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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1answer
44 views

Cardinality of set $\mathbb{N}^{\mathbb{N}}$ and $\{0,1\}^{\mathbb{N}}$

How to show that these two sets have the same cardinality? I know that to show that two sets have the same cardinality I need to show that there is an bijection from one set to another.
1
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1answer
28 views

Proving statements , that require Zorn's lemma , for countable case directly by well-ordering principle of natural numbers

We know that for countable sets , the existence of a choice function is a consequence of the well-ordering principle ; and it is also known that the results like "every vector space has a maximal ...
6
votes
0answers
58 views

Natural Numbers Object and the Axiom of Infinity

It is well known (if you're a topos-theorist, you will call it the definition), that the natural numbers $\mathbb{N}$ together with the zero constant $0$ and the successor function $1\xrightarrow{\ 0\ ...
2
votes
0answers
46 views

Set theory, motivating factors [duplicate]

Set theory seems to pop up in many different fields of mathematics. As someone with a CS degree, I've only encountered very basic set theory; dealing with non-specific sets, and their intersections, ...
2
votes
1answer
31 views

If $\phi$ is a $\Sigma_2$ sentence and $H_{\kappa} \models \phi$, then $V \models \phi$?

In the title question, $\kappa$ is any infinite cardinal. It's easy to see that the result is true if $\phi$ is $\Delta_0$ or $\Sigma_1$. I first tried proving the result for $\Pi_1$, but I don't see ...
0
votes
2answers
61 views

How much foundation do we get for free in $\sf ZFC$?

In $\sf ZFC$ we have the axiom of infinity and thus can define the natural numbers $$\mathbb N \equiv \bigcap\{X:\emptyset\in X\land \forall n(n\in X\implies n\cup\{n\}\in X)\}.$$ From this it's not ...
0
votes
0answers
21 views

Questions of $n$-linked in poset

Let $2\leq n \leq \omega$ and $F$ be a set of size $\leq n$ . Let $\mathbb{P}$ be a poset and $Q \subseteq \mathbb{P}$ an $n$-linked subset. Questions: if $\dot{a} $ is a name for a menber of ...
0
votes
1answer
65 views

Definable real numbers

Reading this Wikipedia page I found this definition: A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language ...
1
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1answer
53 views

About $\kappa$-Knaster and $\kappa$-linked

For an infinite cardinal $\kappa$ and a partial order $\mathbb{P}$, we say: (1) $\mathbb{P}$ has the $\kappa$-linked is a union of $\kappa$-many linked subsets. (2) $\mathbb{P}$ has the ...
4
votes
2answers
82 views

$\omega_2$ is not a countable union of countable sets [duplicate]

Without using axiom of choice, can we show that $\omega_2$ is not a countable union of countable sets? I know this cannot be done for $\omega_1$.
0
votes
1answer
63 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
28 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
29 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 ...
9
votes
3answers
238 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
47 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
72 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
28 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
28 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
154 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
43 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
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0answers
34 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
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2answers
46 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
157 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
62 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
99 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 ...
5
votes
2answers
75 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 ...
2
votes
2answers
48 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 ...
0
votes
0answers
28 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
50 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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0answers
83 views
+50

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
68 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
358 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
81 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: ...
7
votes
0answers
133 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
245 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?
1
vote
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
79 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 ...
5
votes
1answer
76 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 ...
-2
votes
1answer
70 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
66 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
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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 ...
1
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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
97 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
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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 ...