80 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] |
| 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] |
| 10605 | Two functions are the same if they have the same extension [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] |
| 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] |
| 10079 | A 'bijective' function has one-to-one correspondence in both directions [Smith,P] |
| 10077 | A 'surjective' ('onto') function creates every element of the output set [Smith,P] |
| 10078 | An 'injective' ('one-to-one') function creates a distinct output element from each original [Smith,P] |
| 10070 | If everything that a theory proves is true, then it is 'sound' [Smith,P] |
| 10596 | A theory is 'sound' iff every theorem is true (usually from true axioms and truth-preservation) [Smith,P] |
| 10086 | Soundness is true axioms and a truth-preserving proof system [Smith,P] |
| 10069 | A theory is 'negation complete' if one of its sentences or its negation can always be proved [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] |
| 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] |
| 10604 | Incompleteness results in arithmetic from combining addition and successor with multiplication [Smith,P] |
| 10848 | Multiplication only generates incompleteness if combined with addition and successor [Smith,P] |
| 3942 | I do not believe in the existence of anything, if I see no reason to believe it [Berkeley] |
| 3952 | I know that nothing inconsistent can exist [Berkeley] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 3959 | There is no other substance, in a strict sense, than spirit [Berkeley] |
| 3946 | A thing is shown to be impossible if a contradiction is demonstrated within its definition [Berkeley] |
| 3958 | Since our ideas vary when the real things are said to be unchanged, they cannot be true copies [Berkeley] |
| 3943 | If existence is perceived directly, by which sense; if indirectly, how is it inferred from direct perception? [Berkeley] |
| 3931 | Sensible objects are just sets of sensible qualities [Berkeley] |
| 5192 | Berkeley did not deny material things; he merely said they must be defined through sensations [Berkeley, by Ayer] |
| 5174 | Berkeley needed a phenomenalist account of the self, as well as of material things [Ayer on Berkeley] |
| 1103 | 'To be is to be perceived' is a simple confusion of experience with its objects [Russell on Berkeley] |
| 6403 | For Berkelely, reality is ideas and a community of minds, including God's [Berkeley, by Grayling] |
| 3945 | There is nothing in nature which needs the concept of matter to explain it [Berkeley] |
| 3947 | Perceptions are ideas, and ideas exist in the mind, so objects only exist in the mind [Berkeley] |
| 3936 | Time is measured by the succession of ideas in our minds [Berkeley] |
| 3930 | There is no such thing as 'material substance' [Berkeley] |
| 3939 | I conceive a tree in my mind, but I cannot prove that its existence can be conceived outside a mind [Berkeley] |
| 3933 | Primary qualities (such as shape, solidity, mass) are held to really exist, unlike secondary qualities [Berkeley] |
| 3934 | A mite would see its own foot as large, though we would see it as tiny [Berkeley] |
| 3935 | The apparent size of an object varies with its distance away, so that can't be a property of the object [Berkeley] |
| 3937 | 'Solidity' is either not a sensible quality at all, or it is clearly relative to our senses [Berkeley] |
| 3940 | Distance is not directly perceived by sight [Berkeley] |
| 3957 | Immediate objects of perception, which some treat as appearances, I treat as the real things themselves [Berkeley] |
| 3953 | Real things and imaginary or dreamed things differ because the latter are much fainter [Berkeley] |
| 3938 | Geometry is originally perceived by senses, and so is not purely intellectual [Berkeley] |
| 3944 | It is possible that we could perceive everything as we do now, but nothing actually existed. [Berkeley] |
| 3932 | A hot hand and a cold hand will have different experiences in the same tepid water [Berkeley] |
| 3948 | Experience tells me that other minds exist independently from my own [Berkeley] |
| 4871 | A thing is free if it acts only by the necessity of its own nature [Spinoza] |
| 3941 | How can that which is unthinking be a cause of thought? [Berkeley] |
| 5374 | Berkeley probably used 'idea' to mean both the act of apprehension and the thing apprehended [Russell on Berkeley] |
| 3954 | Immorality is not in the action, but in the deviation of the will from moral law [Berkeley] |
| 3950 | There must be a God, because all sensible things must be perceived by him [Berkeley] |
| 3951 | There must be a God, because I and my ideas are not independent [Berkeley] |
| 3949 | It has been proved that creation is the workmanship of God, from its beauty and usefulness [Berkeley] |
| 3956 | People are responsible because they have limited power, though this ultimately derives from God [Berkeley] |
| 3955 | If sin is not just physical, we don't consider God the origin of sin because he causes physical events [Berkeley] |