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

1
vote
1answer
49 views

Closed subsets of $\beta \mathbb R$

Definitions. Suppose $X$ is a topological space. $w(X)=\min\{|\mathcal B|:\mathcal B$ is a base for $X\}+\omega$ $e(X)=\sup\{|D|:D\subseteq X$ is closed and discrete$\}+\omega$ $K(X)$ is the ...
3
votes
1answer
89 views

Definition in Kunen

In Kunen's second edition of set theory he gives the following definition Let $(\mathbb{Q},\leq_\mathbb{Q},\mathbb{1}_\mathbb{Q})$, and $(\mathbb{P},\leq_\mathbb{P},\mathbb{1}_\mathbb{P})$ be forcing ...
1
vote
1answer
76 views

Partitioning an infinite cardinal number

Can we partition an infinite cardinal greater than aleph null, into countable number of cofinal subsets? Can we have restriction on the cardinality of cofinal subsets? For example, suppose $\alpha$ ...
9
votes
1answer
99 views

Choosing elements of linear orders

Is it consistent with ZF that there can be a countable family of linear orders, each isomorphic to $\mathbb Z$ (that is, every element has a unique predecessor and successor, and any two elements have ...
4
votes
1answer
75 views

Model of complete extension of Zermelo set theory

Chang and Keisler's Model theory gives the following exercise problem: Prove that there is a complete extension $T$ of Zermelo set theory which has arbitrary large natural models. (A model ...
5
votes
1answer
65 views

Cardinal Arithmetic: $\beth_\omega$

Is the following equality independent of ZFC: $$\bigcup_{\aleph_{\beta}<\beth_{\omega}}2^{\aleph_{\beta}}=\beth_\omega $$ Consistent: Now that the equality is consistent with ZFC since it holds ...
4
votes
1answer
83 views

How to make an order isomorphism

Two linear orders $A$ and $B$ have starting points $a_0$ and $b_0$, and have cofinalities $\omega_1$. Let $(a_\alpha )_{\alpha<\omega_1}$ and $(b_\alpha )_{\alpha<\omega_1}$ be cofinal ...
0
votes
2answers
59 views

Ordinal existence

Is there any ordinal $\alpha$ such that $\omega ^ {\omega ^ \alpha} = \alpha$? Could you please suggest me how to even try to solve this?
1
vote
0answers
107 views

Does the Math-tea Argument Have Any Relevance to the Method of Forcing?

The Math-tea Argument (i.e. the argument that, for example, there must be real numbers that we cannot describe or define, because there are only countably many definitions, but uncountably many reals) ...
2
votes
2answers
83 views

An exercise in Kunen

The following exercise appears in Kunen; In $M$, let $\mathbb{P}=Fn(\kappa,\lambda)$ (i.e. the finite partial functions), where $\aleph_0\leq\kappa<\lambda$. Then $\lambda$ is countable in $M[G]$, ...
1
vote
1answer
27 views

Class Sizes and Ordering

It is independent of ZFC that every proper class has the same "size". (See Size Of Proper Classes). However, is it necessary that the proper class sizes must be linearly ordered if then are not the ...
2
votes
1answer
91 views

Are All Nonstandard Models of PA Ill-Founded?

We often use sets to represent natural numbers, but we can also use natural numbers to represent sets. For example, we can use the binary expansion of a natural number to represent a set. The number ...
2
votes
1answer
117 views

Best Less-Famous Texts for Forcing

There are many books, papers and lecture notes which give an introduction to forcing (e.g. Jech or Kunen's books) but here I am looking for some possibly less-famous useful comprehensive texts for ...
0
votes
0answers
62 views

Large Cardinals with Elementary Extension Characterization

Question: Which large cardinals $\kappa$ have a characterization in the following form: $\kappa$ is large if and only if for all cardinals $\lambda>\kappa$, $\langle W_{\kappa},\in\rangle\prec ...
6
votes
1answer
109 views

The continuum hypothesis (CH) and its equivalent

For a set $A \subseteq \mathbb{R}^2$ and $x,y \in \mathbb{R}$, we define $A^y=\{x \in \mathbb{R}\mid (x,y) \in A\}$ and $A_x=\{y \in \mathbb{R}\mid(x,y) \in A\}$. Proposition: The continuum ...
0
votes
1answer
56 views

Newbie approach to understand generalized continuum hypothesis

There is this theorem that size of power set constructed from infinite set is "more" infinite than the previous set: $$ \begin{eqnarray*} \aleph_0 &= |\mathbb{N}| \\ \aleph_{n+1} &= ...
2
votes
1answer
107 views

