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Single Idea 10074
[filed under theme 5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
]
Full Idea
A 'total function' is one which maps every element of a domain to exactly one corresponding value in another set.
Gist of Idea
A 'total function' maps every element to one element in another set
Source
Peter Smith (Intro to Gödel's Theorems [2007], 02.1)
Book Ref
Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.8
Related Ideas
Idea 13700
A 'total' function must always produce an output for a given domain [Sider]
Idea 13811
A 'total' function ranges over the whole domain, a 'partial' function over appropriate inputs [Bostock]
The
44 ideas
from Peter Smith
10070
|
If everything that a theory proves is true, then it is 'sound'
[Smith,P]
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10069
|
A theory is 'negation complete' if one of its sentences or its negation can always be proved
[Smith,P]
|
10068
|
Natural numbers have zero, unique successors, unending, no circling back, and no strays
[Smith,P]
|
10073
|
There cannot be a set theory which is complete
[Smith,P]
|
10076
|
The 'range' of a function is the set of elements in the output set created by the function
[Smith,P]
|
10074
|
A 'total function' maps every element to one element in another set
[Smith,P]
|
10077
|
A 'surjective' ('onto') function creates every element of the output set
[Smith,P]
|
10078
|
An 'injective' ('one-to-one') function creates a distinct output element from each original
[Smith,P]
|
10079
|
A 'bijective' function has one-to-one correspondence in both directions
[Smith,P]
|
10075
|
A 'partial function' maps only some elements to another set
[Smith,P]
|
10080
|
'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating
[Smith,P]
|
10081
|
A set is 'enumerable' is all of its elements can result from a natural number function
[Smith,P]
|
10083
|
A set is 'effectively enumerable' if a computer could eventually list every member
[Smith,P]
|
10084
|
A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes)
[Smith,P]
|
10085
|
The set of ordered pairs of natural numbers <i,j> is effectively enumerable
[Smith,P]
|
10086
|
Soundness is true axioms and a truth-preserving proof system
[Smith,P]
|
10596
|
A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation)
[Smith,P]
|
10598
|
A theory is 'negation complete' if it proves all sentences or their negation
[Smith,P]
|
10597
|
'Complete' applies both to whole logics, and to theories within them
[Smith,P]
|
10087
|
A theory is 'decidable' if all of its sentences could be mechanically proved
[Smith,P]
|
10595
|
A 'theorem' of a theory is a sentence derived from the axioms using the proof system
[Smith,P]
|
10088
|
Any consistent, axiomatized, negation-complete formal theory is decidable
[Smith,P]
|
10599
|
For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1)))
[Smith,P]
|
10600
|
Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system
[Smith,P]
|
10601
|
The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent)
[Smith,P]
|
10849
|
Baby arithmetic covers addition and multiplication, but no general facts about numbers
[Smith,P]
|
10850
|
Baby Arithmetic is complete, but not very expressive
[Smith,P]
|
10851
|
Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic
[Smith,P]
|
10852
|
Robinson Arithmetic (Q) is not negation complete
[Smith,P]
|
10602
|
A 'natural deduction system' has no axioms but many rules
[Smith,P]
|
10603
|
The logic of arithmetic must quantify over properties of numbers to handle induction
[Smith,P]
|
10604
|
Incompleteness results in arithmetic from combining addition and successor with multiplication
[Smith,P]
|
10848
|
Multiplication only generates incompleteness if combined with addition and successor
[Smith,P]
|
10605
|
Two functions are the same if they have the same extension
[Smith,P]
|
10608
|
The number of Fs is the 'successor' of the Gs if there is a single F that isn't G
[Smith,P]
|
10609
|
Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof
[Smith,P]
|
10610
|
The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals
[Smith,P]
|
10612
|
An argument is a 'fixed point' for a function if it is mapped back to itself
[Smith,P]
|
10613
|
No nice theory can define truth for its own language
[Smith,P]
|
10615
|
The Comprehension Schema says there is a property only had by things satisfying a condition
[Smith,P]
|
10616
|
Second-order arithmetic can prove new sentences of first-order
[Smith,P]
|
10618
|
All numbers are related to zero by the ancestral of the successor relation
[Smith,P]
|
10617
|
The 'ancestral' of a relation is a new relation which creates a long chain of the original relation
[Smith,P]
|
10619
|
The truths of arithmetic are just true equations and their universally quantified versions
[Smith,P]
|