MTH303 of Monsoon 12023

References:

• Primary reference will be classroom.
• Most of the material is drawn from following textbooks (but may not appear in the same order).
  1. Michael L. O’Leary: A First Course in Mathematical Logic and Set Theory
  2. Herbert B. Enderton: A mathematical introduction to logic.
• Following textbooks are also going to be useful.
  1. P. T. Johnstone: Notes on Logic and Set Theory.
  2. Yu. I. Manin: A Course in Mathematical Logic for Mathematicians.
• A beautiful book (graphic novel!) that gives you a sense of the intellectual turmoil in the late 19th century when logical foudations as we know today were being laid is the following:
  1. Logicomix: An Epic Search for Truth, (writers: Apostolos Doxiadis, Christos Papadimitriou and Artist: Alecos Papadatos, Annie Di Donna). 

Logic §

Russell’s paradox §

Trying to make a definition

Example

Definition A proper set is not a member of itself.

$$A\notin A.$$

Let

$$B=\{A:A\notin A\}$$

now

$$B\in B??$$

Russel’s paradox.

Hilber’s quest §

Works of Hilbert.

---

Prove true/not true §

Take another set

$$C=\{A:A\in A\}$$

Does

$$C\in C?$$

• Proof by contradiction
  • suppose $C\notin C$, then by definition of $C$ $⟹C\notin C$

---

• logical systems.
  • propositional logic (AND, OR, NOT)
  • first order logic (FORALL, etc)
• logical principles.
• symantic system
• symtactic(?)
• meta language = English

Logical system §

A logical system consists of

• “Language”
• Alphabet
  • $A,B,C,(,),\forall$, $\exists$, $\vee ,\wedge$…
• Grammar well formed formulas
• “Truth valuation” semantic
• “Proof methods” syntactic
• Axioms