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Single Idea 8674

[from 'Introducing the Philosophy of Mathematics' by Michèle Friend, in 5. Theory of Logic / L. Paradox / 5. Paradoxes in Set Theory / c. Burali-Forti's paradox ]

Full Idea

The Burali-Forti paradox says that if ordinals are defined by 'gathering' all their predecessors with the empty set, then is the set of all ordinals an ordinal? It is created the same way, so it should be a further member of this 'complete' set!

Clarification

Thus 'fourth' would be {empty, first, second, third}

Gist of Idea

The Burali-Forti paradox asks whether the set of all ordinals is itself an ordinal

Source

Michèle Friend (Introducing the Philosophy of Mathematics [2007], 2.3)

Book Reference

Friend,Michèle: 'Introducing the Philosophy of Mathematics' [Acumen 2007], p.26


A Reaction

This is an example (along with Russell's more famous paradox) of the problems that began to appear in set theory in the early twentieth century. See Idea 8675 for a modern solution.

Related Idea

Idea 8675 Paradoxes can be solved by talking more loosely of 'classes' instead of 'sets' [Friend]