Recursion and Ω -order

This paper examines the recursive definition of an increasing sequence of nested sets by means of a control set whose countably many successive redefinitions leads to a contradictory result that compromises ω-order and the Axiom of Infinity. 1. Recursion and successiveness A recursive definition usually starts with a first definition (basic clause) which is followed by an infinite (usually ω-ordered) sequence of definitions such that each one of them defines an object in terms of the previously defined ones (inductive or recursive clause). For instance, if A = {a1, a2, a3, . . . } is an ω-ordered set, the following recursive definition: A1 = {a1} Basic clause (1) Ai+1 = Ai ∪ {ai+1}; i = 1, 2, 3, . . . Recursive clause (2) defines an ω-ordered increasing sequence 〈Ai〉i∈N of nested sets A1 ⊂ A2 ⊂ A3 ⊂ . . . Recursive definitions as (1)-(2) imply (mathematical) successiveness: the definition of each term (except the first one) must be preceded by the definition of its immediate predecessor. According to the actual 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 (Axiom of Infinity). 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. As we will see in the short discussion that follows there is an elementary way of testing the assumed completion of ω-recursive definitions by means of a set which is successively defined as the successive terms of the sequence. In this way, the control set forces the recursive clause to leave a permanent trace of its assumed actual completion.