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		if A is infinite and . By convention, min(x1, . . ., xk) = min({x1, . . ., xk}) and max(x1, . . . ,xk) = max({ x1, . . ., xk}). For each , A[n] denotes . A - B denotes the set . For , denotes the complement of A, that is, N - A. A  D  B denotes the symmetric difference of A and B, that is, . denotes the disjoint union of A and B, that is, . For each , A = n B means that ; A and B are called n-variants. A =* B means that card(A  D  B) is finite; A and B are called finite variants.

		For each finite set A, we define the canonical index of A to be the natural number . (The canonical index of is thus 0.) The canonical indexing provides a one-to-one correspondence between finite sets and the natural numbers. Dx denotes the finite set whose canonical index is x(see Rogers [l58])

		For natural numbers x and z, let:

.	.
.	.

		Let R denote the set of real numbers. For each a and , let:

.	.
.	.

		Logical Symbols.

		As usual,  ''  and  $  are used for universal and existential quantifiers. , , and  $ ! denote 'for all but finitely many,' 'there exist infinitely many,' and 'there exists a unique.' That is:

		is finite.

		is infinite.

		is a singleton.

		The symbols , , and , respectively, denote logical negation, or and and. Fix an , and, for each , let be a predicate. Then, denotes and denotes .

		Functions.

		Let A and B be arbitrary sets (that is, not necessarily subsets of N). Then (respectively, ) denotes the collection of all partial (respectively, total) functions from A to B. Unless otherwise specified, when we say  y  is a partial function we will mean that  y : and when we say f is a total function, we will mean