21/05/13 11:52:55.24 uhdqO0QU.net
>>889
つづき
In other words, a relation is well founded if
(∀S⊆ X)[S≠ Φ ⇒ (∃m∈ S)(∀s∈ S)￢(sRm)].
Other properties
If (X, <) is a well-founded relation and x is an element of X, then the descending chains starting at x are all finite, but this does not mean that their lengths are necessarily bounded. Consider the following example: Let X be the union of the positive integers and a new element ω, which is bigger than any integer.
Then X is a well-founded set, but there are descending chains starting at ω of arbitrary great (finite) length; the chain ω, n ? 1, n ? 2, ..., 2, 1 has length n for any n.
The Mostowski collapse lemma implies that set membership is a universal among the extensional well-founded relations: for any set-like well-founded relation R on a class X which is extensional, there exists a class C such that (X, R) is isomorphic to (C, ∈).
つづく