Lecture 007

Limit of Human Reasoning

Definitions

Statement: well-formed statement with a truth value

• encoded in string

Axiom: obviously true statement

• should be computable

Deduction Rule: rule allow derive new true statements from other true statement

• need to be simple string manipulation process

Proof: chain of deduction rules, starting from axioms, ending at the statement

GORM: good old mathematics FORM: formalized mathematics

Formalize Deduction and Mathematic System

First Order Logic (FOL): separation syntax and semantics

• Propositional Logic: $\lnot, \land, \lor, \implies, \leftrightarrow$

• Quantifier: $\exists$, $\forall$

• Vocabulary: "constants", "relations", "functions"

FOL Deduction Rules (Hilbert System): simple string manipulations (e.g. From $S$ and $S \rightarrow S'$ we can deduct $S'$)

• Codel's Completeness Theorem: Hilbert System is complete (we only need finite many deduction rules)

Axiomatic System: FOL vocabulary + computable axioms

• Zermelo-Fraenkel-Choice (ZFC): axiomatic system for set theory

• All GORM can be compiled down to Set Theory

Church-Turing Thesis for Mathematics (GORM-to-ZFC Thesis): The ZFC axiomatic system is the right model for GORM (Every GORM-proof compules down to a ZFC-proof - this is mostly true)

Properties of Axiomatic System

Consistency: for every statement $S$, at most one $S$ or $\lnot S$ is provable.

Soundness: If $S$ is provable, then $S$ is true. (We just need axioms to be true because Hilbert System preserve truth)

• Soundness implies Consistency

• soundness cannot be proved because it is outside of mathematic system (cannot be formalized)

Completeness: For every statement $S$, at least one $S$ or $\lnot S$ is provable.

Hilbert's Program:

1. formalize mathematical reasoning
2. prove system is complete
3. prove system is consistent
4. $\text{truth} \equiv \text{provable}$

Incompleteness Theorem

Provable: there is a proof of $S$ Refutable: there is a proof of $\lnot S$ Independent: $S$ is neither provable nor refutable Incompleteness exists an independent $S$

0th Incompleteness Theorem: not everything have finite description, therefore cannot reasoning about:

1st Incompleteness Theorem (Soundness #1)

• Suppose ZFC is sound (and therefore consistent)

• Suppose ZFC is complete

• Then $\text{truth} \equiv \text{provable}$

• Then there is a computation to decide any truth

Theorem: If ZFC is sound, it is incomplete. (there exists TM $M$ such that "$M(\langle{M}\rangle) \text{ accepts}$" is independent of ZFC)

1st Incompleteness Theorem (Soundness #2): find explicits TM $M$ such that $M(\langle{M}\rangle) \text{ accepts}$ is independent of ZFC.

• Theorem: If ZFC is sound, then "$M_X(\langle{M_X}\rangle) \text{ accepts}$" is independent of ZFC

1st Incompleteness Theorem (Consistency): assume only consistency, not soundness Lemma: If a TM halt, then it is provable.

Theorem: If ZFC is consistent, then "$M_X(\langle{M_X}\rangle) \text{ accepts}$" is independent of ZFC

2nd Incompleteness Theorem

• Suppose $S$ is independent (neither $S$ nor $\lnot S$ is provable)

• Suppose: $S \implies S \text{ is provable}$ (this is generally true for every statement)

• Then: $\lnot S$ (contradiction)

Conclusion: Assume "ZFC is consistent" is provable, then $\lnot S$ is provable. However, "ZFC is consistent" also implies $\lnot S$ is not provable (by 1st Incompleteness Theorem). Contradiction.

Theorem: If "ZFC is consistent", then "ZFC is consistent" is not provable.

Nope.

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