[FOM] Hilbert and conservativeness

[praatika@mappi.helsinki.fi]

Aatu Koskensilta <aatu.koskensilta at xortec.fi> wrote:

> At several places Hilbert notes that a finitistic consistency proof for 
> 'ideal' mathematics implies that 'ideal' mathematics is conservative 
> over finitistic mathematics w.r.t. finitistically meaningful ('real') 
> statements.
> 
> What I'm wondering is how Hilbert knew this. Did he note, as we might 
> do today, that if T_1 |- Cons(T_2) then every Pi_1 sentence provable in 
> T_2 is provable in T_1 (provided the theories meet the relevant 
> conditions); or did he simply believe that finitistic mathematics is 
> complete and hence any consistent theory extending it is conservative?



Here is what I wrote on this issue in my paper "Hilbert's program 
revisited": 

* * *

... I think that it is somewhat anachronistic to attribute
such sophisticated logical ideas to Hilbert. For, if such a line of thought
really were behind his consistency program, one would certainly expect
Hilbert to explain it in detail. But there is hardly any hint of such 
reasoning in Hilbert’s work. The only exception is Hilbert’s relatively 
late discussion, in his 1927 Hamburg address (Hilbert 1928), where there 
is indeed an informal sketch of the idea of how the consistency proof 
would allow one to eliminate the infinistic elements from a proof of a 
real sentence. This passage seems to be the only basis for the later 
logical interpretation.

But certainly, if Hilbert’s reasons were as ingenious as the later proof-
theoretical tradition tends to interpret them, there would be more traces
of it in Hilbert’s publications. And yet, for example in his paper ‘On the
Infinite’ from 1926, which is the most extensive mature exposition of 
Hilbert’s program, Hilbert simply stated that a proof of consistency 
amounts to real-soundness and real-conservativity, without a word of 
explanation 
:
:
My hypothesis is the following: I think that Hilbert simply assumed
that finitistic mathematics is deductively complete with respect to the 
real sentences (i.e., is “real-complete”).10 This would have made 
everything smooth: if an ideal theory extending finitistic mathematics 
would prove some real sentence that finitistic mathematics does not prove, 
it would be inconsistent: the real-conservativity and real-soundness 
follow immediately from the consistency. That is, in the presence of real-
completeness of finitistic mathematics, the properties of consistency, 
real-soundness and real-conservativity almost trivially coincide.

Hilbert once remarked that “in my proof theory only the real propositions
are directly capable of verification” (Hilbert 1928, 475), but I
am not certain whether one can interpret this as expressing a commitment
to real-completeness. However, the following statement seems to
do that: “In mathematics there is no ignorabimus. On the contrary, we
can always answer meaningful questions” (Hilbert 1929, 233). And “No
answer” is clearly not an answer. (One should also note that there is, as
such, something odd with the idea of a statement which is meaningful
but does not have a truth-value.) Bernays, in any case, explicitly assumed
real-completeness: “In the case of a finitistic proposition however, the 
determination of its irrefutability is equivalent to determination of its 
truth” (Bernays 1930, 259, my italics). One may presume that this also 
reliably reflects Hilbert’s view.

Further, it is a fact that Hilbert believed that both the axioms of 
elementary arithmetic and those of real analysis are deductively complete 
(see Hilbert 1929, 1931; cf. Bernays 1930). Although he is nowhere that 
explicit with respect to finitistic arithmetic, it is not at all 
implausible to assume that Hilbert believed also in this kind of real-
completeness. Moreover, the former alleged completeness of full arithmetic 
would actually provide a decision method for all real statements, which in 
turn would naturally entail their decidability in finitistic mathematics. 
Finally, the assumptions that first-order logic is decidable and that 
finitistic mathematics can prove the consistency of any consistent theory, 
arguably alleged by Hilbert, both entail that finitistic mathematics is 
complete for real sentences. Therefore, we even have several different 
reasons to assume that Hilbert believed in real-completeness of finitistic 
mathematics.

Reference: Panu Raatikainen, "Hilbert's program revisited", Synthese 137: 
157–177, 2003.


Best, Panu


Panu Raatikainen

Ph.D., Academy Research Fellow,
Docent in Theoretical Philosophy
 
Department of Philosophy
P.O. Box 9
FIN-00014 University of Helsinki
Finland
 
E-mail: panu.raatikainen at helsinki.fi
  
http://www.helsinki.fi/collegium/eng/Raatikainen/raatikainen.htm