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 ...

learn more… | top users | synonyms

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
245 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 ...
12
votes
2answers
795 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
184 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 $\bigcap X=U$, ...
1
vote
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 ...
27
votes
2answers
4k 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 ...
57
votes
6answers
11k 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 ...
3
votes
2answers
273 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
120 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
404 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
434 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
419 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
219 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
259 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 ...
7
votes
1answer
710 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
183 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
198 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
443 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
98 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 ...
1
vote
2answers
431 views

Weakly-compact cardinals

I am reading Jech's Set Theory, in particular the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in ...
4
votes
2answers
422 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
2k 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
537 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
385 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
405 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
869 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 ...
6
votes
4answers
666 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
499 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 ...
3
votes
2answers
633 views

Choice function on $\mathcal P (\mathbb R) \setminus \{ \emptyset\}$

When I was first learning about the axiom of choice, a very helpful and fun exercise was to try to find a rule that maps every nonempty subset of $\mathbb R$ to one of its members. No matter how hard ...
2
votes
1answer
132 views

dense in terms of order and in terms of the order topology

In a densely and totally ordered set, induce a order topology from the order. Are the dense in terms of the order and the dense in terms of the order topology equivalent? Thanks and regards!
2
votes
2answers
344 views

Order on Euclidean space

In 1D case, R1 as a typical case of 1D Euclidean space is totally ordered. I was wondering if any 1D Euclidean space E1 is ordered as well? What is it For higher dimensional case, are both Rn and En ...
12
votes
2answers
740 views

What are large cardinals for?

I've heard large cardinals talked about, and I (think I) understand a little about how you define them, but I don't understand why you would bother. What's the simplest proof or whatever that ...
4
votes
6answers
3k views

Cutting the Cake Problem

The tradition way for two children to divide a piece of cake fairly between them is "you cut, I choose", What process accomplishes this is intuitively clear. Is there a similar process which works for ...
1
vote
1answer
774 views

How to prove that nothing is a member of itself?

I'd like to prove that $\forall x\left(x\not\in x\right)$ in the context of Morse-Kelley set theory. Let's call $A=\left\{y:y\not\in y\right\}$. I can easily prove that $A\not\in A$. In fact, if you ...
8
votes
3answers
617 views

Is the Subset Axiom Schema in ZF necessary?

I am learning the axioms of Zermelo-Fraenkel (ZF) set theory. One axiom schema basically says that given any set S and any formula phi(x), there is a set T consisting of all those elements x of S ...
5
votes
5answers
581 views

What's so special with small categories?

Sometimes one encounter the requirement that the objects of a category needs to be a set. What if it was not, could you provide examples of what could go wrong? (One example per answer).
-1
votes
4answers
411 views

Sets. Classes. …?

Classes are larger than sets. What's larger than classes ?
8
votes
2answers
474 views

Definition of a set

What is a set? I know that results such as Russell's paradox mean that the definition isn't as straight forward as one might expect.
12
votes
4answers
2k views

Cardinality of all cardinalities

Let $C = \{0, 1, 2, \ldots, \aleph_0, \aleph_1, \aleph_2, \ldots\}$. What is $\left|C\right|$? Or is it even well-defined?
3
votes
1answer
107 views

Is the set of all unique (convex) polygons countable? If so, by what bijection to the natural numbers?

Polygons are, in this question, defined as non-unique if they similar to another (by rotation, reflection, translation, or scaling). Would this answer be any different if similar but non-identical ...
31
votes
4answers
2k views

If all sets were finite, how could the real numbers be defined?

An extreme form of constructivism is called finitisim. In this form, unlike the standard axiom system, infinite sets are not allowed. There are important mathematicians, such as Kronecker, who ...
39
votes
11answers
12k views

Why is “the set of all sets” a paradox?

I've heard of some other paradoxes involving sets (ie, "the set of all sets that do not contain themselves") and I understand how paradoxes arise from them. But this one I do not understand. Why is ...
7
votes
6answers
3k views

How do the Properties of Relations work?

This is simply not clicking for me. I'm currently learning math during the summer vacation and I'm on the chapter for relations and functions. There are five properties for a relation: Reflexive - R ...
80
votes
10answers
4k views

Different kinds of infinities?

Can someone explain to me how there can be different kinds of infinities? I was reading "The man who loved only numbers" by Paul Hoffman and came across the concept of countable and uncountable ...