Articles of elementary set theory

Axiom of choice confusion: what does it mean an element to have no distinguishing features?

I’m trying to understand the axiom of choice, but am stuck on this point: How can an element of a set ever have no distinguishing features? Two things which are identical are the same thing – surely? So why would you ever need to invoke the axiom of choice?

Halmos, Naive Set Theory, recursion theorem proof: why must he do it that way?

Summary: I understand the proof Halmos gives in his Naive Set Theory for the recursion theorem, but I don’t understand why he has to do it that way. I give an alternative proof which may be flawed. If so, I want to understand why it’s flawed. The context is the following discussion made by Halmos […]

Context for Russell's Infinite Sock Pair Example

I wanted to verify the following considerations on the context of Russell’s infinite sock pair conundrum. The conundrum pointed out that a rule for choosing from pairs of shoes is possible a-priori. For indistinguishable socks, such rule is not possible a-priori, and has to be assumed. Thus, in Set Theory there are two major avenues […]

The Intersection of Ordered Pairs

I’ve seen that the ordered pair $(a,b)$ is defined as a set that is $(a,b)=\{\{a\},\{a,b\}\}$. Can you explain what do we mean when $(a,b) \cap (b,a) = \{\{a,b\}\}$? I feel that there should be no intersection whenever a is not equal to b.

Definition of functions

This question already has an answer here: It's in my hands to have a surjective function 3 answers

Wanted: example of an increasing sequence of $\sigma$-fields whose union is not a $\sigma$-field

Possible Duplicate: Sigma algebra question The union of a strictly increasing sequence of $\sigma$-algebras is not a $\sigma$-algebra If $F_n$ is an increasing sequence of sigma fields then $F = \bigcup_{n=1}^\infty F_n$ is a field. Please help me find a counter-example to show that $F$ may not be a sigma-field.

How to show that this function is bijective

Possible Duplicate: Proving the Cantor Pairing Function Bijective Assume I define $$ f: \mathbb N \times \mathbb N \to \mathbb N, (a,b) \mapsto a + \frac{(a + b ) ( a + b + 1)}{2} $$ How to show that this function is bijective? For injectivity I tried to show that if $f(a,b) = f(n,m) […]

Closing a subcategory under finite colimits by transfinite induction

Let $\mathcal{C}$ be a locally small category with all finite colimits, and let $\mathcal{A}$ be a small full subcategory. I wish to prove the following: Proposition. There exists a full subcategory $\mathcal{B}$ of $\mathcal{C}$ satisfying these conditions: $\mathcal{A} \subseteq \mathcal{B} \subseteq \mathcal{C}$ Finite colimits exist in $\mathcal{B}$ and are the same as in $\mathcal{C}$. Every […]

Reading, Writing, and Proving Math: Cartesian Product

The following is my attempt at one of my homework assignments. Let A, B, and C be sets. If the statement below is true, prove it. If false, give a counter example. A $\times$ (B $\cap$ C) = (A $\times$ B) $\cap$ (A $\times$ C). I want to say this is true so I went […]

The convergence of a sequence of sets

A sequence $\{A_n : n=0,1,2,…\}$ is said to be monotone nondecreasing if we have $$A_0\subseteq A_1\subseteq \cdot \cdot \cdot \subseteq A_n \subseteq \cdot \cdot \cdot $$ The same sequence is said to be monotone nonincreasing if we have $$A_0\supseteq A_1\supseteq \cdot \cdot \cdot \supseteq A_n \supseteq \cdot \cdot \cdot$$ To specify the type of convergence, […]