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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5
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2answers
40 views

Pigeonhole Principle and Sets

Can anyone point me in the right direction for this homework question? I know what the pigeonhole principle is but don't see how it helps :( Let $n\geqslant 1$ be an integer and consider the set S = ...
1
vote
1answer
20 views

Knaster property

Let $\mathbb{P}$ poset. I want to show $\kappa$-Knaster implice $\kappa$-c.c and Any precaliber poset is $\kappa$-Knaster. can you help me give a suggestion of how to test this as I have had many ...
0
votes
0answers
31 views

Understanding the condition of $\diamondsuit^+$

I try to understand $\diamondsuit^+$, a variant of the diamond principle. It says that there is a sequence $\langle \mathcal{A}_\alpha\rangle_{\alpha<\omega_1}$ of collection of subsets which ...
1
vote
1answer
23 views

No generic is definable in a perfect notion of forcing of a model of Peano Arithmetic

I would like to prove Lemma 6.1.2.2 from The Structure of Models of Peano Arithmetic by Kossak and Schmerl. Let $\mathcal{M}$ be a countable model of Peano Arithmetic and $\mathbb{P}=\langle P, \le ...
2
votes
0answers
22 views

When a $\mathbb{P}$ - generic filter is $\kappa$ - complete?

By definition a $\mathbb{P}$ - generic filter $G$ over a ground model $M$ is $\aleph_0$ - complete because for any finite set of conditions in $G$ there is a condition $p\in G$ such that $p$ is ...
10
votes
2answers
139 views

Is $\text{Hom}(\prod_p \Bbb Z/p\Bbb Z, \Bbb Q) = 0$ possible without choice?

That divisible abelian groups are precisely the injective groups is equivalent to choice; indeed, there are some models of ZF with no injective groups at all. Now, given that $\Bbb Q$ is injective, ...
1
vote
1answer
41 views

Non analytic ideals on $\omega$

I would like to gather examples of NON analytic ideals on $\omega$. However, I have found nothing in the books and papers I have consulted. Could anyone tell me some reference/s?
0
votes
1answer
67 views

Proof that the finite product of nonempty sets is nonempty without axiom of choice from ZF

How do you prove that for $X_{i} \neq \emptyset$, $i \in \{1,...,n\}$ that $\prod_{i=1}^{n} X_{i} \neq \emptyset$ only using the ZF axioms but not the Axiom of Choice? I would like to see a rigorous ...
1
vote
1answer
25 views

Proving that if $A\leq_c B$ then $\chi(A)\leq_o \chi(B)$

By Hartog's Theorem we knoe that for every set $A$ There is a definite operation $\chi(A)$ which associates with each set $A$, a well ordered set $\chi(A) = (h(A),{\leq_{\chi(A)}}),$ such that $h(A) ...
0
votes
0answers
27 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
votes
2answers
147 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 \, ...
0
votes
1answer
36 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 $(\lesssim) ⊆ A×A$ ...
1
vote
0answers
33 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 $|V|=\aleph_0$?
1
vote
2answers
44 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 ...
4
votes
3answers
152 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 ...
1
vote
1answer
58 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, ...
0
votes
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 ...
-1
votes
1answer
91 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 ...
6
votes
2answers
73 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 ...
3
votes
2answers
45 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 ...
1
vote
0answers
25 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
votes
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
votes
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
votes
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 ...
1
vote
2answers
66 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
votes
2answers
356 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
votes
2answers
74 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
votes
0answers
129 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. ...
4
votes
4answers
241 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
vote
1answer
55 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?
0
votes
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 ...
0
votes
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?
1
vote
1answer
50 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 ...
2
votes
2answers
77 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
72 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 ...
-1
votes
1answer
65 views

Questions of Hechler forcing

Shows that Hechler forcing adds Cohen real. A suggestion please. Can you tell me reference Hechler forcing.
6
votes
2answers
65 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
vote
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.
0
votes
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 ...
1
vote
0answers
65 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. ...
8
votes
0answers
96 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 ...
0
votes
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
votes
0answers
69 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
votes
2answers
123 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
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
votes
2answers
120 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
votes
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
votes
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 ...
1
vote
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 ...
7
votes
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 ...