Recursion and the Axiom of Infinity

This paper examines the recursive definition of an increasing sequence of nested sets by means of a testing set whose countably many successive redefinitions as the successive sets of the sequence, leads to some contradictory results involving the Axiom of Infinity. 1. Recursion and successiveness As is well known, an increasing ω-ordered sequence of nested sets is one in which each set has both an immediate successor, of which it is a proper subset, and an immediate predecessor (except the first one). A common way of defining this type of sequences is by recursion. For instance, if N is the set of natural numbers, and A = {a1, a2, a3, . . . } and B = {b1, b2, b3, . . . } are two ω-ordered sets, the following recursive definition: A1 = {a1} Basic clause (1) Ai+1 = Ai ∪ {ai+1, bni}; i = 1, 2, 3, . . . Recursive clause (2) where bni is any element of B − Ai, defines an ω-ordered increasing sequence 〈Ai〉i∈N of nested sets A1 ⊂ A2 ⊂ A3 ⊂ . . . Recursive definitions imply (mathematical) successiveness: the definition of each term (except the first one) must be preceded by the definition of its immediate predecessor. Evidently, recursive definition (1)-(2) defines infinitely many different sequences of nested sets, depending on the particular bni used to define each of the successive terms Ai of the sequence. Each of these sequences 〈Ai〉i∈N will be, therefore, characterized by a particular sequence of natural numbers, n1, n2, n3, . . . , being ni the index of bni in the definition of the i term Ai. Infinitely many of those sequences of indexes, and then of sequences 〈Ai〉i∈N, will be algorithmically irreductible, i.e. random sequences [9]. Evidently, this type of sequences can only be recursively defined, its successive terms must of necessity be successively defined, one after the other. For the sake of clarity, I will term them ω-recursive definitions, and from now on we will assume this is the case of (1)-(2). According to the Axiom of Infinity we assume the completion of all successive definitions of an ω-recursive definition in the same sense we assume the existence of the set N of natural numbers as a complete infinite totality. Consequently, the sequence 〈Ai〉i∈N resulting from (1)-(2) is also a complete infinite totality, as complete and infinite as the set N of natural numbers, which by the way is also recursively defined. Extreme platonic infinitism would surely claim that all sets of any sequence defined in accordance with (1)-(2) exist independently of its recursive definition. But even in this case we need the completion of all its countably many successive definitions in order to make them explicit. As we will see in the short discussion that follows there is an elementary way of testing the assumed completion of the above ω-recursive definition (and of many others) by means of a set which is successively defined as the successive terms of the sequence. In this way,