# Aleph and omega - were these deliberately chosen to represent infinities because of the connection between God (the alpha and the omega) and infinity?

Aleph is used in the names of various cardinal infinities, and omega is used for the first ordinal infinity.

God has often been identified with infinity, e.g. by Cantor himself. And God is often referred to as "the alpha and the omega". Are these connections deliberate?

• No.${}{}{}{}{}$ Jan 19 '14 at 21:00
• You should add the Sunday school homework tag, if it happens to be homework. Jan 19 '14 at 21:02
• Aleph, by the way, comes from the Hebrew alphabet. Cantor did use the letter Tav (which is the last letter of the Hebrew alphabet) to denote the class of cardinals. The notation didn't stick, though. Jan 19 '14 at 21:04
• I am not sure there is no connection. Cantor's views on religion are well documented (Dauben addresses Cantor's belief that his theory of transfinite numbers was communicated to him by god, for instance.) And he did represent the infinities by $\omega$, to contrast them with $\Omega$, the absolute infinity (God). It would not surprise me that the choice of Hebrew characters was intentional as well. I have voted to reopen. Jan 19 '14 at 23:09
• @Andres: The choice of Hebrew letters was intentional, but Cantor claimed it had nothing to do with his Jewish heritage, but rather that the Latin and Greek alphabet were thoroughly used. Jan 19 '14 at 23:55

I'm posting an answer based on Asaf's comments. The following reference addresses this question to some extent:

MR0525577 (80g:01021). Dauben, Joseph Warren. Georg Cantor. His mathematics and philosophy of the infinite. Harvard University Press, Cambridge, Mass.-London, 1979. xii+404 pp. ISBN: 0-674-34871-0. Reprinted: Princeton University Press, Princeton, NJ, 1990. xiv+404 pp. ISBN: 0-691-02447-2.

From page 179 (of the reprint version):

When Cantor finally decided that the transfinite cardinals required a separate notation of their own, he felt that all the usual alphabets, the familiar Greek or Roman letters, were too widely used for other purposes both specific and general. His new numbers deserved something unique. He was always careful about the selection of notation, and it was to be expected that he would make the best choice possible in picking a new symbol for one of the most important concepts of his set theory. Not wishing to invent a new symbol himself, he chose the aleph, the first letter of the Hebrew alphabet. The choice was especially clever, as he was happy to admit, since the Hebrew aleph served simultaneously to represent the number one, and the transfinite numbers, as cardinal numbers, were themselves infinite unities. ${}^{32}$ In addition, it might be said that as the first letter of the Hebrew alphabet, the aleph could be taken to represent new beginnings, and he certainly believed that his theory of transfinite numbers represented a new beginning for mathematics. After all, his set theory had made it possible to deal successfully for the first time with the nature of mathematical infinity.

1. Cantor did so in a letter to Felix Klein: April 30, 1895: (II) 142-143; also #499 in the collection held by the archives of the Universitätsbibliothek, Göttingen.

Here, (II) denotes Cantor's letter-book for 1890 through 1895.

According to this, his notation $\aleph$ was chosen for practical reasons rather than inspired by his heritage or religious reasons.

In page 99, Dauben explains:

The significance of Cantor's new achievement was underscored by his choice of terminology and notation. At last his transfinites were numbers, as “real” mathematically as the real numbers $R$. The new symbol $\omega$ was introduced instead of the familiar $\infty$, Cantor said, in order to emphasize the fact that the transfinite ordinal numbers were completed, actual infinities. ${}^{14}$ The potential infinite, traditionally expressed by the symbol $\infty$, was wholly unsuited to his new purposes. The shift in notation from $\infty$ to $\omega$ thus reflected a significant transformation in his thinking, specifically the advance in status of the transfinites from symbols to numbers.

Note 14 elaborates:

1. Cantor first introduced his $\omega$ in Cantor (1883c). Its appearance can be dated even more accurately in terms of the proof sheets for the Punktmannigfaltigkeitslehre papers, which Cantor sent to Mittag-Leffler for use in preparing the translations for Acta Mathematica. In every case, $\infty$'s were changed to $\omega$'s. The change must have occurred sometime between September and October of 1882, when the first section of the Grundlagen was sent to Teubner for press. The proof sheets Cantor sent to Mittag-Leffler are kept in the upstairs library, in the box marked Cantor, of the Institut Mittag-Leffler, Djursholm, Sweden.

Here, (1883c) is Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen. Leipzig: B. G. Teubner.

The above does not really elaborate on the reasons choosing $\omega$, although it hints at the visual similarity between $\omega$ and $\infty$. In any case, it does not mention any religious connections.

On the other hand, Cantor uses $\Omega$ (rather than $\omega$) for the “Absolute infinite”, and here religious symbolism is definitely present. He writes:

The actual infinite was distinguished by three relations: first, as it is realized in the supreme perfection, in the completely independent, extrawordly existence, in Deo, where I call it absolute infinite or simply absolute; second to the extent that it is represented in the dependent, creatural world; third as it can be conceived in abstracto in thought as a mathematical magnitude, number or ordertype. In the latter two relations, where it obviously reveals itself as limited and capable for further proliferation and hence familiar to the finite, I call it Transfinitum and strongly contrast it with the absolute.

The above is from Cantor's Mitteilungen zur Lehre vom Transfiniten, Zeitschrift für Philosophie und philosophische Kritik 91 (1887), 81–125; 92 (1888), 240–265.

And in a letter to Dedekind from 1899, he adds:

The system $\Omega$ of all numbers is an inconsistent, absolutely infinite multiplicity.

Here, by "number" he means "ordinal number", and the inconsistency he refers to is the fact that $\Omega$ itself, unlike its proper initial segments, cannot be an ordinal (or we would find ordinals $\delta$ greater than themselves).

Dauben discusses Cantor's religious views in some detail (see in particular the last chapter). Another useful reference in this regard is

MR1347523 (96i:03006). Jané, Ignacio. The role of the absolute infinite in Cantor's conception of set. Erkenntnis 42 (1995), no. 3, 375–402.