73 ideas
2930 | The main aim of philosophy must be to determine the order of rank among values [Nietzsche] |
20143 | Scientific knowledge is nothing without a prior philosophical 'faith' [Nietzsche] |
23722 | Objectivity is not disinterestedness (impossible), but the ability to switch perspectives [Nietzsche] |
4417 | Only that which has no history is definable [Nietzsche] |
23715 | Psychologists should be brave and proud, and prefer truth to desires, even when it is ugly [Nietzsche] |
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] |
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] |
10618 | All numbers are related to zero by the ancestral of the successor relation [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] |
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] |
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] |
22518 | The actual must be possible, because it occurred [Aristotle] |
4421 | Philosophers have never asked why there is a will to truth in the first place [Nietzsche] |
23719 | Forgetfulness is a strong positive ability, not mental laziness [Nietzsche] |
4420 | There is only 'perspective' seeing and knowing, and so the best objectivity is multiple points of view [Nietzsche] |
4414 | Philosophers invented "free will" so that our virtues would be permanently interesting to the gods [Nietzsche] |
4419 | People who think in words are orators rather than thinkers, and think about facts instead of thinking facts [Nietzsche] |
4411 | It is a delusion to separate the man from the deed, like the flash from the lightning [Nietzsche] |
16566 | Poetry is more philosophic than history, as it concerns universals, not particulars [Aristotle] |
3793 | We must question the very value of moral values [Nietzsche] |
4408 | The concept of 'good' was created by aristocrats to describe their own actions [Nietzsche] |
23716 | A strong rounded person soon forgets enemies, misfortunes, and even misdeeds [Nietzsche] |
20129 | All animals strive for the ideal conditions to express their power, and hate any hindrances [Nietzsche] |
4409 | Only the decline of aristocratic morality led to concerns about "egoism" [Nietzsche] |
3259 | Nietzsche rejects impersonal morality, and asserts the idea of living well [Nietzsche, by Nagel] |
4416 | Basic justice is the negotiation of agreement among equals, and the imposition of agreement [Nietzsche] |
4418 | A masterful and violent person need have nothing to do with contracts [Nietzsche] |
4407 | Plato, Spinoza and Kant are very different, but united in their low estimation of pity [Nietzsche] |
4415 | Guilt and obligation originated in the relationship of buying and selling, credit and debt [Nietzsche] |
23718 | If we say birds of prey could become lambs, that makes them responsible for being birds of prey [Nietzsche] |
23717 | Modern nihilism is now feeling tired of mankind [Nietzsche] |
23721 | Old tribes always felt an obligation to the earlier generations, and the founders [Nietzsche] |
20142 | The state begins with brutal conquest of a disorganised people, not with a 'contract' [Nietzsche] |
23720 | Punishment makes people harder, more alienated, and hostile [Nietzsche] |
4410 | The truly great haters in world history have always been priests [Nietzsche] |