Cardinalities of Collections of Models

Let $T$ be a complete theory in a countable language (with only infinite models). Recall the spectrum function: $I(\aleph_\alpha,T)=$ the number of non-isomorphic models of $T$ of cardinality ...
0
votes
1answer
59 views

Representative elements in the symmetric difference metric

The symmetric difference is a natural way to quantify the distance between measurable sets: $$d(S,T)=measure([S\setminus T]\cup[T\setminus S])$$ This is a pseudo-metric because there may be ...
2
votes
1answer
35 views

A problem in transitivity of set

Let $D$ be a transitive set with the following property : $$\forall a\in D:\quad a\subseteq B\,\Longrightarrow a\in B$$ Prove that $D\subseteq B$. I think there will be needed axiom of ...
1
vote
1answer
56 views

why $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$ doesn't hold?

this rule doesn't hold for all ordinals $\alpha ,\beta,\gamma$ :$(\beta + \gamma)\alpha = \beta\alpha + \beta \gamma$. I tested many examples but all of them holds for it ! does this hold ? $(B \cup ...
2
votes
1answer
56 views

Axiom of Limitation of Size Reference Request

On the wiki page for the axiom of limitation of size in NGB is states that the axiom of replacement and the axiom of global choice are equivalent to the axiom of limitation of size (see ...
2
votes
0answers
110 views

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? [closed]

If physical space had cardinality larger than $\aleph_1$, would we need new math to describe dynamics? For instance, would the dynamics of a sea of virtual particles of cardinality $\aleph_1$ would ...
2
votes
1answer
47 views

Is the cardinality of the continuum weakly Mahlo?

Is $2^{\aleph_0}$ a weakly Mahlo cardinal? Can it be? That is, are there conditions (such as the negation of the continuum hypothesis or something) under which it is, and other conditions under which ...
2
votes
1answer
77 views

Some Questions Regarding Pointwise Definable Models of ZFC

In their paper "Pointwise Definable Models of Set Theory" Hamkins, Linetsky, and Reitz prove the following theorem: "Every countable model of ZFC has a pointwise definable class forcing extension." ...
0
votes
0answers
23 views

Collection of sets with a given cardinality $\kappa$ is not set [duplicate]

Show that collection of all sets with cardinality $\kappa\neq0$, is not set. I'll state my approach and I need to see whether this idea is precise/precisable or not : First let $K$ be the set ...
2
votes
1answer
59 views

Confusion Regarding the Axiom of Countable Choice

My current understanding of the Axiom of Countable Choice is that the following example needs it in order to work: Let $X$ be a countable family of finite sets. Then there exists a choice function ...
1
vote
3answers
97 views

Proof that $\mathbb{N}\times \mathbb{N}\cong \mathbb{N}$ without $AC_\omega$ or Arithmetic

Proving that $\mathbb{N}\times \mathbb{N} \cong \mathbb{N}$ is incredibly useful for proving that the countable union of countable sets is countable and the fact that finite cartesian products of ...
4
votes
1answer
38 views

Extension of ZFC models preserves cardinals

Let $M \subseteq N$ be countable transitive ZFC set models. Assume that this extension preserves cardinals, i.e. if $\alpha$ is an ordinal number (this notion is absolute) such that $(\alpha \text{ is ...
6
votes
0answers
94 views

Who wants to learn set theory? [closed]

So set theory is something I really want to learn. I found this document that I really like, except the fact that it doesn't prove all of it's theorems in with a lot of detail (a lot of times they say ...
2
votes
1answer
47 views

Adding subsets of regular cardinals (Jech p. 226)

On p. 226 of his ${\it Set}$ ${\it Theory}$, Jech considers adding $\lambda$ many subsets of $\kappa$ to a ground model $M$. He outlines a suitable partial order on the assumption that $M$ satisfies ...
5
votes
1answer
75 views

Partial order on cardinalities without the axiom of choice

Cardinality can still be defined without choice, e.g. as equivalence class of equipotent sets, see Defining cardinality in the absence of choice. Injections define partial order on cardinalities by ...
5
votes
0answers
73 views

“How strong is $\Diamond_\kappa^+$?”

In $\text{ZFC}$, we know that $\Diamond_{\kappa}^+ \implies \Diamond_{\kappa}$ and $\Diamond_{\kappa^+} \implies 2^\kappa = \kappa^+$, so that we may think of $\Diamond^+$, i.e. $\Diamond_\kappa^+$ ...
0
votes
1answer
46 views

Examples of non-trivial determiner formulas of trnsitive models of ZFC

Notation: For each $\{\in\}$-formula $\varphi(x_1,\cdots,x_n)$ and each $\in$-model $M$, define: $$\varphi (M)=\{(a_{1},\cdots,a_{n})\in M^{n}~|~\langle M,\in\rangle\models \varphi ...
1
vote
1answer
80 views

On the number of countable models of complete theories of models of ZFC [duplicate]

Fix the language of set theory $\mathcal{L}=\{\in\}$. Let $\langle M,\in\rangle$ be a set or proper class model of ZFC (e.g. $M$ could be $L$, $HOD$, $V_{\kappa}$ for some inaccessible cardinal ...
2
votes
2answers
155 views

Is it Theoretically Impossible to Demonstrate that Set Theories Are Consistent?

