“Mathematics is the Logic of the Infinite”: Zermelo’s Project of Infinitary Logic

Abstract

In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1

Inconsistency of ℕ and the question of infinity

“…There is a one-to-one correspondence between the numbers-list and a set of the sets-list. So all numbers of the numbers-list are included in a set of the sets-list, that consequently is equal to N. Then on the sets-list there is N. To take all numbers together is in agreement with the axiom of infinity (that is, there is a set with all natural numbers, N) [3] [4] [7]. It should be noted that this one-to-one correspondence is equivalent to that one between n and n + 1 ≡ {0, 1, 2, ...n}, for number definition [5] [6].…”

Inconsistency of ℕ with the set union operation

“…We consider the axiom of infinity [4] [5] [8] , then the existence of N and Peano axioms [3]. Sets are considered with the usual graphical-symbolic notation {0, 1, 2, ...n} (see also [6] [7]).…”