[FOM] Historical notation question

H. Enderton hbe at math.ucla.edu
Wed Feb 25 15:33:11 EST 2004

Brian Postow was asking about the origin of the notation Sigma_i and
Pi_i for the different levels of the arithmetic hierarchy. 
Panu Raatikainen ascribes it to Addison.

Certainly Addison was using this notation by the time of the 1957
Cornell meeting; see his paper from that meeting.  Earlier,
I think Mostowski had been using P_i and Q_i; see his paper
in Fundamenta Mathematicae, vol. 38, pp. 81-112.  Martin Davis
in this 1958 textbook followed that notation.  (I hope he'll
correct me if I'm wrong here.)

I'm not sure whether the Sigma-Pi notation (which, as others have
pointed out, is quite natural) originated with Addison.  I'm pretty
sure that the "lightface/boldface" distinction can be blamed on him!
(I'll cc Addison on this.)

--Herb Enderton

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