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 ...

learn more… | top users | synonyms

0
votes
1answer
27 views

Functions operating in uncountable sets with cardinality $\gt\aleph_1$

A generic function $y=f(x)$ maps a number in the set of real number $X$ in another number in the set $Y$. It's well known that the irrational numbers are not countable. It's also known we can get a ...
6
votes
1answer
54 views

Proper Forcing and Sequence of Names for Reals.

I read something that seems to suggest the following is true: If $\mathbb{P}$ is a proper forcing, $|\mathbb{P}| = \aleph_1$, and $\mathsf{CH}$ holds, then there exists a sequence $\{(p_\xi, ...
0
votes
1answer
55 views

Godel's completeness theorem and formula that states consistency of ZF

Godel's completeness theorem, in original formulation, says that every logically valid statement/formula has finite deduction of a formula. Now then there is Godel's incompleteness theorem. Would this ...
1
vote
0answers
47 views

Are there axiomatizations of first order logic or set theory defined in first order logic or set theory?

There are several axiomatizations for number theory, group theory, and other theories represented in first order logic. Further, these theories are also representable in set theory such as $\sf ZFC$ ...
3
votes
0answers
58 views

Is it possible for infinite sets to exist in ZFC with the negation of the Axiom of Infinity?

The Axiom of Infinity states that at least one inductive set exists. Inductive sets are infinite, but not all infinite sets are inductive. Suppose that we take ZFC with the negation of the Axiom of ...
2
votes
2answers
77 views

Uncountable models for integers

Part of Asaf Karagila's brilliant answer to one of my other questions puzzles me a lot. Namely, I find it hard to understand how there can be a model for ZFC with uncountably many integers. My ...
2
votes
1answer
82 views

Maximal model for $\Bbb R$?

I have not dealt professionally with set theory, so excuse me if my way of formulating this question does not completely follow standard terminology. Actually, my question is about whether or not the ...
2
votes
0answers
40 views

A problem with an assumption in a previous lemma for the proof of Silver´s Theorem on SCH in Jech´s “Set Theory”

In the Jech´s textbook proof of Silver´s Theorem, specifically in the Lemma 8.15, there are an assumption in the beginning of the proof. First, the Lemma 8.15 says that, under the assumption that ...
-2
votes
0answers
56 views

Can we consider each first order structure as a pure set?

Pure sets are the simplest kind of first order structures in the language $\mathcal{L}=\emptyset$. As same as any other generalization, the notion of a first order structure preserves some properties ...
1
vote
0answers
46 views

Is consistency strength order dense?

Definition: Let $\sigma$, $\theta$ be two statement in the language of set theory. We say $\sigma <_{c} \theta$ ($\sigma$ is strictly lower than $\theta$ in consistency strength order) if within ...
3
votes
1answer
46 views

A question about second-order logic and inaccessible cardinals.

Let $\kappa$ denote an inaccessible cardinal, and suppose $T \in V_\kappa$ is a second-order theory. Now consider some mathematical structure $X \in V_\kappa$. Then I think it is clear that $X \models ...
1
vote
1answer
23 views

Writing the ZF formula for the choice function (given Well-ordering)

If every set has a well order, then the axiom of choice follows: Given a well order on $\bigcup_{i \in I} A_i$ we define the choice function in this way $f(i) =$ "the first element of $\bigcup_{i \in ...
-2
votes
1answer
29 views

axiom of regularity and power sets [closed]

So axiom of regularity works for any non-empty set. Is that means that a set like this {aab, +, 50, ), (, **} and all of it's subsets are actually made of empty sets? And not just sets, but power ...
8
votes
1answer
130 views

Why is ZF favoured over NBG

So why is ZF favoured over NBG? Is it historical, but I've read after Gödel's monograph was published, NBG was more prominent. Or is the reason that NBG gets the cold shoulder to do with forcing and ...
1
vote
1answer
52 views

Comprehension and Impredicativity

Wang and McNaughton (Les Systemes Axiomatiques de la Theorie des Ensembles, 1953) discuss briefly the topic of impredicativity in chapter 2 (titled 'Type Theory') of the above mentioned book, but I'm ...
0
votes
1answer
18 views

Redundancy in definiton for forcing poset

The following definition appears in Kunen (2nd edition): For any sets $I,J$ and cardinal $\lambda$: $\text{Fn}_{\lambda}(I,J)$ is the set of all $p\in{[I\times{J}]}^{<\lambda}$ such that $p$ is ...
2
votes
1answer
36 views

What is the definition of the Feferman-Levy model?

Any (reference to) definition of Feferman-Levy model in set theory? I cannot find any... Though I know what is Levy collapse.
2
votes
1answer
16 views

Uniqueness in transfinite recursion.

