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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3
votes
2answers
176 views

Recursive definition isomorphism

If $(X, \lt)$ is a well-ordering I can show by transfinite recursion over the ordinals that the function $f(x) = \text{ran} f |_{\hat{x}}$ exists (where $\hat{x} = \{ y : y \lt x\}$). I have obtained ...
4
votes
1answer
168 views

Recursion on ordinals

I have some kind of "homework" question. I have the following theorem: Theorem (Transfinite Recursion over the class of of ordinals $\mathbf{ON}$: Let $\mathbf{V}$ be the class of all sets. If ...
6
votes
3answers
490 views

Multiplying Cardinal Numbers

I was just reading a proof of the dimension theorem in Steven Roman's Advanced Linear Algebra. In addressing the cases of infinite bases, Roman proceeds to show that if $\mathcal{B}$ and $\mathcal{C}$ ...
7
votes
9answers
818 views

Motivating implications of the axiom of choice?

What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some ...
12
votes
2answers
834 views

Is there an algebraic homomorphism between two Banach algebras which is not continuous?

According to wikipedia, you need the Axiom of Choice to find a discontinuous map between two Banach spaces. Does this procedure also apply for Banach algebras yielding a discontinuous multiplicative ...
6
votes
1answer
985 views

Cardinality of a set that consists of all existing cardinalities

I have taken a look at the following topics: number of infinite sets with different cardinalities Cardinality of all cardinalities Are there uncountably infinite orders of infinity? Types of ...
2
votes
3answers
636 views

Are there many more rational numbers than integers?

The title is somewhat deceptive: I know that $|\mathbb{Q}|=|\mathbb{Z}|.$ But suppose I wanted to compare the sets, knowing that they are of the same cardinality but still wondering if there was ...
2
votes
1answer
231 views

Limiting set theory using symmetry

If my understanding is correct, naive set theory needs to be restricted in order to avoid paradoxes including the Russell paradox. Typically, the restriction is expressed in terms of size. For ...
62
votes
6answers
3k views

Why is $\omega$ the smallest $\infty$?

I am comfortable with the different sizes of infinities and Cantor's "diagonal argument" to prove that the set of all subsets of an infinite set has cardinality strictly greater than the set itself. ...
19
votes
5answers
3k views

Category of all categories vs. Set of all sets

In naieve set theory, you quickly run into existence trouble if you try to do meta-things things like take "the set of all sets". For example, does the set of all sets that don't contain themselves ...
6
votes
1answer
633 views

Banach Tarski Paradox

I know that Banach Tarski Paradox gives that "A three-dimensional Euclidean ball is equidecomposable with two copies of itself" and I have read that doubling the ball can be accomplished with five ...
7
votes
2answers
941 views

How can I write the Axiom of Specification as a sentence?

I began reading Paul Halmos' "Naive Set Theory", and encountered the "Axiom of Specification". To every set $A$ and to every condition $S(x)$ there corresponds a set $B$ whose elements are exactly ...
4
votes
2answers
380 views

Order-isomorphic with a subset iff order-isomorphic with an initial segment

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. I want to show that if $X$ is isomorphic to a subset of Y then $X$ is isomorphic with an initial segment of $Y$. (The other direction is of ...
7
votes
2answers
2k views

Symbol for the cardinality of the continuum

The usual symbol for the cardinality of the continuum (i.e. the real numbers) is Fraktur $\mathfrak{c}$. However, I recall some sources also using $\aleph$ (with no subscript). This usage is not ...
10
votes
2answers
631 views

Surreal and ordinal numbers

Is there a surjective map between the (class of) ordinal numbers On and the set No (Conway's surreal numbers) and is it constructable, In Conway's system we have for example: $\omega_0 = < ...
0
votes
1answer
222 views

Initial segments order-isomorphic

Let $(X, \prec)$ and $(Y, <)$ be two well-ordered sets. Now, I want to understand the proof that $X$ is isomorphic to an initial segment of $Y$ or $Y$ is isomorphic with an initial segment of $X$. ...
4
votes
2answers
296 views

Showing that well-ordered subsets of $P(\omega)$ are countable

I have the following problem: Show that no uncountable subset of $P(\omega)$ is well-ordered by the inclusion relation. I think they want me to do by embedding it in a separable complete dense ...
1
vote
1answer
255 views

Defining $\mathbb{Z}_+$

While reading Munkres' Topology section about integers and reals (Chapter 1, Section 4), he defines the set $\mathbb{Z}_+$ as: Definition: A subset $A$ of the real numbers is said to be ...
11
votes
2answers
2k views

Are there uncountably infinite orders of infinity?

