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Single Idea 10849

[filed under theme 6. Mathematics / B. Foundations for Mathematics / 4. Axioms for Number / b. Baby arithmetic ]

Full Idea

Baby Arithmetic 'knows' the addition of particular numbers and multiplication, but can't express general facts about numbers, because it lacks quantification. It has a constant '0', a function 'S', and functions '+' and 'x', and identity and negation.

Gist of Idea

Baby arithmetic covers addition and multiplication, but no general facts about numbers

Source

Peter Smith (Intro to Gödel's Theorems [2007], 08.1)

Book Ref

Smith,Peter: 'An Introduction to Gödel's Theorems' [CUP 2007], p.51

The 44 ideas from 'Intro to Gödel's Theorems'

10070 If everything that a theory proves is true, then it is 'sound' [Smith,P]
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]
10074 A 'total function' maps every element to one element in another set [Smith,P]
10076 The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
10079 A 'bijective' function has one-to-one correspondence in both directions [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]
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]
10087 A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P]
10086 Soundness is true axioms and a truth-preserving proof system [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]
10596 A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [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]