One can, it seems to me, give a demonstration of the impossibility of an actual infinite in quantity by a series of clear and incontestable principles.
(1) Every multitude, every collection composed of an infinity of terms contains as many ones as it has terms. For it is evident that in a collection each term is counted by a one.
(2) The natural series of numbers stands in place of every collection that has ones; because it is evident that in this assemblage one can designate one, then two, then three, and so in series, until one has gone through all the terms or ones.
(3) As to suppose a collection infinite in terms is nothing other than to suppose an infinite multitude of ones, it is evident that the natural series will be applicable to this collection, or at least that this infinite collection of ones will not possess infinity in another way than the natural series of numbers.
(4) In every progression of the natural series continued to infinity, the succeeding number never rises more than one unit above the preceding number; so there can be no leap from one number to the other, and one cannot reach one from the other, save by continual addition of one and one.
(5) In the natural progression the series of numbers increasing from 1 by the continual addition of unity to unity, and this without end, it follows that there is no assignable term in this series which is not preceded and followed by other terms, from which it differs only by one.
(6) Therefore, since this progression must always continue to infinity, it is impossible for the sequence of terms to arrive at a point where, after any finite state, it does not follow another finite term, and which is not only superior to it by a unit. The passage from the finite to the infinite is therefore not only obscure and incomprehensible, which alone would not be sufficient reason to reject it, but absolutely impossible. This must be demonstrated with the utmost accuracy.
I say, then, that in the natural series of numbers the passage from finite to infinite is impossible. If this passage is possible, there would then be a finite number after which follows the infinite number. This is borne out by the idea of the passage from one to the other, for the natural series beginning with one, and rising by finite numbers, if it reaches infinity, necessarily there is a finite number one at which one passes to infinity. Now, in the natural series, it is impossible for an infinite number to succeed any finite number; for this infinite number must exceed the finite to which it succeeds by one alone, or an indeterminate number of units. If it exceeds by one only, then it is finite, since it has a finite relation to a finite number. If it exceeds it by an indeterminate number of ones, then it is not what immediately follows in the natural progression, contrary to what was supposed; and the natural series always rising by the continual addition of this indeterminate number of ones, in the resulting terms it will always be equally impossible to find the term that ceases to be finite.
Tuesday, April 11, 2017
Gerdil on the Possibility of an Actual Infinite Multitude
An interesting passage from Gerdil's Recueil de dissertations sur quelques principes de philosophie & de religion (1760); it is one part of an argument against the eternity of matter. The translation, which is mine, is just a rough first draft.