FOM: Goedel: truth and misinterpretations

[Kanovei]

> Date: Wed, 25 Oct 2000 08:32:42 +0200
> From: Torkel Franzen <torkel at sm.luth.se>
 
>   >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).
> 
>   What's mystical about arithmetical sentences? 

These sentences are obtained as follows. 
There are two sets of sentences, say X \subseteq Y, 
and we know that X is r.e. but Y is not r.e., hence, the 
difference Y - X is infinite, and the "sentences" are just 
those in the difference. There is no one concrete sentence 
there, all we know is that they do exist in plentitude. 

On the contrary CH is a concrete well defined mathematical 
sentence, as good as any in Y - X as for the matter of 
discussion. This fits my mathematical attitude (to add 
order to disorder) but possibly does not fit the pseudo-philosophical 
attitute to misfind chaos is order. 

> Do you have any objection
> to the assertion that (assuming ZFC to be consistent) there are infinitely
> many true sentences of the form "the Diophantine equation P=0 has no
> solution" which are not provable in ZFC?

As a mathematical fact I have no objection, 
assuming that "there exist infinitely many ... provable in ZFC" 
is understood as a single ZFC formula which describes the intended 
meaning of the sentence in a known proper way. 
But I have objection (mildly speaking) against the intended 
philosophical misinterpretation of the quantifier 
"there exist infinitely many ..." 
in the metalanguage of mathematicians about mathematics, 
even in the veaker form "there exiss at least one ..." (hint to CH).  

> So do you have any objection to the reflection that CH may well be
> true although not provable in ZFC, given that "CH is true" is here a
> mathematical statement mathematically equivalent to CH?

The mathematical statement correctly saying that 
(CH and not-Prov_{ZFC}CH) or (not-CH and not-Prov_{ZFC}not-CH) 
is true in the metalanguage of mathematicians about mathematics, 
i.e. there is a proof of it. 
Neither of the two parts is true in the meta-language UNTIL 
somebody has this established, so are laws of mathematics, 
and there are reasonable doubts that this can be reasonably 
expected.

> Well, whether you use the axioms of ZFC or some other axioms, how would
> you justify those axioms?  Not by proofs, surely?
 
ZFC was taken because its axioms fit mathematical practice, 
this is almost an empiric observation despite it refers to 
nonmaterial objects. 
I admit that there might happen in some future that 
advanced mathematical practice of that hypothetical time will 
strongly require new axioms, actually, as a matter of fact, 
some big logicians claim that this can be the case even now. 
Their justification should normally be the same, that is, 
explicit fit to mathematical practice.  

The REAL question here, let me repeat, is the ratio 
of physical(or empirical)/social/individual in mathematics, 
in view of the most remarkable fact that despite of the 
full absence of any empirical verification (except for 
very special branches of mathematics) and basically individual 
character of mathematical work mathematics has been developing 
mainly as an aggregated whole.

V.Kanovei