71 ideas
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] |
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] |
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] |
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] |
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] |
10852 | Robinson Arithmetic (Q) is not negation complete [Smith,P] |
10851 | Robinson Arithmetic 'Q' has basic axioms, quantifiers and first-order logic [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] |
10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
8502 | Realism doesn't explain 'a is F' any further by saying it is 'a has F-ness' [Devitt] |
8503 | The particular/universal distinction is unhelpful clutter; we should accept 'a is F' as basic [Devitt] |
8501 | Quineans take predication about objects as basic, not reference to properties they may have [Devitt] |
17368 | Essentialism concerns the nature of a group, not its category [Devitt] |
17370 | Things that gradually change, like species, can still have essences [Devitt] |
9354 | Why should necessities only be knowable a priori? That Hesperus is Phosporus is known empirically [Devitt] |
19565 | How could the mind have a link to the necessary character of reality? [Devitt] |
9353 | We explain away a priori knowledge, not as directly empirical, but as indirectly holistically empirical [Devitt] |
9356 | The idea of the a priori is so obscure that it won't explain anything [Devitt] |
19564 | Some knowledge must be empirical; naturalism implies that all knowledge is like that [Devitt] |
5901 | Is 'productive of happiness' the definition of 'right', or the cause of it? [Ross on Bentham] |
5934 | Of Bentham's 'dimensions' of pleasure, only intensity and duration matter [Ross on Bentham] |
5271 | Prejudice apart, push-pin has equal value with music and poetry [Bentham] |
3777 | Pleasure and pain control all human desires and duties [Bentham] |
3554 | Bentham thinks happiness is feeling good, but why use morality to achieve that? [Annas on Bentham] |
3781 | The value of pleasures and pains is their force [Bentham] |
20977 | Natural rights are nonsense, and unspecified natural rights is nonsense on stilts [Bentham] |
3778 | The community's interest is a sum of individual interests [Bentham] |
21003 | Only laws can produce real rights; rights from 'law of nature' are imaginary [Bentham] |
20280 | Large mature animals are more rational than babies. But all that really matters is - can they suffer? [Bentham] |
3779 | Unnatural, when it means anything, means infrequent [Bentham] |
17371 | Some kinds are very explanatory, but others less so, and some not at all [Devitt] |
17369 | We name species as small to share properties, but large enough to yield generalisations [Devitt] |
17367 | Species are phenetic, biological, niche, or phylogenetic-cladistic [Devitt, by PG] |
17372 | The higher categories are not natural kinds, so the Linnaean hierarchy should be given up [Devitt] |
17373 | Species pluralism says there are several good accounts of what a species is [Devitt] |
3780 | We must judge a thing morally to know if it conforms to God's will [Bentham] |