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### Ideas of Peter Smith, by Text

#### [British, b.1944, At the University of Aberystwyth, and then at Cambridge University.]

 2007 Intro to Gödel's Theorems
 1.1 p.1 10068 Natural numbers have zero, unique successors, unending, no circling back, and no strays
 1.1 p.2 10070 If everything that a theory proves is true, then it is 'sound'
 1.1 p.2 10069 A theory is 'negation complete' if one of its sentences or its negation can always be proved
 1.3 p.5 10073 There cannot be a set theory which is complete
 2.1 p.8 10074 A 'total function' maps every element to one element in another set
 2.1 p.8 10076 The 'range' of a function is the set of elements in the output set created by the function
 2.1 p.8 10079 A 'bijective' function has one-to-one correspondence in both directions
 2.1 p.8 10078 An 'injective' ('one-to-one') function creates a distinct output element from each original
 2.1 p.8 10077 A 'surjective' ('onto') function creates every element of the output set
 02.1 n1 p.8 10075 A 'partial function' maps only some elements to another set
 2.2 p.9 10080 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating
 2.3 p.13 10081 A set is 'enumerable' is all of its elements can result from a natural number function
 2.4 p.15 10083 A set is 'effectively enumerable' if a computer could eventually list every member
 2.4 p.16 10084 A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes)
 2.5 p.16 10085 The set of ordered pairs of natural numbers is effectively enumerable
 3.4 p.23 10595 A 'theorem' of a theory is a sentence derived from the axioms using the proof system
 3.4 p.24 10087 A theory is 'decidable' if all of its sentences could be mechanically proved
 3.4 p.24 10598 A theory is 'negation complete' if it proves all sentences or their negation
 3.4 p.24 10086 Soundness is true axioms and a truth-preserving proof system
 3.4 p.24 10596 A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation)
 3.4 p.25 10597 'Complete' applies both to whole logics, and to theories within them
 3.6 p.26 10088 Any consistent, axiomatized, negation-complete formal theory is decidable
 4.5 p.33 10599 For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
 4.7 p.36 10600 Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system
 05 Intro p.37 10601 The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
 8.1 p.51 10849 Baby arithmetic covers addition and multiplication, but no general facts about numbers
 8.3 p.55 10850 Baby Arithmetic is complete, but not very expressive
 8.3 p.55 10851 Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic
 8.4 p.57 10852 Robinson Arithmetic (Q) is not negation complete
 9.1 p.59 10602 A 'natural deduction system' has no axioms but many rules
 10.1 p.71 10603 The logic of arithmetic must quantify over properties of numbers to handle induction
 10.7 p.79 10604 Incompleteness results in arithmetic from combining addition and successor with multiplication
 10.7 n8 p.79 10848 Multiplication only generates incompleteness if combined with addition and successor
 11.3 p.87 10605 Two functions are the same if they have the same extension
 14.1 p.120 10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
 18.1 p.156 10609 Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
 18.2 p.157 10610 The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
 20.5 p.174 10612 An argument is a 'fixed point' for a function if it is mapped back to itself
 21.5 p.180 10613 No nice theory can define truth for its own language
 22.3 p.190 10615 The Comprehension Schema says there is a property only had by things satisfying a condition
 23.4 p.206 10616 Second-order arithmetic can prove new sentences of first-order
 23.5 p.209 10617 The 'ancestral' of a relation is a new relation which creates a long chain of the original relation
 23.5 p.211 10618 All numbers are related to zero by the ancestral of the successor relation
 27.7 p.258 10619 The truths of arithmetic are just true equations and their universally quantified versions