Theorem of transfinite recursion: Given a well-ordered set $A$ let $\varphi(g,y)$ be a ZF formula such that for every $a \in A$ and every function $g$ with domain $I_a$ (where $I_a$ is the initial ...
2
votes
1answer
43 views

If a version of GCH holds for Chang's $\kappa$-constructibility, does a version of GCH hold for $L_{\infty}$?

In C.C. Chang's paper "Sets Constructible Using $L_{\kappa \kappa}$" one can "deduce a version of the GCH, theorem VI [(iv)--my comment], assuming that all sets are $\kappa$-constructible." Now ...
6
votes
1answer
107 views

Why is adding Cohen reals so “uninteresting”?

I read the following in this paper (Otmar Spinas, Proper products. Proceedings of the AMS, 137 (8), (2009), 2767–2772): $ \mathbb{L}^2 $ adds a Cohen real. Thus it was considered uninteresting ...
2
votes
2answers
95 views

Examples of figures whose area depend on axiom of choice

Many times I have heard that there are some 'figures' whose area is not fixed and they depend on axiom of choice. For example, In this answer. In this lecture about probabilty. I did not know ...
0
votes
2answers
56 views

Coming from naive set theory, are the following understanding right in ZF system? [closed]

In ZF system, a subset $S'$ of set $S$ is: If $S'$ is a set and for any object $b$ $b\in S'\implies x\in S$. That is, before being a subset, $S'$ must be a set by itself. The set of integer ...
3
votes
1answer
88 views

What would provable existence of a non-standard natural number imply about ZFC?

Assume that in ZFC we can define a set $x$ in some way and that the following are provable: $x$ is a finite ordinal, (i.e. $x\in\omega$) $x\neq0$ $x\neq1$ $x\neq2$ etc. So to say that the ...
3
votes
1answer
47 views

What are $\in$-well orderable classes?

$V$ is not $\in$-well ordered but $Ord$ is. Is $Ord$ the unique proper class of the universe which is $\in$-well ordered?
-2
votes
0answers
60 views

Is the structure with sets and the ZFC axioms a model of the first order logic?

Wikipedia says ZFC is a one-sorted theory in first-order logic. The signature has equality and a single primitive binary relation, set membership, which is usually denoted ∈. The formula a ∈ b ...
3
votes
0answers
61 views

Expository Papers on Extenders

Extenders are discussed in many set theory text books. Here I am looking for some expository "papers" which are focused on this subject and its connection with forcing and large cardinals. More ...
1
vote
0answers
34 views

Is infinitary Levy hierarchy well-defined?

The well-known Levy hierarchy of formulas consist of two $\omega$-sequences of sets of formulas of different complexity $\langle\langle \Sigma_n:n\in \omega\rangle,\langle \Pi_n:n\in ...
0
votes
0answers
47 views

Similarity of the Universe $V$ and its $\kappa$-Fragments

Intuitively, if a cardinal $\kappa$ is "large" then the $\kappa$-fragment of the universe $V_{\kappa}$ is so "similar" to the entire universe $V$. For example if $\kappa$ is supercompact then ...
1
vote
0answers
34 views

Filters on a set of filters, are they equivalent to just filters?

Let $F(X)$ be the set of all filters (including the improper filter) on a poset $X$, ordered reversely to set-theoretic inclusion of filters. Let $U$ be a set. Is $F(F(\mathscr{P}U))$ order ...
1
vote
0answers
51 views

Existence of a generic ultrafilter over constructible universe

I am referring to the Jech's "Set Theory" book. He wants to show that, if $(P,<)$ is the notion of forcing defined as: $p \in P$ iff $p$ is a finite sequence of ordinals less than $\omega_{1}^{L}$ ...
1
vote
1answer
56 views

infinite Cartesian products

Quote from Wikipedia, infinite set: The Cartesian product of an infinite number of sets each containing at least two elements is either empty or infinite. I know that this is the regime where we ...
1
vote
0answers
58 views

Zuckerman's “Sets and Transfinite Numbers”

I am beginning a study in set theory and I found an old book in my school's library by Martin Zuckeman called Sets and Transfinite Numbers which was published in the 1970's. Has anyone used this text ...
0
votes
0answers
32 views

Flaw in the proof that a set is countable [duplicate]

Q: Let S be the set containing all sequences of 0's and 1's. i.e $S = \{(a_1,a_2,a_3,a_4,\ldots) : a_i = 0 \text{ or } 1\}$ Show that S is countable. Proof(Flawed) : Let $A_i$ be the ...
0
votes
0answers
62 views

Algebraic constructions to add a real to a sub-group of $\langle \mathbb{R}^+,.\rangle$

Consider the group $\langle \mathbb{R}^+,.\rangle$ and let $r$ be a real number (e.g. $r=\sqrt{2}$). I would like to know about all known algebraic constructions to build a sub-group of $\langle ...
5
votes
1answer
48 views

Exact $\Delta$ - System Lemma

The well-known $\Delta$-system lemma states: Each uncountable family of finite sets has an uncountable sub-family with a root. Precisely: $$\forall F~~~(|F|>\aleph_0~\wedge~\forall X\in ...
1
vote
1answer
82 views

difference of infinities [closed]

So I haven't studied this subject In a while, and when i get home, I'm going to consult my book on the Theory of computation to reinforce my understanding, but here is my question. So I understand ...
3
votes
1answer
75 views

$\aleph_\omega$ and large ordinals?

