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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4
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1answer
287 views

Strictly associative coproducts

Background. This question belongs to evil mathematics. It is motivated by this question which links to a paper in which it is claimed that it is an open problem whether there exists strictly ...
11
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1answer
546 views

“Nice” well-orderings of the reals

I have a question which I believe could be easily resolved if I happened to look at the right source - hence my asking it here as opposed to at MathOverflow. I've tried googling it, but I haven't been ...
4
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1answer
160 views

Does there exist a subset of $\mathbb{R}$ of cardinality $2^{\aleph_0}$ that has no perfect subset?

Assuming the axiom of choice, is there a way to construct a subset of the reals of cardinality $2^{\aleph_0}$ that has no perfect subset?
2
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1answer
238 views

Closure of a transitive set under Gödel operations is transitive: why?

I've a small question. If I have $X$ a transitive set, why is its closure under Gödel operations still transitive? Kind thanks,
0
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1answer
221 views

Formula defining the first transfinite ordinal $\omega$

Just as an exercise in formula manipulation, I tried to find the simplest formula $\phi(x)$ with one free variable $x$ in the language of ZFC that defines the first transfinite ordinal $\omega$ (i.e. ...
3
votes
2answers
439 views

Usual notation for Fréchet filter and principal ultrafilters

Given a set $S$ we can define the filter consisting of all complements of finite sets, which is usually called Fréchet filter or cofinite filter. For any $a\in S$ the set $\{A\subseteq S; a\in A\}$ ...
11
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1answer
188 views

Set of points with unique distances

Is there a set of points in $\mathbb{R}^n$ such that every positive distance is realized by exactly one pair of points in the set? I can see that if it exists, the set must be uncountable and ...
2
votes
1answer
293 views

Transfinite Recursion Theorem

I have the following homework assignment: Let $\textbf{V} \models ZFC$ and let $\mathbb{P} = (P, \leq)$ be a forcing partial order in $\textbf{V}$. Define a class function $F: \textbf{V} \to ...
3
votes
1answer
90 views

Determining $\mathbb{P}$-names

Will you look at my answer to the following homework and tell me where I made a mistake? Thanks for your help! Let $\mathbb{P} = (P, \leq)$ be a forcing poset and let $p,q \in P$ denote two different ...
5
votes
1answer
374 views

The “canonical” representative of an order type

An ordinal (in von Neumann sense) can be thought as the "canonical" representative of the order type (i.e. of the class of all order-isomorphic sets) of a well-ordered set. This is very convenient, ...
2
votes
1answer
347 views

Universal set--is its definition circular?

Consider NFU set theory as presented in this: http://math.boisestate.edu/~holmes/holmes/head.pdf On page 15 of that pdf it states that the following is an axiom: The set $\{X\colon X=X\}$ exists. ...
7
votes
1answer
436 views

Addition on ultrafilters is non-commutative

I'm reading a book on ultrafilters and it tells me that showing addition on ultrafilters over the naturals (addition on the set of ultrafilters '$\beta \mathbb{N}$', defined in the standard way) is ...
4
votes
1answer
239 views

cardinality of math theorems

Some naive questions from an interested layman regarding the cardinality of the set of all math theorems (discovered and undiscovered). 1) What branches of math are not contained in ZF set theory + ...
3
votes
1answer
184 views

Showing a $\alpha\times\beta$ is well-ordered for $\alpha,\beta$ ordinals

I am concerned with showing the least element existence in every subset of $\alpha\times\beta$. Below is my attempt. My teacher has used similar argument to show $\mathbb{N}\times\mathbb{N}$ is a ...
1
vote
1answer
378 views

Proof related to an ordinal less than or equal to another ordinal?

Can someone please comment on my solution? I wish to know if my solution is right and every step is well-justified. I will state all propositions that I use in my solution and refer to them later in ...
10
votes
1answer
197 views

Is there a dense subset of $\mathbb{R}^2$ with all distances being incommensurable?

Is there a set $S$ of points on the real plane $\mathbb{R}^2$ such that: there is a point belonging to $S$ in any neighborhood of every point of $\mathbb{R}^2$ (so, $S$ is dense) and ratio of any ...
5
votes
1answer
208 views

Canonical $\mathbb{P}$-name

If $G$ is a subset of some forcing poset $P$ and for $x \in V$ (where I think $V$ is some model of ZFC but I'm not clear what it means) the canonical $\mathbb{P}$-name is defined as $$ ...
9
votes
2answers
439 views

The history of set-theoretic definitions of $\mathbb N$

What representations of the natural numbers have been used, historically, and who invented them? Are there any notable advantages or disadvantages? I read about Frege's definition not long ago, ...
5
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2answers
342 views

Can $\omega_1$ be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?