Given a set $S$, one can easily find a set with greater cardinality -- just take the power set of $S$. In this way, one can construct a sequence of sets, each with greater cardinality than the last. ...
1
vote
1answer
249 views

Specific equivalent to the Axiom of Choice involving the empty set

I'm trying to remember a particular theorem of ZF but unfortunately my memory is quite incomplete. The theorem is of the form (some set operation) is either (expected answer) or the empty set. If ...
13
votes
2answers
858 views

Transfinite Induction and the Axiom of Choice

My question is essentially this: Why does the principle of transfinite induction not suffice to show the axiom of choice when the sets to be chosen from are indexed by a well ordered set? I have read ...
5
votes
1answer
185 views

Where in the analytic hierarchy does V=L start having consequences?

I note that the ordinals of L are the same as V, so I guess that it has no $\Pi_1^1$ consequences. On the other hand Wikipedia tells me that it asserts the existance of a $\Delta_2^1$ non-measurable ...
6
votes
3answers
1k views

Unary intersection of the empty set

In MK (Morse-Kelley) set theory life is easy: $\forall X\forall y\left(y\in\bigcap X\leftrightarrow\forall x\left(x\in X\rightarrow y\in x\right)\right)$. If $X=\left\{\right\}$ then $\bigcap X=U$, ...
2
votes
1answer
1k views

identity and inverse/complement elements in a boolean algebra

In a boolean algebra, 0 (the lattice's bottom) is the identity element for the join operation $\lor$, and 1 (the lattice's top) is the identity element for the meet operation $\land$. For an element ...
32
votes
2answers
5k views

Is there a known well ordering of the reals?

So, from what I understand, the axiom of choice is equivalent to the claim that every set can be well ordered. A set is well ordered by a relation, $R$ , if every subset has a least element. My ...
63
votes
7answers
13k views

Can you explain the “Axiom of choice” in simple terms?

As I'm sure many of you do, I read the XKCD webcomic regularly. The most recent one involves a joke about the Axiom of Choice, which I didn't get. I went to Wikipedia to see what the Axiom of ...
4
votes
2answers
299 views

Separative Quotients, and the Induced Order

Let $P$ be some partial order. We say that $x$ and $y$ are compatible if $\exists r\in P (r\le x \wedge r\le y)$, we denote this by $x \perp y$. Otherwise, we say that $x$ and $y$ are incompatible. ...
1
vote
1answer
122 views

Aggregating multiple fuzzy values for a given observation

Hmmm... this question might be a bit low-brow. I'm no mathematician. Let's say that I have an unordered sequence of fuzzy values $T = (t_1 \ldots t_n)$, $T \in (0,1]^n$. Each value is a positive ...
0
votes
1answer
93 views

Is the union of a family of $LU$-closed sets also $LU$-closed?

I was reading through some notes on closure operations, and the example was given that for a poset $(S,\preceq)$, the operation on $(\mathcal{P}(S),\subseteq)$ given by $E\mapsto LU(E)$ is a closure ...
8
votes
1answer
411 views

ZF is almost finitely axiomatizable

I want to show that there is a finite conjunction $\phi$ of axioms of $ZF$, such that every transitive proper class $M$, which satisfies $\phi$, is already a model of $ZF$. This is an exercise in ...
10
votes
2answers
467 views

Real-measurable cardinals that are not measurable ones

I'm reading Jech's Set Theory, and in the chapter about measurable cardinals there is a theorem that if $\kappa$ is real-measurable but not measurable then it is $\le 2^{\aleph_0}$ and so and so. ...
11
votes
2answers
437 views

Cardinality of $H(\kappa)$

Again I have trouble with some exercises in Kunen's set theory. In the following, let $\kappa > \omega$ a cardinal. Then I want to show that 1) $|H(\kappa)| = 2^{<\kappa}$ 2) ...
1
vote
2answers
221 views

Strict ordering on natural numbers

I'm studying on K. Hrbacek and T. Jech, Introduction to Set Theory. In the third chapter, they prove the usual properties of the strict ordering on natural numbers in the following way: They prove ...
3
votes
1answer
282 views

set of infinite cardinals admits an injective regressive function

Let $A$ be a set of infinite cardinals. Assume that for every regular $\lambda$, the subset $A \cap \lambda$ of $\lambda$ is not stationary. Then I want to prove that there is an injective function ...
8
votes
1answer
742 views

