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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5
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1answer
217 views

How far from ZFC is Cohen's second model?

Recall that Cohen's second model adds sets of generic reals such that there is a countable family of pairs with no choice function, i.e. a surjective function $f: A \to \omega$ such that each fibre of ...
1
vote
1answer
204 views

In NBG set theory how could you state the axiom of limitation of size in first-order logic?

Limitation of size: "For any class $C$, a set $x$ such that $x=C$ exists if and only if there is no bijection between $C$ and the class $V$ of all sets." In Von Neumann–Bernays–Gödel set theory how ...
7
votes
2answers
238 views

What is the largest cardinal which can inject into $\mathbb{R}$ in ZF?

This question takes place in ZF. Assume some mild large cardinals; then it is consistent (in fact, it follows from AD, the consistency of which follows from mild large cardinals) that there are very ...
14
votes
1answer
599 views

Is there an absolute notion of the infinite?

Skolem's paradox has been explained by the proposition that the notion of countability is not absolute in first-order logic. Intuitively, that makes sense to me, as a smaller model of ZFC might not be ...
2
votes
1answer
144 views

$\{0,1\}^\lambda$ and its clopen subsets

Let $\lambda$ be an uncountable cardinal and let $X=\{0,1\}^\lambda$ be endowed with the product topology ( $\{0,1\}$ is discrete). Is there an uncountable chain (with respect to inclusion) of clopen ...
2
votes
3answers
311 views

Finite dimensional subspaces of a linear space

Suppose $V$ is an infinite dimensional vector space. I do not want to assume the axiom of choice, so I will define a vector space $V$ to be infinite dimensional if there is a proper subspace ...
6
votes
1answer
128 views

Why do $2^{\lt\kappa}=\kappa$ and $\kappa^{\lt\kappa}=\kappa$ when the Generalized Continuum Hypothesis holds?

I'm going to assume GCH here. If that holds, then why do we have the equalities $2^{\lt\kappa}=\kappa$ for every $\kappa$, and $\kappa^{\lt\kappa}=\kappa$ for all regular $\kappa$?
11
votes
2answers
406 views

Uncountable chains

$P(\mathbb N)$ = power set of $\mathbb N$. $A \subset P(\mathbb N)$ is a chain if $a,b \in A \implies$ either $a \subseteq b$ or $ b \subseteq a$ That is we have something like this: $$\ldots a ...
5
votes
1answer
527 views

Rank of a set as the supremum of ranks its elements.

In general, for a set $X$, why is it true that $\mathrm{rank}(X)=\sup\{\mathrm{rank}(y)+1\mid y\in X\}$? The definition of rank I have is that $\text{rank}(x)=\text{ the least }\alpha\text{ such that ...
6
votes
2answers
154 views

What is the product of finitely indexed alephs?

I'm simply curious about why the following equality holds: $ \displaystyle\prod_{n\lt\omega}\aleph_n=\aleph_\omega^{\aleph_0}. $ Much thanks!
5
votes
1answer
253 views

Product forcing and generic objects

If we start with a model of $\sf ZFC$, $M$ and $(P,\le)\in M$ is a notion of forcing, $G\subseteq P$ a generic filter, then in $M[G]$ we can define some generic object from $G$. For example, if $P$ is ...
5
votes
1answer
135 views

Filter completeness question

I have a question about filters which I suspect has a very simple answer (hence my asking it here as opposed to MO): Let $F$ be a filter on an infinite set $X$. Then $F$ is "countably closed" if for ...
11
votes
2answers
523 views

Does proving (second countable) $\Rightarrow$ (Lindelöf) require the axiom of choice?

Background: This question came up in my homework (but was not a homework problem). The problem was proving one direction of the Heine-Borel theorem. As with all proofs of compactness, one begins ...
2
votes
2answers
217 views

Forcing partial order $([\omega]^\omega / \mathrm{fin} , \leq)$

I have a partial answer for the following homework and I was wondering if you could tell me if I have it right and help me with part (c) which I'm currently stuck on. Many thanks for your help! Let ...
6
votes
2answers
200 views

What is the product of all nonzero, finite cardinals?

To be specific, why does the following equality hold? $$ \prod_{0\lt n\lt\omega}n=2^{\aleph_0} $$
7
votes
1answer
787 views

Totally ordering the power set of a well ordered set.

Let's say I take a set $S$, where $S$ can be well ordered. From what I understand, one can use that well ordering to totally order $\mathscr{P}(S)$. How does a body actually use the well ordering of ...
6
votes
3answers
168 views

Cardinality of a some linear ordering is at most that of a given cardinal?

This is an intuitive idea that I've used for a while, but don't know how to explain formally. Suppose $(A,\prec)$ is some linear ordering, and each initial segment of $A$ has cardinality strictly ...
2
votes
1answer
78 views

Strictly commutative coproducts

This is a continuation of a previous question. Is it possible to find a map $S : \mathrm{Set} \times \mathrm{Set} \to \mathrm{Set}$ such that $S(X,Y)$ is a coproduct of $X$ and $Y$ (thus it is ...
9
votes
2answers
560 views

axiom of choice: cardinality of general disjoint union

I have this exercise involving the axiom of choice, but I don't understand where it's needed: Let $(X_i)_{i \in I}$ and $(Y_i)_{i \in I}$ be pairwise disjoint sets with $|X_i| = |Y_i|$. Prove, using ...
3
votes
2answers
2k views

Union of Uncountably Many Uncountable Sets

I know that the union of countably many countable sets is countable. Is there an equivalent statement for uncountable sets, such as the union of uncountably many uncountable sets is uncountable? ...
4
votes
1answer
263 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
votes
1answer
522 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
votes
1answer
158 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
votes
1answer
230 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
votes
1answer
211 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
421 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
votes
1answer
187 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
266 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
87 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
338 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
336 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
416 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
227 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
182 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
352 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
192 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
195 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
426 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
votes
2answers
317 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
votes
0answers
207 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
137 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
383 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
671 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
474 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
216 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
233 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
votes
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
299 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
467 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
176 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 ...