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Georg Cantor, when developing the basics of set theory, noted that there are two ways to increase cardinality: power sets and successors (or, in modern terms, the Hartogs operation).1

Eventually the notation for the successors became the $\aleph$ numbers. The power set operation eventually became what we now know as the $\beth$ numbers:

  • $\beth_0(\kappa)=\kappa$,
  • $\beth_{\alpha+1}(\kappa)=2^{\beth_\alpha(\kappa)}$, and
  • $\beth_\alpha(\kappa)=\sup\{\beth_\xi\mid\xi<\alpha\}$ for limit steps.

If $\kappa=\aleph_0$, we simply omit it from the notation and we get the $\beth$ numbers.

What I am trying to find is the origin of the notation for $\beth$ numbers. Kanamori's article in the Handbook only mentions this in relation to the Erdős–Rado paper from 1956, where the notation does not appear.

Jech, which normally has a reasonably thorough historical overview in the 3rd Millennium Edition of "Set Theory", has no mention as to who came up with the notation. The notation does appear in the 1978 first edition (p. 72), but there is no mention of its origin; the notation is also missing from The Axiom of Choice (written in 1973).

So, who came up with the notation and when?

Footnotes

  1. We also have the Lindenbaum operator, similar to Hartogs but with surjections, which can grow "quicker" than the Hartogs and differently from the power set, at least in the absence of choice. But we're not here to discuss $\sf ZF$.
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  • $\begingroup$ Tangential question: $\beth(-)$ is a functor (covariant powerset). Can you devise a way of making $\aleph(-)$ (the Hartogs construction) into a functor? $\endgroup$
    – Paul Taylor
    Commented Jan 25, 2022 at 17:55
  • $\begingroup$ I'm not sure what you mean. $\aleph(X)\leq\aleph(Y)$ whenever $|X|\leq|Y|$, so in some sense it is functorial, no? $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 25, 2022 at 17:59
  • $\begingroup$ I mean acting on functions (or at least inclusions) not just cardinality order. Powerset acts contravariantly on functions as inverse image and covariantly by either of the quantifiers. The Hartogs construction destroys automorphisms. Can you do better than that? $\endgroup$
    – Paul Taylor
    Commented Jan 25, 2022 at 18:08
  • $\begingroup$ The $\beth$ function is not the power set, because it is a cardinal function. So it too will destroy automorphisms, I think. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 25, 2022 at 18:09
  • $\begingroup$ And if you want to preserve inclusions, any non-decreasing cardinal functions preserves inclusions, trivially. Since $\aleph(X)\subseteq\aleph(Y)$ whenever $X\subseteq Y$. If we also assume that AC holds, then there is an injection from $X$ into $\aleph(X)$, and so you can also close the diagram given any $f\colon X\to Y$ by finding a suitable $g\colon\aleph(X)\to\aleph(Y)$ for any $X,Y$ and $f$. Of course, that $g$ is not unique, so it's not quite functorial (if I understand things correctly). $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 25, 2022 at 18:16

1 Answer 1

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Charles Sanders Peirce is credited with the beth notation ℶ, first introduced in a December 1900 letter to Cantor. Apparently, this was then forgotten for half a century.
I reproduce the relevant text from Gregory Moore's Early history of the generalized continuum hypothesis.

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    $\begingroup$ So this use off $\beth$ differs from the modern one not only in that the subscript has to be finite but also in that $\beth_{n+1}$ is defined as $2^{\aleph_n}$, not $2^{\beth_n}$. $\endgroup$
    – Andreas Blass
    Commented Jan 25, 2022 at 13:32
  • $\begingroup$ Interesting! Thanks! As @Andreas says, this seems to deviate from the modern definition, although it may very well have been the principle intention of C.S. Peirce to consider a hierarchy, rather than just the power set of each ordinal. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 25, 2022 at 14:08
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    $\begingroup$ Oh. Moore’s reference is actually to a different Peirce’s manuscript bearing the exact same name, which forms a section of Multitude and Quantity. Now it all makes sense. $\endgroup$
    – Emil Jeřábek
    Commented Jan 25, 2022 at 15:08
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    $\begingroup$ @AndreasBlass This is just a typo in Rowe and McCleary. In Peirce’s paper, as well as in Moore’s description, the hierarchy being defined is $\beth_{n+1}=2^{\beth_n}$. $\endgroup$
    – Emil Jeřábek
    Commented Jan 25, 2022 at 15:12
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    $\begingroup$ Notational history is fascinating. I don't wish to be remembered in the history books, but if I would be mentioned in a paper like that in the mysterious context of some notational convention some hundred years from now, I'd be a very happy dead man. $\endgroup$
    – Asaf Karagila ♦
    Commented Jan 25, 2022 at 15:25

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