Uncountable ordinals without power set axiom

Assume $M$ is a set, in which all axioms of $ZF - P + (V=L)$ hold. Does then $M$ believe that there exists an uncountable ordinal? I mean, why should the class of all countable ordinal numbers be a ...
3
votes
1answer
186 views

A “geometrical” representation for Ramsey's theorem

The [infinite] Ramsey theorem states that Let $n$ and $k$ be natural numbers. Every partition $\{X_1,\ldots ,X_k\}$ of $[\omega]^n$ into $k$ pieces has an infinite homogeneous set. Equivalently, ...
0
votes
1answer
206 views

Set Equality between Lower Bounds and and an Intersection of a Subset and Interval

The following is an example given by my professor, but there is an equality that I don't understand. Let a partially ordered set $(S,\preceq)$ is the union of three sets such that $S=X\cup Y\cup Z$ ...
3
votes
1answer
168 views

Showing the Inclusion is sup-continuous

I fear I over simplified the following problem: For any partially ordered set $(A,\leq)$, let $A^* = A- \{\max A,\min A\}$ if $\max A$ and $\min A$ exist. Show the inclusion ...
5
votes
2answers
452 views

The tree property for non-weakly compact $\kappa$

In my previous question, Weakly-compact cardinals, I was asking about weakly-compact cardinals and equivalent definitions to the basic one, which is $\kappa \to (\kappa)^2_2$. One of which was that ...
1
vote
1answer
75 views

Can one show that $L(E)=\langle s]$ on a partially ordered set?

This is a follow up question to one I posted earlier. I'm trying to decide that if for $(S,\preceq)$ a partially ordered set and $E\subseteq S$, one has $L(E)=\langle s]$ for some $s\in S$ iff $\inf ...
3
votes
1answer
99 views

Does the existence of an infimum imply that the set of lower bounds of a set is totally ordered?

Say we have a partially ordered set $(S,\preceq)$, and some subset $E\subseteq S$ such that $E$ is bounded below and $\inf E$ exists. My question is, since $S$ is not totally ordered is it possible to ...
3
votes
2answers
477 views

On the definition of weakly compact cardinals

I am reading in Jech's Set Theory the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book. I ...
4
votes
2answers
429 views

A nice enumeration of $R(\omega)$

Define $R(0)=0=\emptyset, R(n+1)=P(R(n))$ and $R(\omega) = \cup_{n < \omega} R(n)$. Thus $R(\omega)$ is the set of all sets, which are build out of finitely many braces and $0$. Consider the ...
12
votes
5answers
3k views

Why does the set of all singleton sets not exist?

Proposition: For a set $X$ and its power set $P(X)$, any function $f\colon P(X)\to X$ has at least two sets $A\neq B\subseteq X$ such that $f(A)=f(B)$. I can see how this would be true if $X$ is a ...
9
votes
1answer
555 views

cardinal exponentiation, $k^{<\lambda}$

I have the following well-known exercise in cardinal arithmetics: If $\kappa, \lambda$ are cardinals such that $\lambda$ is infinite, then $\kappa^{<\lambda}$ equals the supremum of the ...
3
votes
3answers
404 views

Do endomaps of sets have interesting properties?

I've been thinking about maps between sets. Injections, surjections and the rest. Often when thinking about some kind of map, it is interesting to say "what about the maps from a set to itself?" Call ...
2
votes
1answer
429 views

Generic sets in ZFC

I'm reading Shelah's book "Proper and Improper Forcing" (the first two chapters were recommended for learning the basics of forcing) Given a quasi-order $P$ we say that $\mathcal{I}$ is a dense ...
5
votes
3answers
915 views

Counting equivalence relations

I am aware that on a finite set the number of equivalence relations is the n-th Bell's Number. On the other hand, the only reference I could find on the web for infinite sets was this: On counting ...
7
votes
4answers
695 views

number of infinite sets with different cardinalities

I would like to know whether the number of infinity sets of different cardinalities is uncountable or countable. Is the proof simple? Thanks.
5
votes
2answers
506 views

large sets. set theory

I haven't studied properly the theory of infinities yet. Let $A_0$ denote the set of natural numbers. Let $A_{i+1}$ denote the set whose elements are all the subsets of $A_i$ for $i=0,...,n,...$ I ...