Mathematical Logic Emma Bird THEORY NOTES Contents Decidability 1 Church's Thesis 2 Church's theorem 3 Theories 3 Properties of theories 4 Theories of arithmetic. 5 Categoricity 6 FOPA - categoricity (lack of) 7 SOPA - categoricity 8 Decidability · A problem is EFFECTIVELY DECIDABLE iff there is an algorithm/effective procedure for resolving (answering 'es' or 'no' whether any given x of some kind k has property p o When assessing decidability, can ignore any accidental limits on time/speed of computation/matter in universe etc · = decidability in principle · ALGORITHM = a mechanical procedure for deciding a problem o = can be given as a finite set of instructions in a stepwise manner, without appeal to random processes or ingenuity o = must be finite, but no set finite upper bound o A problem can be decidable even if the algorithm that makes it so is not known . . The notion of an algorithm is INFORMAL so the notion of decidability is informal Property P is decidable iff the problem it is mentioned in is decidable SETS · DECIDABLE SET = there is an effective decision procedure for determining whether any given object is a member of that set · DECIDABLE RELATION = there is an effective decision procedure for determining whether any given objects, in order, stand in that relation or not · Finite sets = trivially decidable o Can enumerate the members and compare to a test object in term · Example of an undecidable property: o = object X has property P if GC is true and does not have property P if GC is false · Example of a decidable property: o =object X in a domain of numbers (can be infinite) has property P if x is even and does not have property P if x is odd DEDUCTIONS · = finite · = want them to be decidable, so that we can check whether a given deduction is the sequence it is claimed to be Page 1 of 10
Emma Bird Mathematical Logic o Makes derivation a useful notion · In order for deduction to be decidable, must make sure: o The property 'being a sentence' = decidable o The property 'being an axiom' = decidable o The property 'following by a rule of inference' = decidable · Sets of axioms need not be finite o Axiom schemes, like induction in FO PA, illustrate a pattern common to infinitely many axioms o Therefore, decidability of deductions only requires that the number of axiom schemes be finite Nb 'is there a deduction of o from 2?' (not always decidable?) # 'is this sequence of deduction of ¢ from 2?' (decidable) SEMI-DECIDABILITY · = gives you all the 'yes' but not all the 'no' o Sometimes just doesn't give an answer o If it fails to answer for even just one object, then is not decidable · Therefore doesn't decide set membership, property-attribution etc · SEMI-DECIDABLE SET = there is an effective decision procedure which,