# Paul Mahlo's Original Large Cardinal Paper

It is well known that Paul Mahlo (1883-1971) developed a systematic hierarchy of inaccessible cardinals of the type $\pi_{a,b}$ where $\pi_{1,b}$ enumerates the strongly innacessible cardinals, $\pi_{2,b}$ enumerate the fixed points of $\pi_{1,b}$ and so on. My question is, where can I find the original paper of Mahlo in English? If this doesn't exist, are there any good expositional articles on this accessible online?

• have you looked at the references listed on Wikipedia ? The first two are in English. – Carlo Beenakker Jul 17 '18 at 11:37
• Those are just set theory books. I'm looking for the original article. – Elie Ben-Shlomo Jul 17 '18 at 12:18
• it's in German, there is no official English translation, but you could make your own via Google translate – Carlo Beenakker Jul 17 '18 at 12:25
• Why is it so important to read "the original"? While there is some historical benefit in reading the original works, it rarely carries any mathematical or pedagogical benefit. More often than not, the work has been cleaned up, processed and reprocessed repeated, and is presented in modern texts in a way that fits modern thinking on the topic, and using modern terminology (which also has the benefit that it lets you converse with other people on the topic without resorting to stating the definitions every time). – Asaf Karagila Jul 17 '18 at 12:42
• Because in the reprocessing and recasting of results, some ideas and observations are left out and/or deemphasized. If you want to extract and extend the original results using your own perspective, don't read the originals. If you want to expand your perspective and your knowledge of how the results apply and interconnect, attempting to understand someone else's perspective is quite useful, especially the perspective of the pioneer, who may have influences and motivations they tried to communicate in their original paper. Gerhard "Borrow Intuition; Don't Reinvent It" Paseman, 2018.07.17. – Gerhard Paseman Jul 17 '18 at 17:52

Just to show what to expect from a machine translation, I spent a little time on two paragraphs of the 1924 Mahlo paper, which I OCR'd and passed to Google Translate. With all the formulas it requires considerable post-processing, but if one is motivated this should be doable in an afternoon.

## On a property of a sub-type of the continuum

by Paul Mahlo in Recklinghausen

If one considers a set of subsets of the continuum $C$ of this type, which is so simply ordered that each subset contains all of its preceding real subsets, it can be easily proved that its order type is always a sub-type of the continuum; it therefore contains partial types of regular starting numbers or their inverses of at most $\omega$ and $^\ast\omega$. In the following we show that such arrangements of subsets of $C$ of the same type can also contain the partial types $\omega_1$ and $^\ast\omega_1$. In doing so we first connect to a final order of the $\omega$-sequences, belonging to continued fractions of positive irrational numbers $x> 1$.

[...]

After these auxiliary considerations, we now deal with a sub-type of the continuum $C$. We denote by $M$ a nowhere dense perfect part of $C$, but always presuppose this $M$ as bounded; all $M$'s therefore have the same type, and every $M$ has $c$ two-sided and $a$ one-sided limits. There exist nowhere dense perfect subsets $M'$ of $C$, that contain $M$ as nowhere dense part, where each one-sided limit of $M$ is a two-sided limit of $M'$. One can easily construct $a$ perfect sets $M_0$, $M_1$, $\ldots$, $M_n$, $\ldots$ ($n <\omega$), so that each $M_n$ contains only two-sided limits of $M_{n + 1}$. The type of elementary sets of all $M_n$ is called $\pi$; it is independent of the particular choice of $M_n$.

As there is no full answer to this question yet, let me add a partial one which might be helpful:

In a paper of G. H. Müller titled "Reflection in set theory, The Bernays-Levy Axiom System" which is published as part of the collection "Philosophy of Mathematics Today" (edited by Evandro Agazzi and György Darvas), the following passage from Mahlo's 1911 paper is quoted. (See page 158 of the book).

Mahlo, P., Über lineare transfinite Mengen., Leipz. Ber. 63, 187-225 (1911). ZBL42.0090.02.

The text says that the English translation is provided by "D. Reid" but no further reference is added. Also, it is not clear whether it is part of a full translation of the whole paper of Mahlo or just a limited translation for the occasional use in this article. Maybe you can get more information by contacting mentioned people directly.

Therefore we attribute existence to every transfinite number in the definition of which we can find no contradiction if either its equality or its inequality to each otherwise defined transfinite number can be established. This necessary condition is also sufficient, since thus the rank-order between any two arbitrarily chosen transfinite numbers results: otherwise there would have to exist, for a smallest number a, a smallest number $\beta$ for which the size relationship to $\alpha$ could not be established. Were a number $\gamma<\beta$ already greater that $\alpha$, then that would imply that also $\beta>\alpha$, thus establishing the rank-order. Thus $\alpha$ could only be greater than all number $\gamma<\beta$, giving that $\alpha\geq \beta$. If our existence conditions do not imply $\alpha=\beta$, then $\alpha>\beta$, contradicting the assumption that, for $\beta$ as a smallest number, its size relationship to a could not be established. In practice, however, the above condition could remain worthless for a long period, since for example due to our ignorance about certain transfinite numbers, a contradiction could remain hidden in the definition of a proposed new transfinite number. (Transl. by D. Reid.)

• does anyone have an electronic version of the German paper? I would be happy to try a machine translation... – Carlo Beenakker Jul 18 '18 at 11:32
• @CarloBeenakker Here, you can find the German version of the following paper of Mahlo: "Über eine Eigenschaft eines Teiltypus des Kontinuums". See the page 258. Though, I don't have the pdf version of "Über lineare transfinite Mengen". I will appreciate if someone who has, shares it with me via email. – Morteza Azad Jul 18 '18 at 12:31