2007 | Intro to Gödel's Theorems |
01.1 | p.1 | 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays |
01.1 | p.2 | 10070 | If everything that a theory proves is true, then it is 'sound' |
01.1 | p.2 | 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved |
01.3 | p.5 | 10073 | There cannot be a set theory which is complete |
02.1 | p.8 | 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original |
02.1 | p.8 | 10076 | The 'range' of a function is the set of elements in the output set created by the function |
02.1 | p.8 | 10074 | A 'total function' maps every element to one element in another set |
02.1 | p.8 | 10077 | A 'surjective' ('onto') function creates every element of the output set |
02.1 | p.8 | 10079 | A 'bijective' function has one-to-one correspondence in both directions |
02.1 n1 | p.8 | 10075 | A 'partial function' maps only some elements to another set |
02.2 | p.9 | 10080 | 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating |
02.3 | p.13 | 10081 | A set is 'enumerable' is all of its elements can result from a natural number function |
02.4 | p.15 | 10083 | A set is 'effectively enumerable' if a computer could eventually list every member |
02.4 | p.16 | 10084 | A finite set of finitely specifiable objects is always effectively enumerable (e.g. primes) |
02.5 | p.16 | 10085 | The set of ordered pairs of natural numbers <i,j> is effectively enumerable |
03.4 | p.23 | 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system |
03.4 | p.24 | 10086 | Soundness is true axioms and a truth-preserving proof system |
03.4 | p.24 | 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) |
03.4 | p.24 | 10598 | A theory is 'negation complete' if it proves all sentences or their negation |
03.4 | p.24 | 10087 | A theory is 'decidable' if all of its sentences could be mechanically proved |
03.4 | p.25 | 10597 | 'Complete' applies both to whole logics, and to theories within them |
03.6 | p.26 | 10088 | Any consistent, axiomatized, negation-complete formal theory is decidable |
04.5 | p.33 | 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) |
04.7 | p.36 | 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system |
05 Intro | p.37 | 10601 | The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) |
08.1 | p.51 | 10849 | Baby arithmetic covers addition and multiplication, but no general facts about numbers |
08.3 | p.55 | 10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic |
08.3 | p.55 | 10850 | Baby Arithmetic is complete, but not very expressive |
08.4 | p.57 | 10852 | Robinson Arithmetic (Q) is not negation complete |
09.1 | p.59 | 10602 | A 'natural deduction system' has no axioms but many rules |
10.1 | p.71 | 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction |
10.7 | p.79 | 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication |
10.7 n8 | p.79 | 10848 | Multiplication only generates incompleteness if combined with addition and successor |
11.3 | p.87 | 10605 | Two functions are the same if they have the same extension |
14.1 | p.120 | 10608 | The number of Fs is the 'successor' of the Gs if there is a single F that isn't G |
18.1 | p.156 | 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof |
18.2 | p.157 | 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals |
20.5 | p.174 | 10612 | An argument is a 'fixed point' for a function if it is mapped back to itself |
21.5 | p.180 | 10613 | No nice theory can define truth for its own language |
22.3 | p.190 | 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition |
23.4 | p.206 | 10616 | Second-order arithmetic can prove new sentences of first-order |
23.5 | p.209 | 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation |
23.5 | p.211 | 10618 | All numbers are related to zero by the ancestral of the successor relation |
27.7 | p.258 | 10619 | The truths of arithmetic are just true equations and their universally quantified versions |