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
votes
1answer
233 views

Better representaion of natural numbers as sets?

Natural numbers can be represented as $0=\emptyset$ $1=\{\emptyset\}$ $2=\{\{\emptyset\}\}$ $...$ or as $0=\emptyset$ $1=\{0\}=0\cup\{0\}$ $2=\{0,1\}=1\cup\{1\}$ $...$ What are the names of ...
2
votes
1answer
147 views

Prüfer groups are countable

I have read that for any prime number $p$ the Prüfer $p$-group is countable. My question is: where can I find a proof of this fact? Thanks.
2
votes
1answer
182 views

Is the set $S$ of functions $S = \{ f : \mathbb R \to \mathbb R \}$ from the reals to the reals a totally ordered set?

Is the set $S = \{ f : \mathbb R \to \mathbb R \}$ of functions $f$ from the reals to the reals a totally ordered set ? I think the question is clear. Note that there is a bijection to $g(x)$ gives ...
4
votes
4answers
562 views
22
votes
1answer
848 views

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$?

What are some good intuitions for understanding Souslin's operation $\mathcal{A}$? Recall the definition: Let $S = \mathbb{N^{<N}} = \bigcup_{n = 1}^\infty \mathbb{N}^n$ be the set of ...
0
votes
1answer
129 views

Contour Infinites and Vector Spaces

We usually define Hilbert or finite dimensional vector spaces, and even topologies or differential geometry on $\mathbb{R}^n$ , so I wonder what is the implication of doing that on some extended ...
0
votes
2answers
169 views

Why is the class of all sets a stage?

I want to prove that the class of all sets $\mathbb{S}=\{x \mid x=x \}$ is a stage (p. 15) (and then that it is a limit thus that it is the successor of another stage). One way to do it is to proof ...
3
votes
2answers
292 views

Is there a categorification of (infinite) ordinal arithmetic?

Background for the curious reader: An ordinal $\beta$ is a transitive set in the sense that $\alpha\in\beta$ implies $\alpha\subset\beta$. Any ordinal is naturally well-ordered under $\in$ (so any ...
4
votes
1answer
203 views

Why do we need a pullback for the definition or classification of subobjects?

Regarding the subobject classifier construction, why do we need the pullback? Monos from $U$ to $X$ are called subobjects, but I see that there might be injections which just have elements of the X ...
7
votes
1answer
335 views

Existence of non-atomic probability measure for given measure zero sets

Let $\Omega$ be a set and $\Sigma$ be a $\sigma$-algebra of subsets of $\Omega$. Let $N$ be a collection of measurable subsets of $\Sigma$. Question: What conditions on $\Sigma$ and $N$ guarantee ...
5
votes
1answer
173 views

What is the definition of the well-founded part of a model of set theory?

I've been trying to understand John Steel's various notes on inner model theory, but the one thing that trips me up is what he calls the well-founded part of a model of set theory. What exactly is the ...
1
vote
1answer
119 views

A product of 2 Bourbakian posets, ordered by pointwise ordering.

A poset $P$ is called Bourbakian if every order-preserving map $P\rightarrow P$ has a LEAST fixed point. Let $P, Q$ be Bourbakian. Let $P\times Q$ be ordered pointwise, that is $(a,b)\le (c,d)$ if and ...
8
votes
3answers
252 views

Does the category framework permit new logics?

It appears to me that a topos permits a broader concept of subsets than the yes/no decission of a characteristic function in a set theory setting. Probably because the subobject classifier doesn't ...
10
votes
1answer
247 views

Maximal ideal space of $c_{\mathcal{U}}$

Let $\mathcal{U}$ be an filter over $\mathbb{N}$. Define $$c_{\mathcal{U}} = \{{(x_n)\in \ell_\infty\colon \lim_{\mathcal{U}, n}x_n =0\}},$$ which is a C*-algebra. Is there an accessible topological ...
3
votes
1answer
176 views

Ramsey, Erdős-Rado partitions

This a specific question about Ramsey type colorings. The arrow notation: If $\kappa$, $\lambda$, $\mu$ are cardinals and $n<\omega$, then $$\kappa\rightarrow(\lambda)^n_\mu$$ if for any function ...
1
vote
1answer
101 views

What will happen when Ax.Inf is replaced to its negation in ZFC? [duplicate]

Possible Duplicate: What are the consequences if Axiom of Infinity is negated? In ZFC, if we replace Ax.Inf to such a statement that every set is finite, then does this theory satisfiable? ...
1
vote
0answers
90 views

Ultrapowers by extenders of potential premice

I have a problem with an argument in Fine structure and iteration trees by Mitchell and Steel. Let $E$ be a $(\kappa, \lambda)$-extender. Let $\dot E^{\mathcal{M}}$ the a unary predicate with is ...
5
votes
1answer
88 views

No $\Delta-$System on a subset of a singular cardinal.

