76 ideas
| 24146 | All the major problems were formulated before Socrates [Nietzsche] |
| 24142 | What matters is how humans can be developed [Nietzsche] |
| 24143 | Thinkers might agree some provisional truths, as methodological assumptions [Nietzsche] |
| 24125 | Aristotle enjoyed the sham generalities of a system, as the peak of happiness! [Nietzsche] |
| 24147 | Thoughts are uncertain, and are just occasions for interpretation [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] |
| 24137 | Mathematics is just accurate inferences from definitions, and doesn't involve objects [Nietzsche] |
| 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] |
| 24131 | There is no 'being'; it is just the opposition to nothingness [Nietzsche] |
| 24151 | I only want thinking that is anchored in body, senses and earth [Nietzsche] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 24150 | We can only understand through concepts, which subsume particulars in generalities [Nietzsche] |
| 24138 | Strongly believed a priori is not certain; it may just be a feature of our existence [Nietzsche] |
| 24130 | An affirmative belief is present in every basic sense impression [Nietzsche] |
| 24124 | We now have innumerable perspectives to draw on [Nietzsche] |
| 24145 | Mind is a mechanism of abstraction and simplification, aimed at control [Nietzsche] |
| 24144 | A cognitive mechanism wanting to know itself is absurd! [Nietzsche] |
| 24139 | A 'person' is just one possible abstraction from a bundle of qualities [Nietzsche] |
| 24133 | I have perfected fatalism, as recurrence and denial of the will [Nietzsche] |
| 24152 | Fate is inspiring, if you understand you are part of it [Nietzsche] |
| 24129 | We start with images, then words, and then concepts, to which emotions attach [Nietzsche] |
| 24127 | Judging actions by intentions - like judging painters by their thoughts! [Nietzsche] |
| 24149 | Values need a perspective, of preserving some aspect of life [Nietzsche] |
| 24148 | If you love something, it is connected with everything, so all must be affirmed as good [Nietzsche] |
| 24135 | Egoism should not assume that all egos are equal [Nietzsche] |
| 24132 | After Socrates virtue is misunderstood, as good for all, not for individuals [Nietzsche] |
| 24126 | We contain multitudes of characters, which can brought into the open [Nietzsche] |
| 24136 | Who can endure the thought of eternal recurrence? [Nietzsche] |
| 24154 | If you want one experience repeated, you must want all of them [Nietzsche] |
| 24153 | Humans are determined by community, so its preservation is their most valued drive [Nietzsche] |
| 24134 | There is always slavery, whether we like it or not [Nietzsche] |
| 1658 | In early Greece the word for punishment was also the word for vengeance [Vlastos] |
| 24128 | After history following God, or a people, or an idea, we now see it in terms of animals [Nietzsche] |
| 24140 | Cause and effect is a hypothesis, based on our supposed willing of actions [Nietzsche] |
| 24141 | Having a sense of time presupposes absolute time [Nietzsche] |