I have to present on the main realist and non-realist arguments for/against set theory. According to one of my sources, it remains a matter of debate as to whether any of the set theories' (ZF, NF, ...
18
votes
2answers
291 views

Covering $\mathbb R^2$ with function graphs

Suppose we have a countable family of function graphs (each function is $\mathbb R\to\mathbb R$, not necessary continuous). Obviously, they cannot cover the whole plane $\mathbb R^2$, because they ...
10
votes
1answer
75 views

Martin's Axiom and products of c.c.c. spaces

A topological space $X$ has the c.c.c. if every family of pairwise disjoint non-empty open subsets of $X$ is countable. Consider the following statement: $(P)$ The product of two c.c.c. spaces is ...
6
votes
2answers
251 views

Set theoretic realism

What are the main contemporary arguments for and against realism about set theory?
1
vote
2answers
45 views

Image of a proper class under one-one function is a proper class

Working under Zermelo-Franenkel set theory. how I can prove following: Let $V$ be proper class. Let $F : V \rightarrow U$ be a one-one function. Then $U$ has to be a proper class.
1
vote
1answer
46 views

Insistence on limit ordinal in Kunen's proof of reflection.

I'm reading for an exam on set theory using Kunen's first edition, and we've more or less been told that the reflection theorem will come up. The proof constructively yields an ordinal ...
3
votes
2answers
93 views

Introduction to Proper Forcing Reference

What is a good introduction to proper forcing? I am aware of Shelah Proper and Improper Forcing, but I heard this book may be somewhat challenging to read. There is also Devlin's The Yorkshiremen's ...
1
vote
0answers
34 views

$\sigma$-field and Uncountable ordinal

I have been trying to get my head around this question. Any help greatly appreciated. Let $\mathscr{C}$ be any class of subsets $\Omega$ with $\emptyset,\Omega\in \mathscr{C}$. Define ...
0
votes
1answer
142 views

Applications of Countable Infinite Sets and Power Sets

What are the possible applications of Countable Infinite Sets and Power Sets in areas that are not strictly mathematical? Also I want to know the significance they carry. What was not possible ...
2
votes
1answer
72 views

Equivalent (?) definitions of Axiom A

(Why) Are the following definitions of Axiom A equivalent? Soft question: Which one is more common, natural, or usually easier to verify? What was Baumgartner's original definition? ( B ...
1
vote
1answer
94 views

A question regarding Worldly Cardinals and L

For some $L_\kappa$ in the constructible hierarchy, does there exist a $\kappa$ such that $\kappa$ is a worldly cardinal and that $L_\kappa$ contains all of the constructible reals? The motivation ...
1
vote
2answers
62 views

Defining a sequence of models of ZFC satisfying a certain countability requirement

Assuming whatever large cardinal axioms we please, is there a way to define a sequence of sets $M^\omega$ such that for all $n < \omega$, the following hold? $M_n$ is a transitive model of ZFC ...
3
votes
1answer
72 views

The order-theoretic structure of the ordinal numbers in non-standard models

I have read the following. Proposition. Let $T$ denote the first-order theory of $\mathbb{N}$ in the language of arithmetic. Then any countable non-standard model of $T$ is order isomorphic to a ...
3
votes
0answers
85 views

Countable transitive $T \vDash ZFC-P$ with $\approx$ not absolute: why do we need $H(\aleph_3)$?

I am trying to solve Exercise IV.3.31 from Kunen's Foundations of Mathematics. I think I have a solution but I am confused by one of the hints. By request, here is the text of the exercise. ...
0
votes
1answer
44 views

Existence of Certain Names in Iterated Forcing

Suppose $\mathbb{P}$ is a forcing. Let $\dot{\mathbb{Q}}$, $\dot{<_\mathbb{Q}}$, and $\dot{1}_\mathbb{Q}$ be a name such that $1_\mathbb{P} \Vdash_\mathbb{P} ``\langle \dot{\mathbb{Q}}, ...
0
votes
0answers
40 views

Relation between axiom of foundation and constructible set

I have the following problem: In my lecures, the professor states the following theorem: If ZFC is consistent then ZFC+V=L is consistent. (where L are the constructible set). My question is : in ZFC ...