(1) To learn what it means to say ``Mathematics is constructed to be well founded.'' To learn which concepts and assertions depend on which others. To learn what are the most primitive concepts ( = set, $\in$) and the most primitive assertions ( = axioms of set theory).

(2) To learn how to unravel the definitions of ``function'', ``number'', and ``infinite'', through layers of more and more primitive concepts, back to ``set'' and ``$\in$''.

(3) To learn the meanings of, and the distinction between, ``truth'' and ``provability''. To learn proof strategies.

(4) To learn formulas for counting.

Axioms of set theory.

I will occasionally post notes for Math 2001 in the form of flash cards on Quizlet. To join our quizlet class, go to https://quizlet.com/join/mExWGGZqj.

(Test yourself on the Axioms of Set Theory with this Quizlet link: https://quizlet.com/_61ko6h.)

We began discussing `naive' set theory, in which we `define' a set to be an unordered collection of distinct objects. We contrasted this with formal set theory based on the axiom system ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). We discussed
(i) the symbol $\in$.
(ii) the Axiom of Extensionality.
(iii) the Axiom of the Empty Set.
(iv) the Axiom of Pairing.
(iv) the Axiom of Union and the definition of the successor function: $S(x)=x\cup \{x\}$.
(v) the definitions of $0, 1, 2, 3, 4$.
(vi) the recursive definition of addition of natural numbers.
Using these definitions, we proved that $2+2=4$.

Quiz 1 solutions.