Can $\omega_1$ (the first uncountable ordinal) be represented as union of an uncountable collection of cofinal, pairwise disjoint subsets?
3
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0answers
209 views

(countable) Hausdorff maximal principle and induction

I was reading Wilansky's book Modern methods in Topological Vector Spaces and came across this problem on set theory (p. 7): "The maximal axiom for countable posets is equivalent to induction." ...
1
vote
1answer
138 views

Cardinality of set of normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$

What is the cardinality of the set of all normal functions $f \colon \omega_{\alpha} \to \omega_{\alpha}$, where $\omega_{\alpha}$ is the initial ordinal of $\aleph_{\alpha}$?
5
votes
1answer
445 views

How many countable models of ZFC are there?

If we were looking at just an arbitrary binary relation on a countable set then I guess we would be looking at infinite graphs and those are uncountable. However, ZFC places an extra structure on our ...
5
votes
2answers
738 views

How do I choose an element from a non-empty set?

Suppose I have a non-empty set $A$. How do I choose an element $x\in A$? More precisely, I believe I would like to find a formula $P(x,y)$ of ZF such that for every non-empty set $y$ there is ...
3
votes
1answer
536 views

class function question

In NBG set theory, classes are in some sense generalizations of sets. When dealing with sets, we view set functions as specific subsets of a Cartesian product. Is the definition the same for functions ...
1
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2answers
218 views

If a topological space has $\aleph_1$-calibre and cardinality at most $2^{\aleph_0}$ must it be star-countable?

If a topological space $X$ has $\aleph_1$-calibre and the cardinality of $X$ is $\le 2^{\aleph_0}$, then it must be star countable? A topological space $X$ is said to be star-countable if whenever ...
3
votes
1answer
236 views

Set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ is not separable

I want to show the set of ultrafilters $\beta \mathbb{N} - \mathbb{N}$ (where $\beta \mathbb{N}$ is all ultrafilters on $ \mathbb{N}$) is not separable. I know we can take as a base of $\beta ...
17
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2answers
1k views

The set of ultrafilters on an infinite set

After recently learning about filters and ultrafilters, we looked into further problems and properties. I am having trouble with this one: If $X$ is an infinite set, then the set of all ultrafilters ...
7
votes
1answer
304 views

Why do we use groups and not GROUPS?

When a group is defined, we ask that the elements belong to a set. If we allow them to belong to a proper Class we get a GROUP. What is the advantage of working with groups? What properties do we ...
0
votes
2answers
522 views

How far do known ordinal notations span?

What is the largest known ordinal number $\alpha$ such that a uniform notation scheme has been developed for all ordinals up to $\alpha$ (there should be no "gaps" in what ordinals are representable), ...
2
votes
1answer
178 views

Function $f\colon 2^{\mathbb{N}}\to 2^{\mathbb{N}}$ preserving intersections and mapping sets to sets which differs only by finite number of elements

Define on $2^{\mathbb{N}}$ equivalence relation $$ X\sim Y\Leftrightarrow \text{Card}((X\setminus Y)\cup(Y\setminus X))<\aleph_0 $$ Is there exist a function $f\colon 2^{\mathbb{N}}\to ...
8
votes
1answer
444 views

For every $n < \omega$, $\aleph_n^{\aleph_0} = \max(\aleph_n,\aleph_0^{\aleph_0})$

I have proved that if $\aleph_n \leq \aleph_0^{\aleph_0}$, then $\aleph_n^{\aleph_0} \leq \max(\aleph_n,\aleph_0^{\aleph_0})$. Clearly $\aleph_n \leq \aleph_n^{\aleph_0}$ and $\aleph_0^{\aleph_0} \leq ...
2
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4answers
294 views

Union of Uncountably Infinite Sets

How does one notationally describe the set which is the union of uncountably many other sets. For instance, for each x such that a < x < b, where a and b are real numbers, if there is assigned ...
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0answers
83 views

What is the name of this equivalence relation?

Given any sets $X \subseteq Y$, the relation given by: $$1_Y \; \cup \; (X \times X) \;\; \subseteq Y \times Y$$ (where $1_Y = \{ (y, y): y \in Y \}$ ) is an equivalence relation. Is there a name ...
2
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2answers
1k views

A proper formal definition of the Ordinal numbers?

