FOM: Hilbert's second problem

[Martin Davis]

At 01:42 PM 4/18/00 -0500,  Matthew Frank wrote:
>Dear FOMers,
>
>It is not often that people solve one of Hilbert's problems without
>realizing it.  The problem is Hilbert's second problem (properly
>interpreted); the people in question are the two founders of this list.
>
>Hilbert's second problem asked for a proof of the consistency of the
>axioms of arithmetic.  People have often assumed that he meant the
>arithmetic of the integers, and that the problem was therefore answered
>negatively by Godel's incompleteness theorem.  A closer look at Hilbert's
>text (and his reference to replacing the axiom of continuity with
>Archimedes's axiom) makes it clear that he was talking instead about the
>arithmetic of the reals.  So the question might be asked more precisely
>as:
>Can primitive recursive arithmetic [PRA] prove the consistency of the
>theory of real-closed fields [RCF]?

Not in any way to disparage the achievement Mathew Frank celebrates, his 
message is historically inaccurate in two regards:

1. Serious discussion of Hilbert's 2nd problem has always been clear that 
it is the axiomatic theory of real numbers for which he asked a proof of 
consistency. See for example Kreisel's essay in MATHEMATICAL DEVELOPMENTS 
ARISING FROM HILBERT PROBLEMS, Proc. Symp. Pure Math. vol. 28, AMS 1976.

2. Hilbert pointed out that the full (we would say "second order") axiom of 
continuity (i.e. Dedekind completeness) is equivalent to the Archimedian 
axiom **plus the assumption that there is no larger ordered field that 
retains the Archimedian property**. Hilbert does not suggest replacing 
continuity by the Archimedian property without the addition of this second 
order maximality principle. He is quite clear in his statement (which can 
be read in the volume referenced above) that it is the full axiom of 
Dedekind completeness that he is talking about.

Indeed I believe that the context was the felt need to push the reduction 
of analysis to the arithmetic of real numbers one step further by proving 
the consistency of the axioms for the continuum. But this is my surmise 
based on the historical situation and not justified by Hilbert's actual words.

Martin Davis





                           Martin Davis
                    Visiting Scholar UC Berkeley
                      Professor Emeritus, NYU
                          martin at eipye.com
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                        http://www.eipye.com