81 ideas
| 23000 | Vicious regresses force you to another level; non-vicious imply another level [Baron/Miller] |
| 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] |
| 23024 | A traveller takes a copy of a picture into the past, gives it the artist, who then creates the original! [Baron/Miller] |
| 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] |
| 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] |
| 23008 | Grounding is intended as a relation that fits dependences between things [Baron/Miller] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 23018 | How does a changing object retain identity or have incompatible properties over time? [Baron/Miller] |
| 5996 | Critolaus redefined Aristotle's moral aim as fulfilment instead of happiness [Critolaus, by White,SA] |
| 23011 | Modern accounts of causation involve either processes or counterfactuals [Baron/Miller] |
| 23013 | The main process theory of causation says it is transference of mass, energy, momentum or charge [Baron/Miller] |
| 23014 | If causes are processes, what is causation by omission? (Distinguish legal from scientific causes?) [Baron/Miller] |
| 23015 | The counterfactual theory of causation handles the problem no matter what causes actually are [Baron/Miller] |
| 23016 | Counterfactual theories struggle with pre-emption by a causal back-up system [Baron/Miller] |
| 23009 | There is no second 'law' of thermodynamics; it just reflects probabilities of certain microstates [Baron/Miller] |
| 23002 | In relativity space and time depend on one's motion, but spacetime gives an invariant metric [Baron/Miller] |
| 22988 | The block universe theory says entities of all times exist, and time is the B-series [Baron/Miller] |
| 22991 | How can we know this is the present moment, if other times are real? [Baron/Miller] |
| 22992 | If we are actually in the past then we shouldn't experience time passing [Baron/Miller] |
| 22994 | Erzatz Presentism allows the existence of other times, with only the present 'actualised' [Baron/Miller] |
| 22998 | How do presentists explain relations between things existing at different times? [Baron/Miller] |
| 23017 | Presentism needs endurantism, because other theories imply most of the object doesn't exist [Baron/Miller] |
| 23023 | How can presentists move to the next future moment, if that doesn't exist? [Baron/Miller] |
| 22995 | Most of the sciences depend on the concept of time [Baron/Miller] |
| 22993 | For abstractionists past times might still exist, althought their objects don't [Baron/Miller] |
| 23001 | The error theory of time's passage says it is either a misdescription or a false inference [Baron/Miller] |
| 22999 | It is meaningless to measure the rate of time using time itself, and without a rate there is no flow [Baron/Miller] |
| 22986 | The C-series rejects A and B, and just sees times as order by betweenness, without direction [Baron/Miller] |
| 22996 | The A-series has to treat being past, present or future as properties [Baron/Miller] |
| 23007 | The B-series can have a direction, as long as it does not arise from temporal flow [Baron/Miller] |
| 23003 | Static theories cannot account for time's obvious asymmetry, so time must be dynamic [Baron/Miller] |
| 23004 | The direction of time is either primitive, or reducible to something else [Baron/Miller] |
| 23005 | The kaon does not seem to be time-reversal invariant, unlike the rest of nature [Baron/Miller] |
| 23006 | Maybe the past is just the direction of decreasing entropy [Baron/Miller] |
| 23010 | We could explain time's direction by causation: past is the direction of causes, future of effects [Baron/Miller] |
| 22989 | Static time theory presents change as one property at t1, and a different property at t2 [Baron/Miller] |
| 23020 | If a time traveller kills his youthful grandfather, he both exists and fails to exist [Baron/Miller] |
| 23022 | Presentism means there no existing past for a time traveller to visit [Baron/Miller] |
| 22987 | The past (unlike the future) is fixed, along with truths about it, by the existence of past objects [Baron/Miller] |
| 22990 | The moving spotlight says entities can have properties of being present, past or future [Baron/Miller] |
| 22997 | The present moment is a matter of existence, not of acquiring a property [Baron/Miller] |