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criterion for a set to be transitive

Major Section: 
Reference
Type of Math Object: 
Theorem
Parent: 

Mathematics Subject Classification

03E20 no label found

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I know no transitive sets. Please give a concrete example!

pahio writes:

> I know no transitive sets.
> Please give a concrete example!

The empty set!

More generally, any ordinal is a transitive set, if we use von Neumann's definition of ordinals (as we usually do): http://planetmath.org/encyclopedia/VonNeumannOrdinal.html

Some concrete examples are given in the entry with canonical name NaturalNumber. It gives the definition of 0, 1, 2, and 3 as ordinals, and indicates a pattern of how to form the rest.

You may also want to see the entry with canonical name VonNeumannOrdinal.

Tahanks Warren! The natural numbers are sufficiently simple for me =o)

Jussi

On the same topic, I have filed a request for an entry that gives an example of a transitive set that is not an ordinal. I do not know of any such sets.

I've fulfilled this request with an entry about the cumulative hierarchy: http://planetmath.org/encyclopedia/CumulativeHierarchy.html

However, your own entry (to which this thread is attached) also gives examples of such sets, since the power set of an ordinal > 1 is not an ordinal.

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