72 ideas
9123 | Someone standing in a doorway seems to be both in and not-in the room [Priest,G, by Sorensen] |
8720 | A logic is 'relevant' if premise and conclusion are connected, and 'paraconsistent' allows contradictions [Priest,G, by Friend] |
9672 | Free logic is one of the few first-order non-classical logics [Priest,G] |
15945 | Second-order set theory just adds a version of Replacement that quantifies over functions [Lavine] |
9697 | X1 x X2 x X3... x Xn indicates the 'cartesian product' of those sets [Priest,G] |
9685 | <a,b&62; is a set whose members occur in the order shown [Priest,G] |
9675 | a ∈ X says a is an object in set X; a ∉ X says a is not in X [Priest,G] |
9674 | {x; A(x)} is a set of objects satisfying the condition A(x) [Priest,G] |
9673 | {a1, a2, ...an} indicates that a set comprising just those objects [Priest,G] |
9677 | Φ indicates the empty set, which has no members [Priest,G] |
9676 | {a} is the 'singleton' set of a (not the object a itself) [Priest,G] |
9679 | X⊂Y means set X is a 'proper subset' of set Y [Priest,G] |
9678 | X⊆Y means set X is a 'subset' of set Y [Priest,G] |
9681 | X = Y means the set X equals the set Y [Priest,G] |
9683 | X ∩ Y indicates the 'intersection' of sets X and Y, the objects which are in both sets [Priest,G] |
9682 | X∪Y indicates the 'union' of all the things in sets X and Y [Priest,G] |
9684 | Y - X is the 'relative complement' of X with respect to Y; the things in Y that are not in X [Priest,G] |
9694 | The 'relative complement' is things in the second set not in the first [Priest,G] |
9693 | The 'intersection' of two sets is a set of the things that are in both sets [Priest,G] |
9692 | The 'union' of two sets is a set containing all the things in either of the sets [Priest,G] |
9698 | The 'induction clause' says complex formulas retain the properties of their basic formulas [Priest,G] |
15914 | An 'upper bound' is the greatest member of a subset; there may be several of these, so there is a 'least' one [Lavine] |
9695 | An 'ordered pair' (or ordered n-tuple) is a set with its members in a particular order [Priest,G] |
9696 | A 'cartesian product' of sets is the set of all the n-tuples with one member in each of the sets [Priest,G] |
9686 | A 'set' is a collection of objects [Priest,G] |
9689 | The 'empty set' or 'null set' has no members [Priest,G] |
9690 | A set is a 'subset' of another set if all of its members are in that set [Priest,G] |
9691 | A 'proper subset' is smaller than the containing set [Priest,G] |
9688 | A 'singleton' is a set with only one member [Priest,G] |
9687 | A 'member' of a set is one of the objects in the set [Priest,G] |
9680 | The empty set Φ is a subset of every set (including itself) [Priest,G] |
15921 | Collections of things can't be too big, but collections by a rule seem unlimited in size [Lavine] |
15937 | Those who reject infinite collections also want to reject the Axiom of Choice [Lavine] |
15936 | The Power Set is just the collection of functions from one collection to another [Lavine] |
15899 | Replacement was immediately accepted, despite having very few implications [Lavine] |
15930 | Foundation says descending chains are of finite length, blocking circularity, or ungrounded sets [Lavine] |
15920 | Pure collections of things obey Choice, but collections defined by a rule may not [Lavine] |
15898 | The controversy was not about the Axiom of Choice, but about functions as arbitrary, or given by rules [Lavine] |
15919 | The 'logical' notion of class has some kind of definition or rule to characterise the class [Lavine] |
15900 | The iterative conception of set wasn't suggested until 1947 [Lavine] |
15931 | The iterative conception needs the Axiom of Infinity, to show how far we can iterate [Lavine] |
15932 | The iterative conception doesn't unify the axioms, and has had little impact on mathematical proofs [Lavine] |
15933 | Limitation of Size: if it's the same size as a set, it's a set; it uses Replacement [Lavine] |
15913 | A collection is 'well-ordered' if there is a least element, and all of its successors can be identified [Lavine] |
15926 | Second-order logic presupposes a set of relations already fixed by the first-order domain [Lavine] |
15934 | Mathematical proof by contradiction needs the law of excluded middle [Lavine] |
13373 | Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong [Priest,G] |
13368 | The 'least indefinable ordinal' is defined by that very phrase [Priest,G] |
13370 | 'x is a natural number definable in less than 19 words' leads to contradiction [Priest,G] |
13369 | By diagonalization we can define a real number that isn't in the definable set of reals [Priest,G] |
13366 | The least ordinal greater than the set of all ordinals is both one of them and not one of them [Priest,G] |
13367 | The next set up in the hierarchy of sets seems to be both a member and not a member of it [Priest,G] |
13371 | If you know that a sentence is not one of the known sentences, you know its truth [Priest,G] |
13372 | There are Liar Pairs, and Liar Chains, which fit the same pattern as the basic Liar [Priest,G] |
15907 | Mathematics is nowadays (thanks to set theory) regarded as the study of structure, not of quantity [Lavine] |
15942 | Every rational number, unlike every natural number, is divisible by some other number [Lavine] |
15922 | For the real numbers to form a set, we need the Continuum Hypothesis to be true [Lavine] |
18250 | Cauchy gave a necessary condition for the convergence of a sequence [Lavine] |
15904 | The two sides of the Cut are, roughly, the bounding commensurable ratios [Lavine] |
15912 | Counting results in well-ordering, and well-ordering makes counting possible [Lavine] |
15949 | The theory of infinity must rest on our inability to distinguish between very large sizes [Lavine] |
15947 | The infinite is extrapolation from the experience of indefinitely large size [Lavine] |
15940 | The intuitionist endorses only the potential infinite [Lavine] |
15909 | 'Aleph-0' is cardinality of the naturals, 'aleph-1' the next cardinal, 'aleph-ω' the ω-th cardinal [Lavine] |
15915 | Ordinals are basic to Cantor's transfinite, to count the sets [Lavine] |
15917 | Paradox: the class of all ordinals is well-ordered, so must have an ordinal as type - giving a bigger ordinal [Lavine] |
15918 | Paradox: there is no largest cardinal, but the class of everything seems to be the largest [Lavine] |
15929 | Set theory will found all of mathematics - except for the notion of proof [Lavine] |
15935 | Modern mathematics works up to isomorphism, and doesn't care what things 'really are' [Lavine] |
15928 | Intuitionism rejects set-theory to found mathematics [Lavine] |
594 | Speusippus suggested underlying principles for every substance, and ended with a huge list [Speussipus, by Aristotle] |
2632 | Speusippus said things were governed by some animal force rather than the gods [Speussipus, by Cicero] |