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