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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 5 ideas with the same theme [problem arising when we think of the greatest ordinal]:

15895 Russell discovered the paradox suggested by Burali-Forti's work [Russell, by Lavine]
21553 It seems that the ordinal number of all the ordinals must be bigger than itself [Lackey]
13482 The Burali-Forti paradox is a crisis for Cantor's ordinals [Hart,WD]
13366 The least ordinal greater than the set of all ordinals is both one of them and not one of them [Priest,G]
8674 The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal [Friend]