[FOM] R: necessary or sufficient?
Antonino Drago
drago at unina.it
Tue Feb 21 19:54:56 EST 2006
>
> In his message of February 20, I commented:
>
>
> > It is interesting that the same interpretation of the relationship
> > between continuum and a theory can be applied to game theory.
> > Both the lemma of the supporting hyperplane and the lemma of the
> > alternative for matrices can be interpreted as the requirements
> > for connecting the theory to the continuum.
>
> Gabriel Stolzenberg asked:
> Has it really been shown that they are "the requirements," rather
> than merely sufficient? To do so, you would have to show that if
> either lemma is false, then the theory cannot be "connected to the
> continuum." Can you do that?
Sorry, maybe I was unclear. They are requirements for connecting the theory
with such a kind of continuum. Because there exist several kinds of
continuum, i.e. the constructive one, the recursive one, the Weyl's one, the
rigorous one, etc.
Again Stolzenberg:
> Finally, as for "in constructive mathematics they are manifestly
> undecidable," I find it implausible but it would be very exciting
> to find that it is true.
First A.A. Lewis: "Some aspects of effectively constructive mathematics
that are relevant to the foundations of nonclassical mathematical economy
and the theory of games", Math. Soc. Sci., 24 (1992) 209-235 (review by D.S.
Bridges in Math. Rev., 93i:90003) obtained a first result by means of
recursive mathematics.
Then I proved the full result by merely making use of a "fugitive" number.
Moreover, about the same result I saw a pre-print by S. Bridges.
The result is not surprising because the proofs of these two lemmas require
the fixed-point theorem. N. Shioji and K. Tanaka: "Fixed Point theory in
weak second-order arithmetic", J. Pure Applied Logic, 47 (1990) 167-188,
showed that it requires WKL° or equivalent axioms; all more powerful than
constructive mathematics.
Best greetings
Antonino Drago
via Benvenuti 5
Castelmaggiore Calci Pisa 56010
tel. 050 937493
fax 06 233242218
-----Messaggio Originale-----
Da: "Gabriel Stolzenberg" <gstolzen at math.bu.edu>
A: <fom at cs.nyu.edu>
Cc: "Gabriel Stolzenberg" <gstolzen at math.bu.edu>
Data invio: lunedì 20 febbraio 2006 17.50
Oggetto: [FOM] necessary or sufficient?
>
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