FOM: Question grown from Friedman talks

[Harvey Friedman]

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.

The statement I made is, in my opinion, far more rational to believe than
its negation. For instance, I am not aware of any such statement for which
there is the slightest indication of any difficulties in proving it within
ZFC. For example, there could be a technical conjecture in math logic that
is very much like things already known in math logic that would in fact
imply that a particular such statement is independent of ZFC.

With the Cohen thing, it was known that one could do a lot before Cohen if
one could take a countable model of ZFC and add new objects to it to fatten
it, while remaining a model of ZFC. But until Cohen, nobody knew how to do
this. The math logicians had gotten far enough to see this before Cohen.

Ancient problems like four color, sphere packing, FLT, and the like
consistently get solved without using set theory.

In my opinion, independence from ZFC for concrete problems is incredibly
rare until mathematics is pushed into more structurally ambitious
directions than it has been up till now. But my thesis is that once one
pushes things into the new level of structural ambition that is represented
by BRT, then all hell breaks loose, and higher set theory rules.

Furthermore, the mathematical community will completely embrace this new
level of structural ambition in the forseeable future.