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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14
votes
5answers
2k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
2
votes
0answers
38 views

Status of Ultimate L

I have not heard much from Ultimate L development. What is happening now? The last time I heard from it was like 5 years ago, and I haven't heard much since then. Apparently, Hugh Woodin moved from ...
0
votes
0answers
42 views

Prove the Axiom of Replacement implies the Axiom of Specification. [duplicate]

The Axiom of Specification says that: For a set $ A$ and objects $x\in A$,for any object $y$ there exists a set: $$ y \in \{x: P(x) \wedge x \in A\} \implies y \in A \text{ and } P(y)$$ Where $P(x)$ ...
0
votes
1answer
83 views

Books for Ordinals and Cardinals

I am looking for a nice introductory book to read to learn and master ordinals and cardinals. Please help me!
0
votes
1answer
37 views

A road-map through “Combinatorial Set theory: With gentle intro to independence proofs”

I'm going to study independence proofs form Halbeisen's book. It seems that some material is not needed to study independence proofs, so it seems that the book contains more material than my needs. ...
42
votes
13answers
14k views

Why is “the set of all sets” a paradox?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is ...
2
votes
1answer
59 views

Lexicographic Orderings of Aronszajn Trees are Aronszajn Lines

I am trying to prove that every lexicographic ordering of a Aronszajn tree is a Aronszajn Line. If $T$ is a tree, a lexicographic ordering of $T$ is defined as follows: For each ...
3
votes
0answers
70 views

Many $(\kappa+\alpha)$-strong cardinals below a $(\kappa+\beta)$-strong one for $\alpha < \beta < \kappa$

Let $\lambda$ be an ordinal. A cardinal $\kappa$ is $\lambda$-strong iff there is some inner model $M$ and an elementary embedding $$ j \colon V \rightarrow M $$ s.t. $crit(j) = \kappa$ and $V_\lambda ...
11
votes
10answers
646 views

What are the most prominent uses of transfinite induction outside of set theory?

What are the most prominent uses of transfinite induction in fields of mathematics other than set theory? (Was it used in Cantor's investigations of trigonometric series?)
32
votes
3answers
2k views

Set Theoretic Definition of Numbers

I am reading the book by Goldrei on Classic Set Theory. My question is more of a clarification. It is on if we are overloading symbols in some cases. For instance, when we define $2$ as a natural ...
1
vote
1answer
104 views

Why adding a club of $\aleph_1$ collapses $\aleph_1$ to $\aleph_0$?

Let $\{S_n \mid n < \omega\}$ be a partition of $\aleph_1$ into countably many disjoint stationary subsets. Why adding a club of $\aleph_1$ to each $\aleph_1 \setminus S_n$ collapses $\aleph_1$ to ...
3
votes
1answer
65 views

$\operatorname{Fn}(\lambda,2,\lambda)$ collapses $\lambda^+$ to $\operatorname{cf}\lambda$ if $\lambda$ is singular?

It is an exercise problem in Kunen (VII G5). I shall show that $\operatorname{Fn}(\lambda, 2, \lambda)$ adds a map from $\theta = \operatorname{cf}\lambda$ onto $\lambda^+$ for singular $\lambda$. ...
2
votes
0answers
33 views

Help regarding a proof about Dedekind finiteness

I got this one as an exercise. If $F$ is Dedekind finite and $t \notin F$ then prove that $F \cup \{t\}$ is also Dedekind finite. I gave this as an answer: If $F\cup \{t\}$ is Dedekind infinite then ...
1
vote
2answers
96 views

Are Cantor's set theory and the axiom of choice well accepted nowadays? [on hold]

Is there a "more consistent" theory ? Do mathematicians agree about the validity of the results obbteined by this theory ?
3
votes
1answer
66 views

The notations $ M[X] $ and $ V[A] $ and intermediate models

I am confused about Jech's notations $ M[X] $ and $ V[A] $. Let us begin with exercise 13.34 (see [Jec02, p. 199]). Let $ \mathbf{M} $ be a transitive model of $ \mathsf{ZF} $ containing all ...
6
votes
1answer
108 views

Proving that $\sf Add$$(\aleph_\omega , 1)$ collapses cardinals $\leq \aleph_\omega$

First, let me fix some notation. $\sf Fn$$(I, J, \kappa) = $ the poset of all partial functions $p$ such that $|p| < \kappa$, dom$(p) \subseteq I$ and rng$(p) \subseteq J$. $\sf Add$$(\kappa, ...
4
votes
1answer
51 views

A problem with transitivity of ordering of stationary sets

In chapter 8 of Jech's Set Theory in the Third Millenium, he defines the following ordering on stationary sets: $S<T$ if the set $\{\alpha\in T:\alpha\cap S\text{ is stationary}\}$ is stationary. ...
2
votes
1answer
31 views

limit and infinite ordinals: same thing? [duplicate]

I'd like to clarify my understanding re: limit and infinite ordinals. This post says: An initial infinite ordinal is a limit ordinal. Is it true the other way around? That is if I have a limit ...
0
votes
4answers
429 views

Sets. Classes. …?

