By Godfried T. Toussaint

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**Example text**

Note, however, that x, {x}, {{x}}, . . is always a sequence each of whose components is an element of the next following component. 3 The ‘arb’ Operator as the Basis for Proofs by Transfinite Induction The standard (Peano) principle of mathematical induction is equivalent to the statement that every nonempty set s of integers contains a smallest element n0 . For suppose that P (n) is a predicate, defined for integers, for which the implication ∀n ∀m | (m < n) → P (m) → P (n) has been established, that is, for which P (n) must be true for a given n if it is true for all smaller m.

Suppose, more specifically, that P (s) is a predicate, defined for sets, for which the implication ∀s ∀t | (t ∈ s) → P (t) → P (s) has been established. That is, we suppose that P (s) must be true for a given s if it is true for all members of s. Then P (s) must be true for all sets s. For if not, then P (s) must be false for some member s1 of s. Repeating this argument, we see that there must exist a member s2 of s1 for which P (s2 ) is false, then a member s3 of s2 for which P (s3 ) is false, and so forth.

For example, in the von Neumann representation, the next ordinal after an ordinal s is simply s ∪ {s}. To see that s = s ∪ {s} must be an ordinal, note first that each member of a member of s is either a member of a member of s, or directly a member of s; and hence in any case a member of s ; thus s has property (II). The proof that s also has property (I) is equally elementary and is left to the reader. Together these show that s is an ordinal. Other equally elementary results concerning ordinals, whose proof is also left to the reader are: a.