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Single Idea 10069
[filed under theme 5. Theory of Logic / K. Features of Logics / 4. Completeness
]
Full Idea
Logicians say that a theory T is '(negation) complete' if, for every sentence φ in the language of the theory, either φ or ¬φ is deducible in T's proof system. If this were the case, then truth could be equated with provability.
Gist of Idea
A theory is 'negation complete' if one of its sentences or its negation can always be proved
Source
Peter Smith (Intro to Gödel's Theorems [2007], 01.1)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.2
A Reaction
The word 'negation' seems to be a recent addition to the concept. Presumable it might be the case that φ can always be proved, but not ¬φ.
The
44 ideas
from 'Intro to Gödel's Theorems'
10070
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If everything that a theory proves is true, then it is 'sound'
[Smith,P]
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10069
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A theory is 'negation complete' if one of its sentences or its negation can always be proved
[Smith,P]
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10068
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Natural numbers have zero, unique successors, unending, no circling back, and no strays
[Smith,P]
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10073
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There cannot be a set theory which is complete
[Smith,P]
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10076
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The 'range' of a function is the set of elements in the output set created by the function
[Smith,P]
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10074
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A 'total function' maps every element to one element in another set
[Smith,P]
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10077
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A 'surjective' ('onto') function creates every element of the output set
[Smith,P]
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10079
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A 'bijective' function has one-to-one correspondence in both directions
[Smith,P]
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10078
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An 'injective' ('one-to-one') function creates a distinct output element from each original
[Smith,P]
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10075
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A 'partial function' maps only some elements to another set
[Smith,P]
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10080
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'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating
[Smith,P]
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10081
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A set is 'enumerable' is all of its elements can result from a natural number function
[Smith,P]
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10083
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A set is 'effectively enumerable' if a computer could eventually list every member
[Smith,P]
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10084
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A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes)
[Smith,P]
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10085
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The set of ordered pairs of natural numbers <i,j> is effectively enumerable
[Smith,P]
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10087
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A theory is 'decidable' if all of its sentences could be mechanically proved
[Smith,P]
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10086
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Soundness is true axioms and a truth-preserving proof system
[Smith,P]
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10596
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A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation)
[Smith,P]
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10598
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A theory is 'negation complete' if it proves all sentences or their negation
[Smith,P]
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10597
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'Complete' applies both to whole logics, and to theories within them
[Smith,P]
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10595
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A 'theorem' of a theory is a sentence derived from the axioms using the proof system
[Smith,P]
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10088
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Any consistent, axiomatized, negation-complete formal theory is decidable
[Smith,P]
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10599
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For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
[Smith,P]
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10600
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Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system
[Smith,P]
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10601
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The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
[Smith,P]
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10849
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Baby arithmetic covers addition and multiplication, but no general facts about numbers
[Smith,P]
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10850
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Baby Arithmetic is complete, but not very expressive
[Smith,P]
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10851
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Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic
[Smith,P]
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10852
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Robinson Arithmetic (Q) is not negation complete
[Smith,P]
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10602
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A 'natural deduction system' has no axioms but many rules
[Smith,P]
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10603
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The logic of arithmetic must quantify over properties of numbers to handle induction
[Smith,P]
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10604
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Incompleteness results in arithmetic from combining addition and successor with multiplication
[Smith,P]
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10848
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Multiplication only generates incompleteness if combined with addition and successor
[Smith,P]
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10605
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Two functions are the same if they have the same extension
[Smith,P]
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10608
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The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
[Smith,P]
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10609
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Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
[Smith,P]
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10610
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The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
[Smith,P]
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10612
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An argument is a 'fixed point' for a function if it is mapped back to itself
[Smith,P]
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10613
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No nice theory can define truth for its own language
[Smith,P]
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10615
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The Comprehension Schema says there is a property only had by things satisfying a condition
[Smith,P]
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10616
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Second-order arithmetic can prove new sentences of first-order
[Smith,P]
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10618
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All numbers are related to zero by the ancestral of the successor relation
[Smith,P]
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10617
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The 'ancestral' of a relation is a new relation which creates a long chain of the original relation
[Smith,P]
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10619
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The truths of arithmetic are just true equations and their universally quantified versions
[Smith,P]
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