Classes can be considerd as "larger" than sets in the sense that any set is a class. Is there mathematical object which is "larger" than classes ?
2
votes
2answers
23 views

Multiplying two ordinals where one has been raised to power of $\omega$. Term order matters?

When multiplying two ordinals that are both raised to some power, my book says that one adds the exponents. But what happens if one of the exponents is $\omega$ ? Does the order of the terms matter ? ...
8
votes
2answers
149 views

Why do we know that Gödel sentences are true in the standard model of set theory, but do not know if the continuum hypothesis is?

By what methods can we identify sentences that are true in the standard model of set theory, but not in other models? In particular, how do we prove that Gödel sentences are true in the standard ...
0
votes
0answers
27 views

example of multiplication of ordinals with infinite cardinality with larger value on right where we dont' take the max?

I recall reading about a rule for multiplying ordinals where at least one is infinite, and where the cardinality of the multiplier (on right) is larger than the multiplicand (on left). If I recall ...
6
votes
1answer
95 views

Good resources for studying independence proofs

I've finished most of Enderton's set theory. And I intend to spend some time studying independence proofs. I'm more interested in independence of axiom of choice not CH. From I know so far, there ...
-4
votes
0answers
33 views

What is the definition of the following concepts and how I can characterize each concept. [closed]

What is the definition of the following concepts and how I can characterize each concept. A set $A\subseteq 2^{\omega}$ is Lebesgue measurable zero if ? A set $A\subseteq \omega^{\omega}$ is ...
3
votes
2answers
38 views

How do you multiply this

How can you multiply these ordinal numbers: $(\omega+1)(\omega+1)(\omega2+2)$ I tried and have gotten to this: $(\omega^2+1)(\omega2+2)$ Is that the correct way, or did i made a mistake?
9
votes
6answers
369 views

Principle of Transfinite Induction

I am well familiar with the principle of mathematical induction. But while reading a paper by Roggenkamp, I encountered the Principle of Transfinite Induction (PTI). I do not know the theory of ...
0
votes
1answer
41 views

any example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$?

I am working from a book that gives the above problem, but no solution ;^(. That is: Show an example of non-zero ordinals $a,b,c$ for which $a < b$, but $a^c \ge b^c$. Exponentiation here ...
5
votes
1answer
79 views

Aronszajn lines

Exercise 32 of chapter 2 of Kunen (1980) tells me to show that there exists a total ordering with no $\omega_1$ strictly increasing/decreasing sequencies such that every separable subspace is nowhere ...
7
votes
1answer
146 views

What is the significance of having Prime Ideal Theorem in models for failure of Axiom of Choice?

Prime Ideal Theorem says: PIT: Every ideal on a Boolean algebra can be extended to a prime ideal. It follows from Axiom of Choice but is weaker than it. In many cases I saw that people check ...
1
vote
3answers
77 views

Can we generalize Aleph numbers to non integer values? [duplicate]

I'm really new to those kind of arguments so don't call me mad but I was wondering if there is a way to define an infinite set which cardinality is an Aleph-number like: $\aleph_{\frac 12}$. I have a ...
4
votes
1answer
77 views

Set with cardinality of Aleph 2

What is a simple or commonly known set that has a cardinality equal to Aleph 2 or greater?
0
votes
0answers
41 views

Duals of filters, an explicit formula for meet?

