(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.

We concluded by defining the symbols $\cap$ and $\bigcap$ for intersection and we stated without proof that the class $\bigcap \emptyset$ is not a set. In this way, intersection behaves differently than union: $\bigcup\emptyset$ IS a set. (Check: $\bigcup\emptyset=\emptyset$.)

Quiz 2 solutions.

We explained how to determine the kernel of a function if we are given the coimage of the function.

Quiz 3 solutions.

We next began discussing induction, recursion, and the arithmetic of $\mathbb{N}$ following these slides.

1. Induction
2. Recursion
3. Arithmetic on $\mathbb N$.

Quiz 4 solutions.

Midterm Review Sheet.

Midterm Review Sheet.

Midterm Review Sheet.

Quiz 5 solutions.

Midterm Review Sheet.

No quiz next Monday. No HW assignment due next Wednesday.

We began a discussion of logic following these slides. We showed how to write Goldbach's Conjecture as a formal statement. We discussed the first 5 pages of these slides about Propositional Logic. The topics considered include:

1. tautologies and contradictions,
2. logical equivalence, logical inequivalence, and logical independence,
3. disjunctive normal form (DNF), and
4. the fact that $\{\wedge, \vee, \neg\}$ is a complete set of logical connectives.

XKCD on Tautologies.

Quiz 6 solutions.

We discussed the first half of this handout. We started with examples where we converted informal sentences to formal sentences. The examples we considered were:

• $f\colon \mathbb R\to \mathbb R$ is continuous.
• There are at least three elements.
• There are exactly three elements.
• (About an ordered set) Any two elements have a least upper bound.
• If $a$ is a child of $A$ and $b$ is a child of $B$ and $A$ and $B$ are siblings, then $a$ is a cousin of $b$.

Arnie has no respect for those who have no respect for logic.

We started by explaining how to formalize the sentence
Someone in class used to play the guitar, but no longer plays the guitar
in a $2$-sorted language (people, time) with predicates for ``$p$ is a member of this class'', ``$p$ plays guitar at time $t$'', ``$t_{\textrm{NOW}}$'', and ``$s<t$'' (an ordering symbol for time). Then we explained how to avoid the use of multisorted structures by adding additional predicates.

We next focused on assigning tables to primitive and compound operations and predicates following these notes.