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 ...
3
votes
1answer
33 views
In the absence of GCH, what can ZFC tell us about $f(\alpha,\beta)$?
Lets temporarily adopt the (non-standard) definition that for all cardinals $\kappa$ and $\nu$ we have $[\kappa,\nu) = \{\mbox{cardinals } \eta \mid \kappa \leq \eta < \nu\}.$ Note the word ...
2
votes
1answer
53 views
Cardinality definitions in variants of ZF theory, yet again
As has been discussed here before, it is possible to define cardinality (a function $|\cdot|$ from the universal class to a class of "cardinals" such that $x\approx y\iff|x|=|y|$) using either AC or ...
2
votes
2answers
35 views
$\aleph_a^{cf(\aleph_a)} = \aleph_{a+1}$ for $\aleph_a$ regular assuming GCH
$\aleph_a^{cf(\aleph_a)} = \aleph_{a+1}$ for $\aleph_a$ regular assuming GCH
$\aleph_a$ regular, so $cf(\aleph_a) = \aleph_a$.
Assuming GCH $2^{\aleph_a} = \aleph_{a+1}$ holds.
But I'm missing the ...
11
votes
2answers
202 views
Is there an axiom of ZFC expressing that GCH fails as badly as possible?
The GCH axiom basically says that for all infinite cardinal numbers $\kappa$, the number of cardinals lying strictly between $\kappa$ and $2^\kappa$ is as small as possible. Namely, there are none.
...
1
vote
1answer
49 views
The comprehension axioms follows from the replacement schema.
I hope to show that the comprehension axioms follows from the replacement schema.
This is a solution that professor wrote.
$P(u,u)$: every set $u$, exists an unique $u$ such that $\psi(u)$.
Then ...
2
votes
1answer
62 views
Prerequisites for understanding Borel determinacy
I have just learned that Gale-Stewart game is determined for open and closed sets, so naturally I'm interested to understand Borel determinacy which seems to be on a totally different level. What's ...
5
votes
2answers
130 views
Sets question, without Zorn's lemma [duplicate]
Is there any proof to $|P(A)|=|P(B)| \Longrightarrow |A|=|B|$ that doesn't rely on Zorn's lemma (which means, without using the fact that $|A|\neq|B| \Longrightarrow |A|<|B|$ or $|A|>|B|$ ) ?
...
1
vote
4answers
54 views
How to prove the bijection from unit interval to unit-square constructed by Cantor is discontinuous?
Can it be proved with mathematical analysis?
1
vote
1answer
82 views
Is there any binary relation operator that has these properties in any objects?
Consider binary relation operators
d b
q p
(with a direct correspondence by generalization of:
< >
≮ ≯
these are a ...
1
vote
1answer
45 views
Proving $\alpha+(\beta+\gamma) = (\alpha+\beta)+\gamma$ for ordinals
I am following Jech's construction, by definition $\alpha+0 = \alpha, \alpha+(\beta+1)=(\alpha+\beta)+1$, and for limit $\beta$ we define $\alpha+\beta = \cup\{\alpha+\xi: \xi<\beta\}$.
Jech's ...
2
votes
1answer
62 views
A club in $\omega_1$ induced by a bijection
I'm trying to prove that if $f:\omega_1\to\omega_1\times\omega_1$ is a bijection, then the set $X_f=\{\alpha\in\omega_1:f[\alpha]=\alpha\times\alpha\}$ is a club in $\omega_1.$
So I think I see that ...
4
votes
1answer
49 views
Why is this cardinal regular?
I have the following problem in front of me.
Show that if $\kappa$ is the least cardinal such that $2^\kappa>2^{\aleph_0},$ then $\kappa$ is regular.
I've scribbled this:
Suppose ...
3
votes
2answers
107 views
An uncountable famliy in $2^\omega$ linearly ordered by inclusion without passing to $2^{\Bbb Q}$.
I was solving the problem of finding an uncountable, linearly ordered subset of $2^\omega$ ordered by inclusion. I was having a lot of trouble before I realized that I can substitute anything ...
3
votes
3answers
65 views
Which ordinals embed in $2^\omega$ ordered by inclusion?
Which ordinals can be embedded in the power set of $\omega$ ordered by inclusion?
I see that $\omega\cdot\omega$ can (and therefore anything less than that): we can partition $\omega$ into $\omega$ ...
2
votes
0answers
66 views
Formalizing model-theoretical large cardinals in a formal system for ZFC
I work with the Metamath formal system, which is expressive enough to define ZFC and prove some nontrivial stuff, but my current goal is to define large cardinals, and I'm hitting a wall somewhere ...
5
votes
1answer
76 views
In ZF, does there exist an ordinal of provably uncountable cofinality?
Question is in the title. In ZFC, one can prove that $\aleph_{\alpha+1}$ is regular, so there is a large source of cardinals with uncountable cofinality, but in ZF, it is consistent that ${\rm ...
