural consider NBG set theory as the foundation ...
0
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1answer
37 views

Function from an expression

How do you formalize, in ZFC set theory, the process of forming a function from an expression? Intuitively, I want to say something like this (I am guessing some vocabulary): There is an expression ...
4
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1answer
64 views

Prove that $\forall\alpha\geq\omega$, $|L_\alpha|=|\alpha|$ without AC

Without using Axiom of Choice, prove that $$\forall\alpha\geq\omega,~~|L_\alpha|=|\alpha|,$$ in which $\alpha$ is an ordinals, $\omega$ is the set of natural numbers, $L_\alpha$ is the $\alpha$-th ...
4
votes
1answer
102 views

Are there logical arguments against modern $\sf ZFC$ set theory?

As of asking this question, my knowledge of set theory is quite pedestrian. I've read about it in numerous nontechnical papers and even worked through three chapters of Jech - Set Theory, but in terms ...
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0answers
56 views

Choice-like axioms close to AC & rank-into-rank hypothesizes

Consider the following folklore theorems within ZF, Theorem 1: $V=L$ implies "$0^{\sharp}$ doesn't exist". Theorem 2: $V=L[U]$ implies "$0^{\dagger}$ doesn't exist". Theorem 3: $AC$ ...