Mathematical Logic, Lecture Notes- Maths, Study notes of Mathematics

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B1a Logic

Dr Jochen Koenigsmann

Oxford, MT 2010

Introduction

1. What is mathematical logic about?

• provide a uniform, unambiguous language for mathematics
• make precise what a proof is
• explain and guarantee exactness, rigor and certainty in mathematics
• establish the foundations of mathematics

B1 (Foundations) = B1a (Logic) + B1b (Set theory)

N.B.: Course does not teach you to think logically, but it explores what it means to think logically

3. Hilbert’s Program

1. find a uniform (formal) language

for all mathematics

2. find a complete system of

inference rules/ deduction rules

3. find a complete system of

mathematical axioms

4. prove that the system 1.+2.+3. is

consistent, i.e. does not lead to contradictions

⋆ complete: every mathematical sentence can be proved or disproved using 2. and 3. ⋆ 1., 2. and 3. should be finitary/effective/computable/algorithmic so, e.g., in 3. you can’t take as axioms the system of all true sentences in mathematics ⋆ idea: any piece of information is of finte length

4. Solutions to Hilbert’s program

Step 1. is possible in the framework of ZF = Zermelo-Fraenkel set theory or ZFC = ZF + Axiom of Choice (this is an empirical fact) ; B1b Set Theory HT 2011

Step 2. is possible in the framework of 1st-order logic: G¨odel’s Completeness Theorem ; B1a Logic - this course

Step 3. is not possible (; C1.1a): G¨odel’s 1st Incompleteness Theorem: there is no effective axiomatization of arithmetic

Step 4. is not possible(; C1.1a): G¨odel’s 2nd Incompleteness Theorem

6. Why mathematical logic?

1. Language and deduction rules are tailored for mathematical objects and mathemati- cal ways of reasoning N.B.: Logic tells you what a proof is, not how to find one
2. The method is mathematical: we will develop logic as a calculus with sen- tences and formulas ⇒ Logic is itself a mathematical discipline, not meta-mathematics or philosophy, no ontological questions like what is a number?
3. Logic has applications towards other areas of mathematics, e.g. Algebra, Topology, but also towards theoretical computer sci- ence