Assuming the GCH, each time you take the powerset of an aleph number, the subscript increases by one. So there is $\aleph_0, \aleph_1, \aleph_2\ldots \aleph_\omega\ldots \aleph_{\epsilon_0}\ldots$ and ...
1
vote
1answer
49 views

A question about $cov(I)$

Given a set $X$ and an ideal $I \subseteq \mathcal P(X)$, $cov(I) := \min \{ |\mathcal A| | \mathcal A \subseteq I (\bigcup \mathcal A = X)\}$. My question is: isn't it possible that for each ...
1
vote
0answers
77 views

A Question Regarding Consistent Fragments of Naive (Ideal) Set Theory

It is well known that Naive (to some, otherwise known as Ideal) set theory, that is, the set theory generated by the axioms: (EXT) (x)(x $\in$ A iff x $\in$ B) iff A=B (COMP) ($\exists$y)(x)(x ...
1
vote
1answer
52 views

Could induction give us an infinite sequence of sets $X_1 \subseteq X_2 \subseteq \cdots$ or do we need the axiom of choice?

Suppose at each step $n$ of a proof by induction, we have a set $X_n$. Also, $X_{n} \subseteq X_{n+1}$. Is induction enough to give us $\bigcup_{n \in \mathbb{N}} X_n$? Or rather, when is it enough? ...
0
votes
2answers
112 views

Is it true that $\omega=\{0,1,2,3,\ldots\}$ in ZFC?

This is a bit of a philosophical question. According to "Set Theory" by Jech, the set $\omega$ of natural numbers is defined as the least nonzero limit ordinal. After thinking about this definition ...
10
votes
1answer
106 views

Partitions of the Cantor space into parities

Call a partition of $2^\mathbb{N} = A\cup B$ a parity partition if, for any $n\in\mathbb{N}$, flipping the $n$th bit of any element of $A$ results in an element of $B$, and vice-versa. Given a choice ...
0
votes
1answer
31 views

What is the meaning of the notation $cf(add(\mathcal I))$?

I am trying to prover the following claim: Assume that $\mathcal I$ is an ideal over some infinite set $X$. Then, $cf(add(\mathcal I))=add(\mathcal I)$. Doe anyone know what is the meaning of the ...
1
vote
0answers
54 views

Other Types of Order Types

Consider following fundamental fact of ZFC. Fact ‎‎$‎‎‎\forall ‎‎~\text{well-ordering}~\langle ‎A,<‎\rangle~~~‎\exists ‎!\alpha~~~‎‎‎‎‎\langle A,<‎‎\rangle‎\cong‎\langle‎\alpha‎,\in‎‎\rangle‎$‎ ...
3
votes
0answers
48 views

Exercise on preservation of cardinals

I'm trying to solve exercise 12.1 in Prof. Monk's lectures on set theory which asks me to show that $Fn(\omega_1, 2, \omega_1)$ preserves cardinals larger than $\omega_2$. Now, the only method in the ...
7
votes
1answer
102 views

Constructing a Banach space of cardinality $\beth_{\omega+1}$

This is related to yesterday's question Constructing a vector space of dimension $\beth_\omega$; it's the next exercise (I.13.35 (a)) in Kunen's Set Theory. Let $B_0 = \ell^1$ and let $B_{n+1} = ...
3
votes
2answers
98 views

Constructing a vector space of dimension $\beth_\omega$

I'm trying to solve Exercise I.13.34 of Kunen's Set Theory, which goes as follows (paraphrased): Let $F$ be a field with $|F| < \beth_\omega$, and $W_0$ a vector space over $F$ with $\aleph_0 ...
0
votes
1answer
35 views

Model of a system of set theory

I was trying to understand Easton's theorem's proof. But I am not able to understand a few terms- model of a system of set theory, ranked variables, free variables, abstraction term, set constant, ...
-1
votes
1answer
38 views

Functions with cardinals as domain

How do we actually define a function with cardinals as domain? For example, take domain of the function as $\aleph_1$. Do we define it for all cardinal numbers strictly less than $\aleph_1$?
2
votes
0answers
38 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 ...