I've been making my way through the new Kunen and I've come across an exercise that I can't work out. The question is this: Let $\kappa$ be a singular cardinal. Show that there is a collection $A$ ...
2
votes
3answers
325 views

Other ways of proving that the set of all countable ordinals is uncountable

I know that the standard way of proving that the set of all countable ordinals is uncountable is by stating that if the set is countable, then it incurs Burali-Forti paradox. Is there other ways of ...
0
votes
1answer
908 views

Division by two in set theory

Let $A,B$ be two sets such that $2A \cong 2B$ (here $2A := A \coprod A$). Then $A \cong B$. This can be proven without the axiom of choice, which means that one can explicitly construct a bijection $A ...
3
votes
1answer
191 views

Question on the use of a parametric version of Transfinite Recursion Theorem in Introduction to Set Theory 3rd ed. by Hrbacek and Jech

My question concerns a proof given on page 118 in the text Introduction to Set Theory 3rd ed. by Hrbacek and Jech. The authors on page 117 prove a version of the transfinite recursion theorem ...
3
votes
1answer
74 views

What is $\cap_{i\in \emptyset}A_i$?

I tried this: $x\in \cap_{i\in \emptyset}A_i\iff x\in A_i\forall i\in \emptyset$ and the right hand side is vacuously ture--right? So this means it is equivalent to $x$ being an element of ... any ...
5
votes
2answers
332 views

Which are “big theorems” of descriptive set theory?

Question: If one were to fully understand 10 theorems in DST, or 15,20,25,30 theorems, which ones would be the most important to understand in order to work towards an understanding of descriptive set ...
4
votes
2answers
113 views

Relation between inacessible cardinals and CH

I know that the two statements “There exists an inaccessible cardinal” and “Continuum Hypothesis” are both independent of ZFC. Now, are those two statements independent of each other? That older ...
16
votes
2answers
965 views

Where is axiom of regularity actually used?

Where is axiom of regularity actually used? Why is it important? Are there some proofs, which are substantially simpler thanks to this axiom? This question was to some extent provoked by Dan ...
5
votes
3answers
1k views

The real numbers and the Von Neumann Universe

So I'm going to prefix this question by saying that I probably don't have a great understanding of what I'm asking. We build the cumulative hierarchy as follows: $V_0=\emptyset$ For every $\alpha$, ...
7
votes
3answers
948 views

Ultra Filter and Axiom of Choice

Some person said me: "The fact that Ultra Filters exist is equivalent to the Axiom of choice". Is this correct? I nees some good references about the subject, please help me. Thanks
4
votes
2answers
464 views

Solovay's Model and Choice

Reference; Foundation for analysis without axiom of choice? Please let me know if I'm misunderstanding something and I hope you explain this with relatively easy words. I am eager to learn, but I ...
4
votes
2answers
147 views

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$?

Is $2^{|\mathbb{N}|} = |\mathbb{R}|$? If so, how? I was reading the Wiki page on the , and it says "Moreover, $\mathbb{R}$ has the same number of elements as the power set of $\mathbb{N}$", but I ...
3
votes
1answer
102 views

Two naive questions about sets

Can every set have a power set ? Does there exist a set A such that there always is a surjection of A onto B , where B is any arbitrary set? (note that positive answers to both the questions lead to ...
0
votes
1answer
263 views

Induction, Cantor Set, Ternary representation

Cantor Set defined by sequence Let $C$ be a Cantor Set. I'm trying to show that "$x\in C ⇒$ There exists a ternary representation (base $3$) of $x$ such that every term is $0$ or $2$." Here's my ...
0
votes
1answer
515 views

Cantor Set defined by sequence

http://www.scribd.com/mobile/doc/76236535 page 49-50 Exercise 3.19 Let $A=\{0,2\}$ and $C$ be the Cantor Set. Define $x(\alpha) = \sum_{n=1}^\infty (\alpha_n / {3^n})$ for all $\alpha \in ...
4
votes
0answers
113 views

In NFU, is there a bijection between the set of all sets and the set of all one-element sets?

