FOM: Goedel: truth and misinterpretations
Kanovei
kanovei at wmwap1.math.uni-wuppertal.de
Tue Oct 24 14:03:42 EDT 2000
> Date: Tue, 24 Oct 2000 09:31:04 +0200
> From: Torkel Franzen <torkel at sm.luth.se>
> ... after all it is mathematically
> provable that if ZFC is consistent then there is a true arithmetical
> sentence A such that the canonical translation A* of A into the
> language of ZFC is not a theorem of ZFC. Apparently you don't regard
> this particular result as even meaningful. Why not?
I wrote that the result is widely misinterpreted by philosophers
(and philosophying mathematicians). The nature of misinterpretation
is that the wording of true mathematical result is interpreted in
terms of relations between a human-mathematician (or the mathematical
community) and the mathematical universe.
It seems that you do not buy my conceptual explanation of the
yesterday post, so let me explain my point in simpler, but
more practical fascion.
You write:
> true arithmetical sentence A such that ...
True -- where ? in which sense ?
Let's forget Goedel's mystical "sentences" and consider CH
which everybody knows what it means (at least formally).
neither CH nor not-CH is a theorem of ZFC -- and this is
a mathematical fact (I let aside the assumption of Con ZFC).
Now, the misinterpretation which I am talking about reduces
to the following:
BUT either CH or not-CH is TRUE ,
which is here nothing but an example of the excludedmiddle.
Now let me ask again: True -- where ? in which sense ?
Are you going to answer anything more meaningful here than
just claiming CH or not-CH as a ridiculous successor of the
Danish prince ? No, you are more reasonable
> that's no obstacle to statements that refer only to a
> determinate part of that indeterminate structure being determinately
> true or false.
For instance, astronomers know that there is a mid-size black hole
in the center of the Milky Way -- despite possibly there is no
rigorous and fully determined concept of the Milky Way (its boundaries,
for example). This is because there is factual information enough
by the standards of astronomy to make the above conclusion.
But we have reasonable doubts can the concept of the Milky Way
ever be reasonably specified to make the statement that
"it has either even or odd number of stars" to have any real
meaning (I even do not say any verifiable meaning).
And this is about the empirically existing structure, what it remains
to say about the mathematical universe whose empirical core consists
of things like finite combinatorics and observable shapes and the
rest are just "ideas", i.e., axioms and their logical consequences~?
Is there anything more than just a dogma, in saying that either CH
is true or not-CH is true (for a mathematician, in the mathematical
universe) ?
And if yes then -- again, in which sense other than
the meaningless in this context "CH-or-not-CH"? Any answer ?
> >Since Euclid, "we *know*" well that a mathematical statement is true
> >if there is a (mathematically sound) PROOF of the statement -- this
> >is by the way why Mathematics has been called EXACT science.
>
> Right, but what is your view of the axioms used in this proof?
This is the point on which there sometimes has been
substantial disagreement between even "peers" of
mathematics (the "Sinq lettres" between Borel, lebesgue, and
Hadamard can be mentioned, let alone the Berkeley critics of
the use of infinitesimals).
My view is that ZFC concentrates, in the form of a few
simply formulated principles, everything which has been
practiced in mathematics, although, perhaps, in more
generality that 99.9/100 of mathematics really needs.
Perhaps we can trace ZFC to finite combinatorics of
empiric collections of pebbles, that would give it some
real substance.
There are writings by those who fully know the subject
e.g. Shoelfield's article in the "Handbook of Math. Logic".
V.Kanovei
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