Fix some set $U$. Recall that filters on $U$ are nonempty sets $F$ such that $A\cap B\in F \Leftrightarrow A\in F\land B\in F$. Replacing every element of $F$ with its complement and simultaneously ...
10
votes
2answers
155 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
29 views

A filter concentrates on a set

Given a filter $\mathcal F$ on some nonempty set $X$ and some $Y \subseteq X$, people often say that "$\mathcal F$ concentrates on $Y$". Questions: Does this simply mean $$\forall Z \subseteq X ...
1
vote
1answer
44 views

Non well founded model of ZF

I read that there are $(M,E)$ well-founded model of ZF that in fact not to be. However I don't understand in what sense "not really to be ". I think that $(M,E)$ is well-founded because $(M,E)\vDash ...
1
vote
1answer
44 views

Inequality similar to König's Lemma

If $I$ is an set containing more than 2 elements, $A_i,\,B_i$ are sets for every $i \in I$, and $\#A_i < \#B_i$ for every $i \in I$ then $$\sum_{i\in I}\#A_i < \prod_{i\in I}\#B_i$$ Will ...
2
votes
1answer
81 views

Equivalent condition for CH

I would like to know why the following condition $\otimes$ is equivalent with the Continuum Hypotheses. $\otimes$ There exists a sequence $\langle A_{\alpha} | \alpha < \omega_1\rangle$, such ...
1
vote
4answers
90 views

$A=\{A,\emptyset\}$ and axiom of regularity

The axiom of regularity says: (R) $\forall x[x\not=\emptyset\to\exists y(y\in x\land x\cap y=\emptyset)]$. From (R) it follows that there is no infinite membership chain (imc). Consider this set: ...
0
votes
0answers
18 views

onto map implies existence of one one map and AC [duplicate]

Let us assume the fact: $f: A \to B$ onto function implies there exist $1-1$ function from $B$ to $A$. Would it imply AC? I know every surjective function has right inverse this fact is ...
7
votes
2answers
106 views

Well ordering agreeing with ordinal ordering on a cardinal

This is an exercise from Kunen book - Set theory, an introduction to independence proofs. Let $\kappa$ be an infinite cardinal and $\triangleleft$ any well-ordering of $\kappa$. Show that there is an ...
1
vote
1answer
36 views

Dually Dedekind Set and Weakly Dedekind set

$A$ is dually Dedekind infinite (dD-infinite), if there is a surjective non-injective map from $A$ onto $A$. $A$ is weakly Dedekind infinite (wD-infinite), if there is a surjective map from $A$ onto ...
1
vote
1answer
50 views

dual Dedekind-infinity may not imply Dedekind-infinite without AC

It is written in wikipedia: https://en.wikipedia.org/wiki/Dedekind-infinite_set It is not provable (in ZF without the AC) that dual Dedekind-infinity implies that A is Dedekind-infinite. (For ...
1
vote
1answer
53 views

Cohen forcing factoring

I start from $M$ a transitive countable model of $ZFC + \mathbb V= \mathbb L$ and I add a single Cohen generic $G$. Now if $A \in M[G]$ is also Cohen generic over $\mathbb L$ and $M[A] \ne M[G]$, can ...
0
votes
1answer
38 views

The order isomorphism theorem without Replacement

In $\sf ZF$, there is the following theorem: Theorem: For any well-ordered set $\langle A,\prec\rangle$, there is a (unique) order isomorphism $F$ from $\langle\alpha,\in\rangle$ to $\langle ...
3
votes
1answer
67 views

Is there an flat unordered pairing function in ZFC?

Is there an unordered pairing function that does not increase rank whenever the max rank is infinite, in ZFC? An unordered pairing function is one such that $f(x,y)=f(z,w)$ iff $(x=z \wedge y=w) \vee ...
1
vote
2answers
52 views

Does the principle of schematic dependent choice follow from ZFCU?

Let ZFCU be ZFC modified in the usual way to allow for urelements but without an axiom stating that there is a set of all urelements. Let the principle of Schematic Dependent Choice (SDC) be: ...
-1
votes
1answer
65 views

Can we unify every pair of inner models of ZFC by a same hierarchy?

Definition: Fix a ground model $V$ of ZFC. Let $F:V\rightarrow V$ be a definable class function (we call it an operator). The hierarchy $W^F$ corresponding to $F$ is defined as follows: ...
2
votes
0answers
33 views

On Levy's formal definition of class terms

I've been reading Levy's Basic Set Theory and it has recently been drawn to my attention a certain problem with Levy's definition of formulas and terms in his extended language (section I.4.1) (well, ...
2
votes
0answers
37 views

Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?

Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that is well-founded. This means that for every non-empty set $a$ there is a set $b\in a$ such that $cRb\implies c\notin a$. Here ...
3
votes
1answer
68 views

Are there weak versions of the axiom of choice equivalent to weak versions of Zorn's lemma and similar principles?

I recalled reading about other weaker forms of $AC$, for example countable choice, where we could make choices from a sequence $(S_{k})_{k \in \mathbb{N}}$ of non-empty sets. I also recalled mention ...