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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100 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? ...
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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 ...
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1answer
84 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 $A$ ...
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3answers
302 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
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1answer
881 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 ...
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1answer
189 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
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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 ...
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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 ...
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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 ...
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2answers
932 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 ...
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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 $\alpha$, ...
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3answers
926 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
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2answers
462 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 ...
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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 ...
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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 ...
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1answer
262 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 ...
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1answer
510 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 ...
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0answers
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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 ...
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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 ...
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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 ...
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1answer
506 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.
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2answers
254 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
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2answers
311 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 ...
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3answers
328 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 ...
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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 $\beta\mathbb{N}$ ...
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1answer
186 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 ...
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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
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1answer
386 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 $\exists X (\forall Y (Y \notin X))$
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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 ...
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2answers
130 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 ...
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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? ...
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1answer
373 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 ...
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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?
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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
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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 ...
8
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2answers
269 views

Existence in ZF of a set with countable power set

Is it consistent with ZF for there to exist a set $S$ such that the power set $P(S)$ is countable? If so, what is the weakest form of the axiom of choice needed to prove that no such set exists?
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4answers
627 views

Is there a set that is both a sigma algebra and a topology but not a powerset?

Is there a set that is both a sigma algebra, $\Sigma$, and a topology, $\tau$, but not a powerset, $\mathcal{P}$?
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1answer
145 views

$(\beth_{\omega})^\omega=\beth_{\omega+1}$

I'm trying to show that $(\beth_{\omega})^\omega=2^{\beth_\omega}$. This is an exercise in Kunen where he suggests to encode subsets of $\beth_\omega$ with functions from ...
6
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1answer
231 views

Intuition behind extenders

What's the best way to think about extenders? For instance if we have a normal $\kappa$-complete ultrafilter on $\kappa$, call it $D$, $M$ the ultrapower given by $D$, and if we look at $j: V \to M$, ...
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1answer
431 views

Visualizing Infinity discerning countable and uncountable

This is rather a philosophical question. Although it uses topological notions, it isn't any precise mathematics, so maybe one cannot take it very seriously. Sometimes I try to picture an infinite set ...
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0answers
3k views

Lists of open problems in set theory

Are there any publicly available lists of open problems in set theory besides the following ones? (And if so, what are they?) http://www.math.wisc.edu/~miller/res/problem.pdf ...
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3answers
320 views

How does the Axiom of Extensionality prove the uniqueness of Specified sets?

http://en.wikipedia.org/wiki/Axiom_schema_of_specification says: Note that there is one axiom for every such predicate φ; thus, this is an axiom schema. To understand this axiom schema, note ...
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6answers
2k views

Is the statement $A \in A$ true or false?

In last week's discrete math homework, one question had us evaluate the truthness of several set notation statements, where $A$ was defined as an arbitrary set. One such statement was $$A \in A$$ ...
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2answers
154 views

Cardinal arithmetic and first uncountable cardinal

Cardinal arithmetic does not seem to open its way to the existence of $\aleph_1$ that is not $2^{\aleph_0}$, as any operation on $\aleph_0$ would lead to $\aleph_0$ or $2^{\aleph_0}$ and ...
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0answers
72 views

Looking for good online write-up on Countability (Set Theory)

Anyone have any suggestions for a good write-up on Countability (from an intro to set theory perspective)?
2
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1answer
410 views

Why PROP (Set of all propositions) is a set by ZF axioms?

In Propositional Logic when we define the set of all propositions inductively how we can prove such a set(smallest with such properties) does exists? means that the set (of all sets with these ...
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3answers
454 views

The cardinality of a countable union of sets with less than continuum cardiality

Can continuum be a countable union of sets with cardinality, less than continuum? I can prove it for a finite union by mathematical induction from this: $$\mathbb R = A_1 + A_2 \implies |\mathbb R| = ...
4
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1answer
927 views

What is the generating set of the Vietoris topology?

Here is what Kechris says about the Vietoris topology, Let X be a top. space. We denote by K(X) the space of all compact subsets of X equipped with the Vietoris topology, i.e., the one generated by ...
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1answer
312 views

The number of equivalence classes of finite symmetric difference relation

Let $\Sigma$ be an infinite set. Let $A,B \subseteq \Sigma$ be of finite symmetric difference iff they have a finite difference, more formally: $A \sim B$ iff $|A \Delta B| \in \mathbb{N}$ How ...
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2answers
434 views

Understanding a proof of an implication of the axiom of choice

Can someone please clear my doubts regarding a result concerning the axiom of choice: Prove that (i) implies (ii) where: (i) For every nonempty set whose elements are non empty sets there exists a ...