Combining Texts

Ideas for 'Confessions', 'Causation in a Physical World' and 'Intro to Gdel's Theorems'

expand these ideas     |    start again     |     choose another area for these texts

display all the ideas for this combination of texts

29 ideas

5. Theory of Logic / A. Overview of Logic / 7. Second-Order Logic
10616 Second-order arithmetic can prove new sentences of first-order [Smith,P]
5. Theory of Logic / E. Structures of Logic / 5. Functions in Logic
10076 The 'range' of a function is the set of elements in the output set created by the function [Smith,P]
10605 Two functions are the same if they have the same extension [Smith,P]
10075 A 'partial function' maps only some elements to another set [Smith,P]
10074 A 'total function' maps every element to one element in another set [Smith,P]
10612 An argument is a 'fixed point' for a function if it is mapped back to itself [Smith,P]
5. Theory of Logic / E. Structures of Logic / 7. Predicates in Logic
10615 The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P]
5. Theory of Logic / E. Structures of Logic / 8. Theories in Logic
10595 A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P]
5. Theory of Logic / H. Proof Systems / 4. Natural Deduction
10602 A 'natural deduction system' has no axioms but many rules [Smith,P]
5. Theory of Logic / I. Semantics of Logic / 2. Formal Truth
10613 No nice theory can define truth for its own language [Smith,P]
5. Theory of Logic / J. Model Theory in Logic / 2. Isomorphisms
10078 An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P]
10077 A 'surjective' ('onto') function creates every element of the output set [Smith,P]
10079 A 'bijective' function has one-to-one correspondence in both directions [Smith,P]
5. Theory of Logic / K. Features of Logics / 3. Soundness
10070 If everything that a theory proves is true, then it is 'sound' [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]
5. Theory of Logic / K. Features of Logics / 4. Completeness
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]
10069 A theory is 'negation complete' if one of its sentences or its negation can always be proved [Smith,P]
5. Theory of Logic / K. Features of Logics / 5. Incompleteness
10609 Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P]
5. Theory of Logic / K. Features of Logics / 7. Decidability
10080 'Effective' means simple, unintuitive, independent, controlled, dumb, and terminating [Smith,P]
10087 A theory is 'decidable' if all of its sentences could be mechanically proved [Smith,P]
10088 Any consistent, axiomatized, negation-complete formal theory is decidable [Smith,P]
5. Theory of Logic / K. Features of Logics / 8. Enumerability
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]
10601 The thorems of a nice arithmetic can be enumerated, but not the truths (so they're diffferent) [Smith,P]
5. Theory of Logic / K. Features of Logics / 9. Expressibility
10600 Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P]