In the set theory NFU (described by M. Randall Holmes in "Elementary Set Theory with a Universal Set"), it is possible to define the set of all sets, and the set of all one-element sets. An object is ...
0
votes
1answer
130 views

How to associate non-natural numbers with set features/relations?

Pythagoras said "everything is number". Therefore, it might seem that sets (everything) can be described/identified using numbers. But does it really make sense to associate non-natural numbers with ...
0
votes
3answers
102 views

Existence of function (AC)

I want to make such a construction: Let $X$ be an infinite set. Put: $Y_0 = X \\ y_0 \in Y_0$ $Y_1 = Y_0 - \{y_0\} \\ y_1 \in Y_1$ ... $Y_n = Y_{n-1} - \{y_{n-1}\} \\ y_n \in Y_n$ ... for every ...
3
votes
1answer
508 views

Importance of Kripke–Platek set theory

What would be importance of Kripke-Platek set theory? I know that Saul Kripke is one of the most important philosophers in the world, but curious in what place he is in set theory.
-2
votes
2answers
255 views

How large is the infinity of real numbers [closed]

Umm ... Can someone disprove my proof that there are aleph-1 number of real numbers? Even comments to make my proof more rigorous are welcome. https://www.dropbox.com/sh/1fz28jlwrprh4jv/rhA7Ad7OtX
1
vote
2answers
316 views

ZFC set theory,first order theory [duplicate]

Possible Duplicate: What is the difference between Gödel's Completeness and Incompleteness Theorems? what is the relationship between ZFC and first-order logic? I am a bit ...
0
votes
3answers
335 views

which of the following sets are countable

are the following sets countable? set of all sequences of non-negative integers. The set of all sequences of non-negative integers with only a finite number of non zero terms could any one tell ...
0
votes
2answers
209 views

Ultrafilters in $\omega$

Let $U$ be a non-principal ultrafilter in $\beta \mathbb{N}$. Can it have a countable character as a point in this topological space? Is there decreasing chain of clopen subsets of $\beta\mathbb{N}$ ...
1
vote
1answer
188 views

Is this category essentially small?

Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in ...
12
votes
2answers
2k views

Axiom of Regularity

I am having difficulty understanding http://en.wikipedia.org/wiki/Axiom_of_regularity Every non-empty set A contains an element B which is disjoint from A So from my understanding if I have a ...
3
votes
1answer
389 views

Definition for Empty Set in Logic

$\exists X (\forall Y (\neg(Y \in X)))$ is whats given in my lecture, but I was wondering, is it the same as $\exists X (\forall Y (Y \notin X))$
10
votes
4answers
2k views

Example for fintely additive but not countably additive probability measure

A probability measure defined on a sample space $\Omega$ has the following properties: For each $E \subset \Omega$, $0 \le P(E) \le 1$ $P(\Omega) = 1$ If $E_1$ and $E_2$ are disjoint subsets $P(E_1 ...
1
vote
2answers
131 views

On a Surjection

For any set $B$ let $\mathcal{P}(B)$ denote the set of all subsets of $B$. Let $A$ be an infinite set and suppose there exists a surjection $f : A \mapsto \mathcal{P}(A)\setminus A$. Consider the ...
1
vote
2answers
198 views

Implications of the Zorn lemma for Zariski closed sets

By definition, a Zariski closed subset of $\operatorname{Spec}A$ is a set of the form $V(I) = \{P \in \operatorname{Spec}A \mid I \subset P \}$. What if we work in a ZF model where AC is violated? ...
2
votes
1answer
381 views

Ordinals with uncountable cofinality

How to construct an ordinal with uncountable cofinality? All the very "large" ordinals I can think of, such as $\omega_\omega^{\omega_\omega}$, still seem to have countable cofinality. I need a better ...
9
votes
3answers
2k views

What are Aleph numbers intuitively?

I cannot get my head around the concept of the `types' of Aleph infinity. What is an easy intuitive way to see when you are given the integer numbers $\aleph_0$ the $\aleph_1$ will follow?
2
votes
2answers
262 views

Constructing the reals from the rationals

Dr. H. Jerome Keisler, in his book Elementary Calculus: An Infinitesimal Approach, states on page 24: Just as the real numbers can be constructed from the rational numbers, the hyperreal numbers ...
2
votes
2answers
195 views

statement that is consistent in ZFC but the negation of it can be both consistent and inconsistent in ZFC or vice versa

Is there any known case where ZFC system is known to be consistent with a statement, but is also consistent with the negation of the statement? Or vice versa. Also, when we say ZFC is consistent with ...