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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2
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2answers
15 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 ? ...
0
votes
0answers
19 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
2answers
95 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 ...
-4
votes
0answers
31 views

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

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 ...
4
votes
1answer
63 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 ...
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?
0
votes
1answer
38 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 ...
1
vote
1answer
40 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 ...
9
votes
9answers
522 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?)
9
votes
6answers
355 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 ...
4
votes
1answer
68 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 ...
3
votes
1answer
73 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?
1
vote
3answers
74 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 ...
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 ...
7
votes
1answer
115 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 ...
4
votes
1answer
26 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
43 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 ...
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 ...
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 ...
0
votes
1answer
37 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 ...
1
vote
2answers
51 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
vote
1answer
51 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 ...
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 ...
2
votes
0answers
30 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, ...
3
votes
1answer
66 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 ...
0
votes
1answer
48 views

Is there a constructible flat pairing function?

In ZFC set theory, is there a Skolem function f such that ZFC can prove f is a flat pairing function? And if so, can someone explicitly give me a formula?
4
votes
1answer
84 views

Can every ccc forcing of size $2^{\aleph_0}$ be $\sigma$-linked?

It is well known that $MA_{\aleph_1}$ implies that every ccc forcing of size $< 2^{\aleph_0}$ is $\sigma$-linked (in fact - a countable union of filters). On the other hand, a separative forcing ...
2
votes
0answers
35 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 ...
-1
votes
0answers
51 views

Question about 2-step iterations [closed]

Let $\mathbb{P}$ be poset. If $\mathbb{\dot{Q}}=\dot{G}$, the $\mathbb{P}$-name of the $\mathbb{P}$-generic filter, as a suborder of $\mathbb{P}$ then $\Vdash_{\mathbb{P}} \mathbb{\dot{Q}} ...
1
vote
1answer
44 views

Why $\forall{f\in \omega^{\omega}\cap V}$ $ \forall^{\infty}n f(n)\neq f_{G}(n)$

Eventually different forcing, $\mathbb{E}=\{\langle s,H \rangle:s \in \omega^{\omega}\wedge H\subseteq [\omega^{\omega}]^{<\omega}\}$. ordered by $(s',H')\leq (s,H)$ iff $s \subseteq s'$ and $H ...
1
vote
1answer
53 views

Which ZFC axiom schemes are reducible to a single axiom?

It is a remarkable fact that the $\in$-induction scheme (i.e. the claim that $\phi(x)$ holds for any $x$, whenever $\phi(\emptyset)$ and $\forall x(\forall y\in x \phi(y)) \Rightarrow \phi(x)$) is ...
7
votes
2answers
96 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 ...
0
votes
2answers
22 views

Help with defining binary relation image in ZFC

I need to define in ZFC the following things: image and domain of a binary relation ($\{ x \mid (x,y)\in f \}$ would be a definition of domain, but it is a class for which is for me is not quite ...
4
votes
1answer
55 views

Does there exist a model of $ZF¬C$ in which there is a function on $\mathbb R$ which is sequentially continuous at a point where it is not continuous? [duplicate]

Does there exist a model of $ZF¬C$ in which there is a function $f:\mathbb R \to \mathbb R$ such that $f$ is sequentially continuous at some $a \in \mathbb R$ but not $\epsilon-\delta$ continuous , ...
0
votes
1answer
31 views

$V[G]=V[f_{G}]$ if $f_{G}=\bigcup\{s:(s,H)\in G\}$.

Eventually different forcing, $\mathbb{E}$, consists of pairs $(s,F)$, where $s \in \omega^{<\omega}$ and $F$ is a finite set of reals. $(s,F)\leq (t,G)$ iff $t \subseteq s$ and $G \subseteq F$ ...
1
vote
0answers
51 views

Isomorphic as sets. Do they mean bijective? (Kashiwara's Categories & Sheaves)

Here's the book. On page 10 it says: A set is called $\mathcal{U}$-small if it is isomorphic to a set belonging to $\mathcal{U}$. and on page 11 it says: A category $C$ is called a ...
0
votes
0answers
32 views

If $\alpha \leq \delta$ and cf$(\lambda)\geq\kappa$, then $\mathbb{P}_{\lambda}\simeq \text{limdir}_{\alpha<\lambda}\mathbb{P}_{\alpha}$.

let $\kappa$ be an infinite cardinal, $\delta$ an ordinal and $\mathcal{I}=\{C\subseteq \delta :|C|<\kappa \}$. Consider an $\mathcal{I}$-support iteration $(\langle \mathbb{P}_{\alpha \leq ...
5
votes
1answer
78 views

Ramsey Combinatorics and linear order

Prove that the following are equivalent for an infinite cardinal $\kappa$. (1) $\kappa \to (\kappa)^2_2$ (2) In any linearly ordered set of cardinality $\kappa$ there is either a well-ordered or a ...
4
votes
1answer
69 views

Closed and unbounded set on a specific set or ordinals.

I'm having problems on showing the following: Let $\kappa$ be a strongly inaccesible cardinal. Show that the set of all ordinals $\alpha <\kappa$ such that $V_\alpha \models ZFC$ contains a club. ...
5
votes
2answers
106 views

A construction of sigma-algebras - surely not new, right?

I know no descriptive set theory. I've stumbled on something that must be well known, being so simple. But it contradicts something I've been told by smart people; the question is whether it's well ...
4
votes
1answer
64 views

First Uncountable Ordinal Cofinality: Needs AC?

Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset). This ...
2
votes
1answer
63 views

Are there collections of sets that are neither a set nor a definable proper class?

Working in ZFC, are there collections of sets that are neither a set nor a definable proper class? I mean if some collection of sets is not a set can we necessarily conclude that it is a definable ...
4
votes
1answer
53 views

What conditions must be checked for that $c$ is Cohen over $V$.

$\textbf{Hechler forcing} $ Let $\mathbb{D}=\{(s,f): s \in \omega^{<\omega},f\in \omega^{\omega} \text{and} s \subseteq f\}$, ordered by $(t,g)\leq (s,f)$ if $s \subseteq t$, $g$ dominates $f$ ...
3
votes
0answers
56 views

References for Introductory Model Theory focusing on applications other than algebra

I would appreciate suggestions for references (books, lecture notes, articles etc...) on Model Theory (at an introductory level) that don't focus mostly on algebra when giving examples and/or applying ...
2
votes
1answer
43 views

ZF and the Existence of Finitely additive measure on $\mathcal{P}(\mathbb{R})$

My understanding is that Solovay (1970)'s relative consistency shows that if ZFC+I has a model then ZF+DC has a model in which every subset of the reals is Lebesgue measurable (and hence ...
5
votes
2answers
85 views

Proving that $\alpha\approx|\alpha|$ is not constructive

$\sf ZF$ tells us that for every ordinal $\alpha$ there is a bijection $f:\alpha\to|\alpha|$, more or less by the definition of $|\alpha|$, the cardinal of $\alpha$. What I would like to show is that ...
6
votes
2answers
120 views

The non-existence of non-principal ultrafilters in ZF

In Hrbacek and Jech (1999, p.205), they point out that "it is known that the theorem [the extension of any filter to an ultrafilter] cannot be proved in Zermelo-Fraenkel set theory alone." And in Jech ...