[FOM] Fwd: invitation to comment
Arnon Avron
aa at tau.ac.il
Sat May 21 16:35:49 EDT 2011
> FPP likely implies Con(PA) over some weak base formal theory B. So if PA is
> in-consistent FPP is false. This does not imply that PP is false and that
> Perelman's argument is flawed.
If so then why on earth you (and Voevodsky) are claiming that Godel
has proved that no proof of the consistency of PA is possible???
According to your views,
Godel has proved nothing about the consistency of PA. All he proved
was that a certain formal sentence Con(PA) is not provable in PA.
It is perfectly possible (according to *your* views) that Con(PA)
is unprovable (or even FALSE), while one can give a perfectly valid
mathematical proof of the consistency of PA.
And of course one can indeed give a perfectly valid mathematical
proof of the consistency of PA (even though Con(PA) is true).
The proof is very simple: N is a model of these axioms and
so they are consistent. This type of argument
has always been the standard method
of proving consistency and independence. Would anybody say that
the ordinary proofs of the consistency of various Non-Euclidean
Geometries via models are "non-mathematical"??
> In order to claim that FPP is equivalent to PP people use various
> non-mathematical arguments. I cannot imagine a *mathematical* argument that
> could fill this gap.
But you can imagine that it can be filled in the case of Godel Theorems, right?
This discussion brings me back to a subject I wrote about in the past: every
reservation Rodin and others may have about translating mathematical
propositions into formal languages apply in an even stronger way
to their translations into "natural languages" (such translations
do not exist, by the way. No mathematical book is written
in any ``natural languages". All of them contain a *lot* of formal
stuff, not belonging to any "natural languages"). Whenever someone
tries to express a mathematical proposition in some language,
the resulting expression only represents the intended proposition,
and something is lost in this translation. Something is lost
also when a proposition that was initially represented in German
(say) is translated into English. We can never be sure (according to
Rodin) that the translation represents the original formulation
(to say nothing about the original proposition), and "I cannot imagine
a *mathematical* argument that could fill this gap". It has always been
a mystery to me why this point is raised by some philosophers only with
regard to formal languages. Representing Mathematical propositions
in properly designed "artificial languages" is
much more reliable then representing them in "natural languages" -
which has been the reason for inventing and using "artificial languages"
in the first place. (Similarly, algorithms can much better and more
faithfully be represented in formal
languages than in ``natural languages". Just try
to formulate the algorithm for finding derivatives of elementary
functions in a purely "natural language"!).
Arnon Avron
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