[FOM] history of modern notation for sets defined by a property

Allen Hazen allenph at unimelb.edu.au
Wed Feb 25 23:51:38 EST 2009

Sorry to be tardy in replying; hope you have found the answer by now.
Feys and Fitch, "Dictionary of Symbols of Mathematical Logic"
(North-Holland, Studies in Logic, 1969) gives one of the variants
(circumflexed variable before parenthesized formula) as their preferred
form, then mentions that J.B. Rosser, in "Logic for Mathematicians" (1953)
mentions some alternative notations, including the "modern" one you ask
about: Feys and Fitch then go on to credit Rosser with an extension of the
notational convention typified by
for the class of numbers which are sums of x and y satisfying ...  .
I don't know if Rosser credits the "alternatives" to particular writers.

Kelley's "General Topology" (1955) says that what you call the modern
standard notation "is the one most widely used today," and mentions one
alternative: Capital E with the abstracted variable written below it as an
abstraction operator.

Sorry: I thought that Feys and Fitch would have an answer to your question,
but they don't really take us beyond what you already knew!

Allen Hazen
Philosophy Department (PASI)
University of Melbourne

On 12/2/09 11:47 PM, "Arnold Neumaier" <Arnold.Neumaier at univie.ac.at> wrote:

> Does anyone know who invented the notations {x in A | P(x)} and
> {x in A : P(x)} for sets defined by a property P(x)?
> McShane, Integration, 2nd ed. (1947) uses {x | S},
> I don't have access to the first edition from 1944
> - is it perhaps already there?
> Halmos, Measure Theory (1950) uses {x: pi(x)}.
> These are the earliest occurrences of the modern notations known
> to me (communicated to me by Viktor Losert).
> Maybe someone is able to trace back the use even further.
> Please send information directly to me (Arnold.Neumaier at univie.ac.at);
> the newest state of the investigation will always be visible on my
> web page http://www.mat.univie.ac.at/~neum/contrib/set.txt
> Note that variants of the modern notations are used in the older
> literature. For example, a paper by Von Neumann 1928 uses M(x; E(x)),
> and in ''On rings of operators'' (1936), he uses (x; epsilon(x)).
> I am interested only in the modern forms.
> Arnold Neumaier
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