73 ideas
| 12223 | It is a fallacy to explain the obscure with the even more obscure [Hale/Wright] |
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [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] |
| 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] |
| 10615 | The Comprehension Schema says there is a property only had by things satisfying a condition [Smith,P] |
| 10595 | A 'theorem' of a theory is a sentence derived from the axioms using the proof system [Smith,P] |
| 12230 | Singular terms refer if they make certain atomic statements true [Hale/Wright] |
| 10602 | A 'natural deduction system' has no axioms but many rules [Smith,P] |
| 10613 | No nice theory can define truth for its own language [Smith,P] |
| 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] |
| 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] |
| 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] |
| 10609 | Two routes to Incompleteness: semantics of sound/expressible, or syntax of consistency/proof [Smith,P] |
| 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] |
| 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] |
| 10600 | Being 'expressible' depends on language; being 'capture/represented' depends on axioms and proof system [Smith,P] |
| 10631 | If 'x is heterological' iff it does not apply to itself, then 'heterological' is heterological if it isn't heterological [Hale/Wright] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [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] |
| 10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
| 10068 | Natural numbers have zero, unique successors, unending, no circling back, and no strays [Smith,P] |
| 10603 | The logic of arithmetic must quantify over properties of numbers to handle induction [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 8784 | Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright] |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| 10629 | If structures are relative, this undermines truth-value and objectivity [Hale/Wright] |
| 10628 | The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright] |
| 8788 | Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright] |
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright] |
| 8783 | Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright] |
| 12225 | Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright] |
| 12224 | Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright] |
| 12226 | The identity of Pegasus with Pegasus may be true, despite the non-existence [Hale/Wright] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 12229 | Maybe we have abundant properties for semantics, and sparse properties for ontology [Hale/Wright] |
| 18443 | A successful predicate guarantees the existence of a property - the way of being it expresses [Hale/Wright] |
| 10626 | Objects just are what singular terms refer to [Hale/Wright] |
| 6375 | The taste of chocolate is a 'finer-grained' sensation than the taste of sweetness [Polger] |
| 6381 | The mind and the self are one, and the mind-self is a biological phenomenon [Polger] |
| 6378 | Teleological functions explain why a trait exists; causal-role functions say what it does [Polger] |
| 6380 | Identity theory says consciousness is an abstraction: a state, event, process or property [Polger] |
| 10630 | Abstracted objects are not mental creations, but depend on equivalence between given entities [Hale/Wright] |
| 8786 | One first-order abstraction principle is Frege's definition of 'direction' in terms of parallel lines [Hale/Wright] |
| 12227 | Abstractionism needs existential commitment and uniform truth-conditions [Hale/Wright] |
| 12228 | Equivalence abstraction refers to objects otherwise beyond our grasp [Hale/Wright] |
| 12231 | Reference needs truth as well as sense [Hale/Wright] |
| 10627 | Many conceptual truths ('yellow is extended') are not analytic, as derived from logic and definitions [Hale/Wright] |
| 6379 | A mummified heart has the teleological function of circulating blood [Polger] |
| 6377 | Teleological notions of function say what a thing is supposed to do [Polger] |