Ideas of Peter Smith, by Theme

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

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4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / a. Axioms for sets
There cannot be a set theory which is complete
5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
Second-order arithmetic can prove new sentences of first-order
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
A 'total function' maps every element to one element in another set
A 'partial function' maps only some elements to another set
Two functions are the same if they have the same extension
The 'range' of a function is the set of elements in the output set created by the function
An argument is a 'fixed point' for a function if it is mapped back to itself
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
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
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
A 'natural deduction system' has no axioms but many rules
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
No nice theory can define truth for its own language
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
A 'bijective' function has one-to-one correspondence in both directions
An 'injective' ('one-to-one') function creates a distinct output element from each original
A 'surjective' ('onto') function creates every element of the output set
5. Theory of Logic / K. Features of Logics / 3. Soundness
If everything that a theory proves is true, then it is 'sound'
Soundness is true axioms and a truth-preserving proof system
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
A theory is 'negation complete' if it proves all sentences or their negation
'Complete' applies both to whole logics, and to theories within them
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
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
Gödel's First Theorem sabotages logicism, and the Second sabotages Hilbert's Programme
5. Theory of Logic / K. Features of Logics / 7. Decidability
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating
A theory is 'decidable' if all of its sentences could be mechanically proved
Any consistent, axiomatized, negation-complete formal theory is decidable
5. Theory of Logic / K. Features of Logics / 8. Enumerability
A set is 'enumerable' is all of its elements can result from a natural number function
A set is 'effectively enumerable' if a computer could eventually list every member
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes)
The set of ordered pairs of natural numbers <i,j> is effectively enumerable
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
Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system
6. Mathematics / A. Nature of Mathematics / 3. Numbers / a. Numbers
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
6. Mathematics / A. Nature of Mathematics / 3. Numbers / g. Real numbers
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
6. Mathematics / A. Nature of Mathematics / 3. Numbers / q. Arithmetic
The truths of arithmetic are just true equations and their universally quantified versions
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / a. Axioms for numbers
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
All numbers are related to zero by the ancestral of the successor relation
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / b. Baby arithmetic
Baby Arithmetic is complete, but not very expressive
Baby arithmetic covers addition and multiplication, but no general facts about numbers
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / c. Robinson arithmetic
Robinson Arithmetic (Q) is not negation complete
Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / d. Peano arithmetic
Natural numbers have zero, unique successors, unending, no circling back, and no strays
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / f. Mathematical induction
The logic of arithmetic must quantify over properties of numbers to handle induction
6. Mathematics / B. Foundations for Mathematics / 3. Axioms for Number / g. Incompleteness of Arithmetic
Incompleteness results in arithmetic from combining addition and successor with multiplication
Multiplication only generates incompleteness if combined with addition and successor
8. Modes of Existence / A. Relations / 4. Formal Relations / c. Ancestral relation
The 'ancestral' of a relation is a new relation which creates a long chain of the original relation