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Single Idea 13366

[filed under theme 5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox ]

Full Idea

Burali-Forti: φ(x) is 'x is an ordinal', and so w is the set of all ordinals, On; δ(x) is the least ordinal greater than every member of x (abbreviation: log(x)). The contradiction is that log(On)∈On and log(On)∉On.

Gist of Idea

The least ordinal greater than the set of all ordinals is both one of them and not one of them

Source

Graham Priest (The Structure of Paradoxes of Self-Reference [1994], §2)

Book Ref

-: 'Mind' [-], p.27

The 37 ideas from Graham Priest

9123 Someone standing in a doorway seems to be both in and not-in the room [Priest,G, by Sorensen]
9672 Free logic is one of the few first-order non-classical logics [Priest,G]
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]
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]
9687 A 'member' of a set is one of the objects in the set [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]
9688 A 'singleton' is a set with only one member [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]
9680 The empty set Φ is a subset of every set (including 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]
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]
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]
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]
13373 Typically, paradoxes are dealt with by dividing them into two groups, but the division is wrong [Priest,G]
8720 A logic is 'relevant' if premise and conclusion are connected, and 'paraconsistent' allows contradictions [Priest,G, by Friend]