[FOM] Historical notation question

H. Enderton hbe at math.ucla.edu
Mon Mar 1 15:02:00 EST 2004

Panu Raatikainen quoted Kleene as saying "that the metamathematical use
of the notations Sigma^0_i and Pi^0_i to describe the levels in the
arithmetical hierarchy were introduced by Addison 1958 and Mostowski

I asked John Addison about that.  The full notation (including the
choice of those particular symbols) is indeed due to Addison.  His 1954
dissertation dealt in part with the analogies between the Borel
hierarchy and the arithmetical hierarchy.  It was after writing it that
he worked out his systematic notation for these and other hierarchies,
now well known.  Then in the spring of 1957 he was in Warsaw on an NSF
postdoc.  In e-mail Addison writes, "That spring I do recall presenting
the case for the new notation very forcefully to Mostowski and
eventually getting him to agree to use it."  (Those of you who know
John can imagine those conversations!)  As I noted previously,
Mostowski had been using P and Q.

Addison formally introduced his systematic notation in a 1959
Fundamenta paper (JSL XXIX 60).

The historical point is interesting.  Its bearing on FoM is perhaps
only that there are foundational issues involving these hierarchies,
and the language -- and the notation -- we use to discuss them must
be completely clear.

--Herb Enderton

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