User:RJGray/Cantor1887E

4. Communication on the Theory of the Tranfinite.

In the preceding paper, prompted by certain older and more recent works written against the possibility of infinite numbers, I made the attempt to tie the most questions on the actual infinite according to key differences from the most general point of view, in this way to get an overview of the main positions that can be taken in relation to this topic. The A.-U. was differentiated according to three relationships: first, it is realized in the highest perfection, in the completely independent, extra-worldly being, in God, where I call it Absolute-infinite or briefly Absolute; secondly, provided that it is represented in the dependent, animal world;  thirdly, insofar as it can be understood as a mathematical quantity, number or order type from thinking in the abstract. In the last two relationships, where it appears capable of limited, still further increase and in this respect is related to the finite A.-U., I call it Transfinite and set it strictly in contrast with the Absolute.

In each of the three relationships, the possibility of the actual infinity can be affirmed or denied; from this follows a total of eight different points of view, all of which are represented in philosophy and from which I take the one which is absolutely affirmative, with regard to all three considerations.

If it is particularly speculative Theology whether to investigate the Absolute infinite and determine what can be said about it on the human side, then on the other hand, the questions on the Transfinite mainly fall into the areas of Metaphysics and Mathematics; they are the ones I have been occupied with for years.

Since I was lucky enough to be able to correspond with several scholars who have dedicated a friendly interest in my work, this has given me the opportunity to explain and clarify what has been published in a generally understandable manner, so in this material, which is the result of a lively exchange of ideas, I have suitable starting points for further explanations that are of interest to a larger audience. I would therefore like to first publish several of these letters I have written without making any major changes to them. However, where it seems necessary to me, I will give explanations in the notes below the text.

I would like to begin by stating the following as an introduction to Letters I, III, IV and VIII.

On I and VIII. Here you will find the way of interpreting the whole numbers and order types as universals, which I have done for about four years and I have taught many times in my lectures, and which refer to sets and result from them when the nature of the elements is abstracted. Every set of well-differentiated things can be viewed as a single thing in itself, in which those things are components or constitutive elements. If one abstracts from the nature of the elements, as well as from the order of their existence, one obtains the cardinal number or power of the set, a general concept in which the elements, as so-called ones, have grown together organically to a certain extent in such a way that none of them has a preferred ranking. It follows from this, when we consider carefully, that two different sets have the same cardinal number if and only if they are what I call equivalent, and there is no contradiction if, as this often occurs with  infinite  sets, two sets, one of which is a  part  or constituent  of the other, have  completely identical  cardinal numbers. In failing to appreciate this fact I see the main obstacle to the introduction of infinite numbers, which has been brought up from time immemorial.