(Fwd) Re: FOM: Question grown from Friedman talks

[ivhtmon@ifa.au.dk]

>Martin Davis wrote Fri, 02 Feb 2001 20:57:34:

>>At 12:22 PM 2/2/01 -0500, Harvey Friedman wrote:
>>>I believe that no conjectures in the existing literature within 
>normal >>mathematics are independent of ZFC.
>>
>>I wonder whether this is simply a hunch, or whether Harvey has some
>>rational ground for this belief. As is well known, G\"odel did not 
>believe >this.

>With all due respect, what is the evidence that Godel disagreed with 
>me on this? We are talking about the actual real world real life 
>hard nosed materially existing normal mathematical literature in our 
>libraries.

Here is Gödel himself adressing the matter (in a talk given in 1951, 
reprinted in the _Collected Works_ Vol. III, ed. Feferman et al., 
Oxford University Press 1995, 309):

"It is true that in the mathematics of today the higher levels of 
this [set theoretical] hierarchy are practically never used. It is 
safe to say that 99.9% of present-day mathematics is contained in the 
first three levels of this hierarchy. So for all practical purposes, 
all of mathematics _can_ be reduced to a finite number of axioms. 
However, this is a mere historical accident, which is of no 
importance for questions of principle. Moreover it is not altogether 
unlikely that this character of present-day mathematics may have 
something to do with another character of it, namely, its inability 
to prove certain fundamental theorems, such as, for example, 
Rimeann's hypothesis, in spite of many years of effort. For it can be 
shown that the axioms for sets of higher levels, in their relevance, 
are by no means confined these sets, but, on the contrary, have 
consequences even for the 0-level, that is, the theory of integers."

So, Harvey Friedman is right that Gödel did not exhibit an example 
from the (then) existing literature of a question that would be 
unsolvable in ZFC, but Martin Davis is right that Gödel took such 
questions to exist in what he calls "objective" mathematics (in the 
same paper), meaning mathematics as independent of human 
mathematicians. This should resolve the exegetical questions 
concerning Gödel.

Terese M. O. Nielsen
Ph.D. student in Philosophy of Mathematics
Department of History of Science, Aarhus University, Denmark