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Browsing MathOverflow I came across a question about analytical sets. Through the discussion following a comment made by our very own Asaf, I learned that bold face $\mathbf{\Sigma^1_1}$ and light face $\Sigma^1_n$ are different things.

Now, I am not asking about the definition or the distinction (I can look them up myself in a set theory book if I have time). I am asking about the history of this, in my not-so-humble opinion, ill-advised notation.

Does anyone know who is responsible for (or the history behind the development of) the convention that the same symbol in bold face means the projective hierarchy and in light face means the analytical hierarchy?

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The title seems a little mismatched with the question. Boldface $\Sigma^1_1$ sets are analytic", and so are lightface ones. Lightface $\Pi^1_1$ sets can be analytical (i.e. in the analytical hierarchy) but not analytic. Is the question just about lightface and boldface? –  Carl Mummert Apr 27 '12 at 12:19
    
@Carl: yes to your final question. –  Willie Wong Apr 27 '12 at 13:28

2 Answers 2

up vote 5 down vote accepted

Jech, in the book Set Theory, 3rd Millennium edition points the blame to Addison in the historic notes to the first chapter about descriptive set theory (i.e. Borel and Analytic sets).

The definition of the projective hierarchy was given by Luzin in 1927. In 1955 Kleene published his hierarchy of analytical predicates. Several years later, Addison published a paper binding the two and gave the distinction of boldface/lightface hierarchies.

Here is a link and a MR link:

J. W. Addison. "Separation principles in the hierarchies of classical and effective descriptive set theory". Fundamenta Mathematicae, vol. 46 no. 2 (1959), pp. 123-135.

MathSciNet link.

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@t.b.: I knew you'd come around! :-) –  Asaf Karagila Apr 27 '12 at 12:27
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See also footnote 19 on page 48 of Moschovakis's Descriptive Set theory (server seems currently down) where he attributes the notation to Addison (the one you link to) and Shoenfield, The problem of predicativity, Essays on the foundations of mathematics, Magnes Press, Hebrew University Jerusalem, pp. 132–139 (1961). –  t.b. Apr 27 '12 at 12:31

I found an answer with a lucky search. John Addison claims in his paper "Warsaw 1957: Memories of Mostowski" that he invented the $\Pi^n_t$ and $\Sigma^n_t$ notation, along with the boldface/lightface distinction. This was in the early 1950s.

You're right that boldface is not a particularly good way to make a distinction in notation. In particular, typewriters cannot produce bold and photocopiers can reduce the distinction between bold and regular text. A separate convention developed to write a tilde under the $\Sigma$ or $\Pi$ to denote boldface, particularly in typewritten works.

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Thanks Carl! Unfortunately I can only pick one answer. And you were heads. :-( –  Willie Wong Apr 27 '12 at 13:30

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