2
votes
2answers
65 views
Concrete Mathematics Iversonian Set Relation Clarification
Sorry for asking a very dumb question, but in Concrete Mathematics(Graham,Knuth,Patashnik), chapter 2 section 4, Knuth talks about this formula called "Rocky Road".
This is the formula to use when ...
2
votes
1answer
53 views
existence of a subset that carries all measure of a chain
Suppose $m: \mathcal{P}(X) \to [0, 1]$ is a $\sigma$-additive measure on all subsets of $X$ and $\mathcal{L} = \{A_\alpha: \alpha < \beta\}$ is a well ordered by inclusion chain of subsets of $X$ ...
3
votes
2answers
61 views
Stationary sets
Currently learning about stationary sets, came across this problem:
Let $\{S_\alpha : \alpha<\omega_1\}$ be disjoint pairwise sets with $S_\alpha \subseteq \omega_1$ non-stationary for each ...
2
votes
1answer
35 views
Axiom UB on Grothendieck Universes
I am having problems understanding Grothendieck's second axiom on universes: In SGA it reads "Let $R\{x\}$ be a relation and ${\mathscr U}$ be a universe. If there exists $y\in{\mathscr U}$ such that ...
1
vote
0answers
56 views
Tarski's axiom implies a proper class of inaccessible cardinals
I'm trying to prove this theorem, but is it even true? A Tarski's class is a set $T$ such that:
For each $y\in T$, ${\cal P}(y)\subseteq T$ and ${\cal P}(y)\in T$, and
for each $A\subseteq T$ such ...
6
votes
3answers
147 views
More than continuum many functions
Suppose we have a function $f_\alpha:\mathbb{N} \to o$ for each $\alpha<c^+$ (successor cardinal of $c=2^{\aleph_{0}}$), where $o$ is some ordinal. Show that there exists a set $S \subseteq c^+$ ...
2
votes
2answers
77 views
A confusion on Axiom of infinity
I'm currently working "the elements of advanced mathematics" by steven g. krantz, currently on Chapter 5.
I came to "Axiom of Infinity" which roughly states:
$$\exists A \; s.t. \; \phi \in A \; and ...
1
vote
4answers
99 views
cardinality of all real sequences
I was wondering what the cardinality of the set of all real sequences is. A random search through this site says that it is equal to the cardinality of the real numbers. This is very surprising to me, ...
8
votes
1answer
147 views
Infinite combinatorial games
Hercules vs. Hydra: Recall the story where every time Hercules cuts of a head, two more heads grow instead. Now suppose the following: The hydra starts off with one head, but every time Hercules cuts ...
2
votes
1answer
69 views
Similar to Fodor lemma
Let $\lambda>\aleph_0$ be a regular cardinal such that $S \subseteq \lambda$ is not a stationary subset. Prove that there exists a regressive function $f:S \to \lambda$ such that ...
0
votes
1answer
30 views
Question Involving Transitive Sets
In Jech's Set Theory, we are asked to show the following two statements:
1.3 If $X$ is inductive, then the set $\{x \in X : x \subset X\}$ is inductive. Hence $N$ is transitive, and for each $n$, ...
2
votes
1answer
43 views
Understanding the Relationship Between Natural Language Predicates and First-Order Formulae
Consider the class $C$ defined in terms of first-order formula $\phi(x,p_1, \ldots , p_n)$. That is, $C = \{x : \phi(x, p_1, \ldots , p_n)\}$.
Now this is all well and good, but I'm trying to ...
4
votes
1answer
43 views
Understanding the Definition of the Axiom Schema of Specification
Consider the Axiom Schema of Separation:
If $P$ is a property (with paramter $p$), then for any $X$ and $p$
there exists a set $Y = \{u \in X : P(u,p)\}$ that contains all those
$u \in X$ that ...
2
votes
1answer
48 views
Understanding Formulas in First Order Logic
I'm reading a text on Set Theory which states that any formula, say $\phi$, is ultimately built up from atomic sentences of form $x \in y$ and $x = y$ via the logical connectives.
So then my question ...
0
votes
1answer
30 views
Definition of ordinal exponentiation
I found that the usual ordinal exponentiation $\alpha^{\beta}$ is the set of functions from $\beta$ to $\alpha$ with finite support, ordered by antilexicographic order. (least significant position ...
2
votes
1answer
75 views
Tetration of Alephs
Does this even mean anything?
$\underbrace{x^{x^{x^{...^{x^x}}}}}_n$
Where $n = \aleph_0$?
Because I "know" it converges when (say) $x = .5$ and $n \to \infty$
3
votes
1answer
82 views
Is it possible to prove $|V_\kappa|=\kappa$ for strongly inaccessible $\kappa$ without AC?
First, let me note that my definition of "strongly inaccessible" is that a nonzero ordinal $\kappa$ is strongly inaccessible if ${\rm cf}(\kappa)=\kappa$ and $\forall\alpha<\kappa, {\cal ...
