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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0answers
40 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
51 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 ...
6
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3answers
259 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 ...
5
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2answers
96 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 ...
1
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1answer
70 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
95 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 ...
5
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2answers
1k views

How to understand the regular cardinal? [closed]

How to understand the regular cardinal? Could someone give me some examples?
2
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2answers
100 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
40 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 ...
5
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2answers
121 views

Set theory by Julia Robinson

I used to have a set theory textbook downloaded free from the internet. I lost my laptop in the airport of a city in Eastern Europe, and it was not found (or perhaps “not found”) by the airport ...
3
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1answer
64 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 ...
4
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1answer
191 views

Recommended books/articles for learning set theory

What is the recommended reading for thoroughly learning set theory? I'm currently studying Kunen's book [1]. But what then, and in what order? One needs to learn large cardinals, inner models and ...
3
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1answer
43 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
90 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 ...
6
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4answers
356 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 ...
19
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5answers
2k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
3
votes
2answers
366 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} ...
3
votes
2answers
100 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: ...
29
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2answers
813 views

When should I be doing cohomology?

Background: I'm a logic student with very little background in cohomology etc., so this question is fairly naive. Although mathematical logic is generally perceived as sitting off on its own, there ...
5
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4answers
277 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 ...
1
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1answer
43 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 ...
9
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4answers
322 views

Partition of $\mathbb{R}^+$ into two semigroups

Can the semigroup $(\mathbb R^+ ,+)$ be partitioned into two semigroups? I have been trying but haven't found anything, please help.
0
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1answer
48 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?
1
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1answer
53 views

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

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 ...
3
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1answer
65 views

Addition on well ordered sets not-commutative by showing $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$

My goal is to show that addition on well ordered sets are non-commutative by showing that, $[0,1) +_o \mathbb{N} =_o \mathbb{N} \neq_o \mathbb{N} +_o [0,1)$ Some definitions (let A and B be ...
9
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3answers
319 views

Is Zorn's lemma necessary to show discontinuous $f\colon {\mathbb R} \to {\mathbb R}$ satisfying $f(x+y) = f(x) + f(y)$?

A UC Berkeley prelim exam problem asked whether an additive function $f\colon {\mathbb R} \to {\mathbb R}$, i.e. satisfying $f(x + y) = f(x) + f(y)$ must be continuous. The counterexample involved ...
6
votes
2answers
78 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 $\widehat X$, ...
4
votes
1answer
122 views

Is the anti-foundation axiom considered constructive?

In the area of theoretical computer science that I am interested in, constructive mathematics is of practical interest because it gives algorithms that can be implemented on a computer. However, ...
2
votes
2answers
95 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 ...
4
votes
1answer
81 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
81 views

Questions of Hechler forcing

Shows that Hechler forcing adds Cohen real. A suggestion please. Can you tell me reference Hechler forcing.
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1answer
59 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
47 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 ...
1
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0answers
68 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. ...
11
votes
3answers
404 views

Axiom of Choice: What exactly is a choice, and when and why is it needed?

I'm having trouble understanding the necessity of the Axiom of Choice. Given a set of non-empty subsets, what is the necessity of a function that picks out one element from each of those subsets? For ...
8
votes
0answers
98 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 ...
7
votes
4answers
664 views

Simple (even toy) examples for uses of Ordinals?

I want to describe Ordinals using as much low-level mathematics as possible, but I need examples in order to explain the general idea. I want to show how certain mathematical objects are constructed ...
0
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0answers
25 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 ...
2
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0answers
80 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 ...
3
votes
1answer
75 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 ...
4
votes
1answer
136 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 ...
1
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2answers
153 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
votes
1answer
56 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 ...
3
votes
0answers
52 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 ...
8
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3answers
271 views

Is the “domain of discourse” in axiomatic set theory also a “set”?

The domain of discourse is defined by Wikipedia as the "set of entities over which certain variables of interest in some formal treatment may range." However, I believe we could not call the domain of ...
10
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2answers
209 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 ...
0
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0answers
29 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 ...
1
vote
1answer
68 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 ...
8
votes
2answers
302 views

Are there non-equivalent cardinal arithmetics?

‎Generalizing a concept in mathematics is always a problematic situation. In most cases there are several ways to generalize a notion and it is not easy to decide if a particular generalization is ...
2
votes
1answer
79 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 ...