Maybe I'm just being a bit stupid with where I'm searching here, but I can't actually find a proper definition of these things wherever I look, books or online. The definition I find most is that it's ...
15
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1answer
951 views

Proofs given in undergrad degree that need Continuum hypothesis?

Or alternative you need to assume CH is false. I know several proofs that use axiom of choice. Heine Borel theorem is the best example I can think off. Zorns lemma is heavily used in the non ...
12
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2answers
1k views

Is the axiom of universes 'harmless'?

Usually when you start studying category theory you see the usual definition: a category consists of a class $Ob(\mathcal{C})$ of objects, etc. If you take ZFC to be your system of axioms, then a ...
5
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2answers
646 views

Countable or uncountable set 8 signs

Let S be a set of pairwise disjoint 8-like symbols on the plane. (The 8s may be inside each other as well) Prove that S is at most countable. Now I know you can "map" a set of disjoint intervals in R ...
11
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1answer
639 views

Surreal numbers without the axiom of infinity

Let $ZF^\times$ denote the set of axioms of Zermelo-Fraenkel set theory without the axiom of infinity. The set $V_\omega$ of all hereditarily finite sets is a model of $ZF^\times$, and ...
1
vote
1answer
115 views

Another question on saturated models of ZFC

Let $M$ be a saturated model of ZFC and let $\kappa$ be any cardinal in $M$. Now let $$\begin{align*} p_\kappa(f) &= \{ “f \text{ 1-1 function”} \wedge \operatorname{dom}(f) \subseteq \omega ...
6
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1answer
251 views

Probabilistic proof of existence of an integer

The prime number theorem (PNT) says that an integer $n$ is prime with probability $\frac{1}{\ln n}$. Using only PNT, it's conceivable that each integer upto $10^{10^{10}}$ is non-prime. However using ...
3
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1answer
341 views

Properties of computable numbers

If we enumerate* all the computable numbers, those for which there exist a turing machine that outputs its digits to arbitrary precision. What is known about the asymptotic density of rationals, ...
14
votes
1answer
878 views

Is it true that a connected graph has a spanning tree, if the graph has uncountably many vertices?

I found a proof that every connected graph (possibly infinite) has a spanning tree in Diestel's Graph Theory (Fourth Edition), Ch. 8 that uses Zorn's Lemma, but at a crucial step it seems to be ...
1
vote
1answer
319 views

iterate random forcing

I am studying Kunen's article Random and Cohen forcing [1], and I meet a problem. On page 904, Theorem 3.13 states that if $M$ is a countable transitive model of ZFC, $I,J,K \in M$ such that $I$ is a ...
11
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3answers
1k views

Axiom of choice, non-measurable sets, countable unions

I have been looking through several mathoverflow posts, especially these ones http://mathoverflow.net/questions/32720/non-borel-sets-without-axiom-of-choice , ...
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0answers
219 views

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable?

Is there a ccc but not separable space $X$ with a zeroset-diagonal, that isn't submetrizable? separable = $X$ has a countable dense subset. A space $X$ has a zeroset-diagonal when there is a ...
13
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2answers
341 views

Does $2^X \cong 2^Y$ imply $X \cong Y$ without assuming the axiom of choice?

A friend of mine told me that $X \cong Y \Rightarrow 2^X \cong 2^Y$ ($X$ and $Y$ being sets), which is very easy to prove, but he was wondering about the converse in ZF, i.e., can one take logarithms? ...
7
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1answer
376 views

Can forcing push the continuum above a weakly inacessible cardinal?

There is a famous quote of Paul Cohen which reads $\lt\lt$ A point of view which the author feels may eventually come to be accepted is that the continuum hypothesis is obviously false ... The ...
8
votes
2answers
451 views

Implications of continuum hypothesis and consistency of ZFC

I've been a bit confused whilst doing some reading. I think the confusion arises because I am trying to read Wikipedia on topics without being able to work along. Anyway, I have read that, of course, ...
8
votes
4answers
701 views

Fractional cardinalities of sets

Is there any extension of the usual notion of cardinalities of sets such that there is some sets with fractional cardinalities such as 5/2, ie a set with 2.5 elements, what would be an example of such ...
1
vote
3answers
203 views

Equivalence relations and bijections without ordered pairs

Is it correct that — opposed to general relations and functions — equivalence relations and bijective functions can be defined without reference to ordered pairs? Especially, do the ...