2
votes
2answers
53 views
Weakly inaccessible cardinals and Discovering Modern Set Theory
So I've been trying to teach myself some set theory and I've come across some exercises in Just and Weese's Discovering Modern Set Theory. To whit:
Pg. 180
Definition 20: A cardinal $\kappa$ is ...
2
votes
2answers
42 views
Question Regarding the Replacement Schema
For each formula $\phi(x,y,p)$, we have the following axiom:
$\forall x \forall y \forall z (\phi(x,y,p) \wedge \phi(x,z,p) \rightarrow y = z) \rightarrow \forall X \exists Y \forall y (y \in Y ...
5
votes
1answer
59 views
What is cofinality of $(\omega^\omega,\le)$?
Let us consider the set $\omega^\omega$ of all maps $\omega\to\omega$ with the pointwise ordering. By cofinality of $(\omega^\omega,\le)$ I mean the smallest cardinality of the subfamily $\mathcal B$ ...
3
votes
5answers
163 views
Does universal set exist?
Take any set $A$. Is it true that the complement of $A$ is non empty?
I am unable to find which axiom in set theory leads to answer affirmative of this question.
3
votes
1answer
50 views
Base change and ordinals
Problem. Define the operation base change from $k$ to $m$: to make the operation for natural number $n$ we should write $n$ in the base-$k$ numeral system and read this in the base-$m$ numeral ...
7
votes
3answers
215 views
Does there exist a function $f : X \to Y$ such that $f \in Y$?
Does there exist a function $f : X \to Y$ such that $f \in Y$?
I think this is related to Russell's paradox, but I'm not exactly sure how.
Added Later: As Brian points out, given any function ...
8
votes
1answer
103 views
Exercise 24.13 of T. Jech's *Set Theory*
Having struggled my way through most of chapter 24 of Jech's Set Theory, I'm stuck on the very last part of the very last question, 24.13:
Let $I=I_{NS}$ be the nonstationary ideal on $\omega_1$, ...
1
vote
2answers
65 views
The existence of the set of all functions on A into B
I need to prove that the set of all functions on A into B, denoted by $B^{A}$, exists.
I think if $|A| = m$ and $|B| = n$ then $|B^{A}|=n^{m}$ because for each element of $A$ there are $n$ ...
0
votes
1answer
29 views
Isomorphisms between Orderings
If $h$ is an isomorphism between $(P,<)$ and $(Q,\prec)$ then show $h^{-1}$ is an isomorphism between $(Q,\prec)$ and $(P,<)$
DEFINITION: $h$ is an isomorphism between $(P,<)$ and ...
1
vote
1answer
53 views
Cardinality of the set of an atmost countable number of functions
If $\beth_0$ is the cardinality of the natural numbers, $\beth_1$ the cardinality of the continuum, $\beth_2$ the cardinality of the set of all functions from $\mathbb{R}$ to $\mathbb{R}$, where ...
3
votes
1answer
123 views
Possible mistake in Specker's thesis
From Specker's thesis "Zur Axiomatik der Mengenlehre (Fundierungs- und Auswahlaxiom)":
Take $A$ to be a countable infinite set of atoms, $G$ the group of all permutations of the universe $\mathcal ...
4
votes
3answers
78 views
Which set theories without the power set axiom are used occasionally?
To get a set theory without the power set axiom, I could just take an existing set theory like ZF or ZFC, and remove the power set axiom. However, perhaps I would have to be careful how to formulate ...
2
votes
2answers
78 views
Book recommendations for studying mathematical areas based on set theory
I am at the end of my studies with set theory, and I would like to
continue in fundamental fashion, and study for example calculus based
on set theory. So, I am talking about not calculus the way it ...
12
votes
2answers
98 views
How far is it true that statements dependent on Axiom of Choice are not constructive.
Axiom of Choice is often used in mathematics to construct various objects, such as basis of $\mathbb{R}$ as a vector space over $\mathbb{Q}$, unmeasurable subset of $\mathbb{R}$, or a non-principal ...
8
votes
2answers
108 views
weak* separable question
(In another question Nate Eldredge said I should ask this.)
Let $X$ be a Banach space, $X^\ast$ the dual space, and $B_{X^\ast}$ the unit ball of $X^\ast$. In the weak* topology for $X^\ast$, does ...
1
vote
1answer
68 views
Permutation models: when are they isomorphic?
Let $M, M'$ be two permutations models with atoms $A,A'$. Assume $A$ and $A'$ have the same cardinality so that there is a bijection $f: A \to A'$. Now assume $M$ and $M'$ are defined in terms of the ...
4
votes
0answers
108 views
Largest provably existing ordinal in ZF without power set
If I remove the power set axiom from ZF, but retain the axiom of infinity, what will be the largest ordinal that can still be proven to exist. Or, if no largest such ordinal exist, what is the limit ...
