Foundations of a general theory of sets on Superphysics

The Infinite

Mon, 01 Jan 0001 00:00:00 +0000

“Grundlagen einer allgemeinen Mannigfaltigkeitslehre”

My theory of sets depends on extending the notion of actual integers beyond the previous limits.

Ordering and Well-ordered Sets

Mon, 01 Jan 0001 00:00:00 +0000

Another great gain that can be attributed to the new numbers consists for me in a new concept, unmentioned previously by anyone else, regarding the “ordinal-number” of elements of a well-ordered infinite set.

Rules concerning ordering

Mon, 01 Jan 0001 00:00:00 +0000

The concept of the well-ordered set is fundamental for the whole theory of sets.

It is always possible to bring any well-defined set into the form of a well-ordered set.

Extending the current notions of the infinite

Mon, 01 Jan 0001 00:00:00 +0000

The extended integer sequence can, if required, be easily completed into a continuous set of numbers by adding to every integer α all real numbers x that are greater than zero and less than one.

The justification for extending the infinite

Mon, 01 Jan 0001 00:00:00 +0000

When I spoke of traditions, this was not to understand them in the narrow sense of what was experienced, but rather to follow them back to the founders of modern philosophy and natural sciences. In order to assess the question that is at stake here, I cite only a few of the most important sources, such as:

Properties of infinite integers

Mon, 01 Jan 0001 00:00:00 +0000

If there are difficulties in conceiving of immeasurably large, self-contained, integers whose magnitudes are comparable to each other, and also to finite numbers, where their relationships to each other and to finite numbers are governed by fixed rules, then these difficulties are connected with a perception that while the new numbers may certainly have the properties of the traditional ones in many respects, in several other respects they have a very strange nature. This perception may be due to the fact that different properties can occur together in a single new number, but which never appear together in a traditional number; they only appear separately in the traditional numbers.

Diversity of views on the infinite

Mon, 01 Jan 0001 00:00:00 +0000

Although in § 5 I cited many passages from Leibniz’s works in which he speaks out against infinite numbers by saying, amongst other things:

“Il n’y a point de nombre infini ni de ligne ou autre quantité infini, si on les prend pour des touts veritables.” (There is no infinite number, line or any other infinite quantity, if we take them to be true entire entities).

Defining the real numbers

Mon, 01 Jan 0001 00:00:00 +0000

Great importance has been attached to so-called real, rational and irrational numbers in the theory of sets.

I want to state here the most important things regarding their definitions. I will not go into the establishment of the rational numbers, since rigorously arithmetical representations of them have been developed several times.

The reality of existence of transfinite numbers

Mon, 01 Jan 0001 00:00:00 +0000

We can speak of the reality or the existence of integers in two ways, the finite and the infinite, but strictly speaking, any concepts or ideas that contemplate their reality apply equally to both. On the one hand, we may regard the integers as real insofar as they occupy a very specific place in our understanding on the basis of definitions, and are well-delineated from all our other mental concepts, and are in certain relationships with them and thus affect our thoughts in a certain way; allow me to call this kind of reality of our numbers their intra-subjective or fundamental reality.[Footnote13]

The Continuum

Mon, 01 Jan 0001 00:00:00 +0000

The concept of the “continuum” has not only played an important role everywhere in the development of the sciences, but has also always caused the greatest differences of opinion and even angry disputes. Perhaps this is due to the fact that its conceptual basis appears quite different to the dissenters, and the reason for this is that the precise and complete definition of the concept has not been communicated to them. It is perhaps also the case, which seems most probable to me, that the concept of the continuum originated with the Greeks, but without the necessary clarity and completeness that would obviate the possibility that subsequent thinkers would perceive it in diverse ways. Thus we see that Leucippus, Democritus and Aristotle regard the continuum as a composite, which consists of endlessly divisible particles (ex partibus sine fine divisibilibus), whereas Epicurus and Lucretius consider that it is a composite of atoms that are finite things. From this a great dispute later arose among philosophers, some of whom followed Aristotle, while others followed Epicurus, while others, in an attempt to steer clear of the dispute, sided with Thomas Aquinas[Footnote17] in the claim that the continuum consists neither of an infinite number nor of a finite number of parts, but of no parts at all. This opinion seems to me more of a tacit admission that one has not got to the root of the matter, preferring to avoid it in a polite manner, rather than being any explanation of the facts. Here we observe the medieval-scholastic origin of a view that we still find to be held today, according to which the continuum is an indivisible concept or, as others put it, a notion that is purely a priori, and hardly amenable as a basis for any definition. Every attempt at an arithmetical determination of this mystery is regarded as an inadmissible interference and rejected with all due intensity, and those of a diffident disposition get the impression that the “continuum” is not a mathematical-logical notion but rather a religious dogma.

Ordinal-numbers

Mon, 01 Jan 0001 00:00:00 +0000

It will now be shown how we progress to the definition of the new numbers, and how the natural segmentation into what I call number-classes results from the absolutely infinite sequence of actual integers. To this discussion I will only add the main propositions about the second number-class and its relation to the first.

Cardinal-numbers

Mon, 01 Jan 0001 00:00:00 +0000

The first thing we now have to show is the theorem that the new number-class (II) has a cardinal-number that is different from that of the first number-class (I).

This proposition results from the following proposition:

Succession of cardinal-numbers

Mon, 01 Jan 0001 00:00:00 +0000

I now come to the promised proof that the cardinal-numbers of (I) and (II) follow upon one another directly, so that there are no other cardinal-numbers in between. If one chooses a set (α′ ) of different numbers α′ from the number-class (II) according to some rule, i. e., any set (α′ ) contained within (II), then such a set always has peculiarities, which can be expressed in the following sentences:

Operations on transfinite numbers

Mon, 01 Jan 0001 00:00:00 +0000

In conclusion, I will now consider the numbers of the second number-class (II) and the operations that can be carried out with them, but on this occasion I will limit myself to what is most obvious, and reserve the publication of detailed investigations on them for later.