Statement: well-formed statement with a truth value
Axiom: obviously true statement
Deduction Rule: rule allow derive new true statements from other true statement
Proof: chain of deduction rules, starting from axioms, ending at the statement
GORM: good old mathematics FORM: formalized mathematics
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')
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)
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:
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 ZFC2nd 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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