88 ideas
| 10073 | There cannot be a set theory which is complete [Smith,P] |
| 10676 | The Axiom of Choice is a non-logical principle of set-theory [Hossack] |
| 10686 | The Axiom of Choice guarantees a one-one correspondence from sets to ordinals [Hossack] |
| 23623 | Predicativism says only predicated sets exist [Hossack] |
| 23624 | The iterative conception has to appropriate Replacement, to justify the ordinals [Hossack] |
| 23625 | Limitation of Size justifies Replacement, but then has to appropriate Power Set [Hossack] |
| 10687 | Maybe we reduce sets to ordinals, rather than the other way round [Hossack] |
| 10677 | Extensional mereology needs two definitions and two axioms [Hossack] |
| 10616 | Second-order arithmetic can prove new sentences of first-order [Smith,P] |
| 23628 | The connective 'and' can have an order-sensitive meaning, as 'and then' [Hossack] |
| 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] |
| 23627 | 'Before' and 'after' are not two relations, but one relation with two orders [Hossack] |
| 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] |
| 10671 | Plural definite descriptions pick out the largest class of things that fit the description [Hossack] |
| 10666 | Plural reference will refer to complex facts without postulating complex things [Hossack] |
| 10669 | Plural reference is just an abbreviation when properties are distributive, but not otherwise [Hossack] |
| 10675 | A plural comprehension principle says there are some things one of which meets some condition [Hossack] |
| 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] |
| 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] |
| 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] |
| 10673 | Plural language can discuss without inconsistency things that are not members of themselves [Hossack] |
| 10599 | For primes we write (x not= 1 ∧ ∀u∀v(u x v = x → (u = 1 ∨ v = 1))) [Smith,P] |
| 10680 | The theory of the transfinite needs the ordinal numbers [Hossack] |
| 10610 | The reals contain the naturals, but the theory of reals doesn't contain the theory of naturals [Smith,P] |
| 10684 | I take the real numbers to be just lengths [Hossack] |
| 10619 | The truths of arithmetic are just true equations and their universally quantified versions [Smith,P] |
| 23626 | Transfinite ordinals are needed in proof theory, and for recursive functions and computability [Hossack] |
| 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] |
| 10674 | A plural language gives a single comprehensive induction axiom for arithmetic [Hossack] |
| 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] |
| 10681 | In arithmetic singularists need sets as the instantiator of numeric properties [Hossack] |
| 10685 | Set theory is the science of infinity [Hossack] |
| 23621 | Numbers are properties, not sets (because numbers are magnitudes) [Hossack] |
| 23622 | We can only mentally construct potential infinities, but maths needs actual infinities [Hossack] |
| 10668 | We are committed to a 'group' of children, if they are sitting in a circle [Hossack] |
| 10617 | The 'ancestral' of a relation is a new relation which creates a long chain of the original relation [Smith,P] |
| 23708 | Humeans see properties as having no more essential features and relations than their distinctness [Friend/Kimpton-Nye, by PG] |
| 23709 | Dispositions are what individuate properties, and they constitute their essence [Friend/Kimpton-Nye] |
| 23707 | Powers are properties which necessitate dispositions [Friend/Kimpton-Nye] |
| 23714 | Dispositional essentialism (unlike the grounding view) says only fundamental properties are powers [Friend/Kimpton-Nye] |
| 23711 | A power is a property which consists entirely of dispositions [Friend/Kimpton-Nye] |
| 23712 | Powers are qualitative properties which fully ground dispositions [Friend/Kimpton-Nye] |
| 23698 | Dispositions have directed behaviour which occurs if triggered [Friend/Kimpton-Nye] |
| 23699 | 'Masked' dispositions fail to react because something intervenes [Friend/Kimpton-Nye] |
| 23700 | A disposition is 'altered' when the stimulus reverses the disposition [Friend/Kimpton-Nye] |
| 23701 | A disposition is 'mimicked' if a different cause produces that effect from that stimulus [Friend/Kimpton-Nye] |
| 23702 | A 'trick' can look like a stimulus for a disposition which will happen without it [Friend/Kimpton-Nye] |
| 23703 | Some dispositions manifest themselves without a stimulus [Friend/Kimpton-Nye] |
| 23704 | We could analyse dispositions as 'possibilities', with no mention of a stimulus [Friend/Kimpton-Nye] |
| 10664 | Complex particulars are either masses, or composites, or sets [Hossack] |
| 10678 | The relation of composition is indispensable to the part-whole relation for individuals [Hossack] |
| 10665 | Leibniz's Law argues against atomism - water is wet, unlike water molecules [Hossack] |
| 10682 | The fusion of five rectangles can decompose into more than five parts that are rectangles [Hossack] |
| 23710 | Dispositionalism says modality is in the powers of this world, not outsourced to possible worlds [Friend/Kimpton-Nye] |
| 10663 | A thought can refer to many things, but only predicate a universal and affirm a state of affairs [Hossack] |
| 23706 | Hume's Dictum says no connections are necessary - so mass and spacetime warping could separate [Friend/Kimpton-Nye] |
| 10683 | We could ignore space, and just talk of the shape of matter [Hossack] |