### Ideas of Peter Smith, by Theme

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

green numbers give full details    |    back to list of philosophers    |     expand these ideas
###### 4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
 10073 There cannot be a set theory which is complete
###### 5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
 10616 Second-order arithmetic can prove new sentences of first-order
###### 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
 10075 A 'partial function' maps only some elements to another set
 10605 Two functions are the same if they have the same extension
 10612 An argument is a 'fixed point' for a function if it is mapped back to itself
 10076 The 'range' of a function is the set of elements in the output set created by the function
 10074 A 'total function' maps every element to one element in another set
###### 5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
 10615 The Comprehension Schema says there is a property only had by things satisfying a condition
###### 5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
 10595 A 'theorem' of a theory is a sentence derived from the axioms using the proof system
###### 5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
 10602 A 'natural deduction system' has no axioms but many rules
###### 5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
 10613 No nice theory can define truth for its own language
###### 5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
 10078 An 'injective' ('one-to-one') function creates a distinct output element from each original
 10077 A 'surjective' ('onto') function creates every element of the output set
 10079 A 'bijective' function has one-to-one correspondence in both directions
###### 5. Theory of Logic / K. Features of Logics / 3. Soundness
 10070 If everything that a theory proves is true, then it is 'sound'
 10086 Soundness is true axioms and a truth-preserving proof system
 10596 A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation)
###### 5. Theory of Logic / K. Features of Logics / 4. Completeness
 10598 A theory is 'negation complete' if it proves all sentences or their negation
 10597 'Complete' applies both to whole logics, and to theories within them
 10069 A theory is 'negation complete' if one of its sentences or its negation can always be proved
###### 5. Theory of Logic / K. Features of Logics / 5. Incompleteness
 10609 Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
###### 5. Theory of Logic / K. Features of Logics / 7. Decidability
 10080 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating
 10087 A theory is 'decidable' if all of its sentences could be mechanically proved
 10088 Any consistent, axiomatized, negation-complete formal theory is decidable
###### 5. Theory of Logic / K. Features of Logics / 8. Enumerability
 10081 A set is 'enumerable' is all of its elements can result from a natural number function
 10083 A set is 'effectively enumerable' if a computer could eventually list every member
 10084 A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes)
 10085 The set of ordered pairs of natural numbers is effectively enumerable
 10601 The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
###### 5. Theory of Logic / K. Features of Logics / 9. Expressibility
 10600 Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / a. Numbers
 10599 For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
###### 6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / g. Real numbers
 10610 The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
###### 6. Mathematics / A. Nature of Mathematics / 4. Using Numbers / f. Arithmetic
 10619 The truths of arithmetic are just true equations and their universally quantified versions
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / a. Axioms for numbers
 10618 All numbers are related to zero by the ancestral of the successor relation
 10608 The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic
 10849 Baby arithmetic covers addition and multiplication, but no general facts about numbers
 10850 Baby Arithmetic is complete, but not very expressive
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / c. Robinson arithmetic
 10851 Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic
 10852 Robinson Arithmetic (Q) is not negation complete
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / d. Peano arithmetic
 10068 Natural numbers have zero, unique successors, unending, no circling back, and no strays
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / f. Mathematical induction
 10603 The logic of arithmetic must quantify over properties of numbers to handle induction
###### 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / g. Incompleteness of Arithmetic
 10604 Incompleteness results in arithmetic from combining addition and successor with multiplication
 10848 Multiplication only generates incompleteness if combined with addition and successor
###### 8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
 10617 The 'ancestral' of a relation is a new relation which creates a long chain of the original relation