[FOM] PA Incompleteness

[Harvey Friedman]

On 10/14/07 8:03 PM, "Feng Ye" <yefeng at phil.pku.edu.cn> wrote:

> I am always wondering what will be the answer if 'normal mathematics' is
> replaced by 'mathematics with potential applications in sciences', or
> 'mathematics relevant to this physical universe'. The problem is that all
> current independent propositions are related to fast growing functions, but
> the scope of the physical universe that sciences are dealing with is so
> 'ridiculously small' from the mathematical point of view.
> 
I have a couple of remarks about this.

1. The idea that time may never end, or may go on for a very long time, is
now present in a lot of discussions by cosmologists. In particular, I have
seen numbers indicating how long all of the black holes will take to
evaporate, etcetera, and if I recall properly, those numbers were something
like 10^(10^100), considerably larger than what you usually see in
cosmology.

2. I now work on Pi01 independence results. I can sometimes show, or hope to
show, that for somewhat reasonable n, if we restrict the Pi01 statement to
[1,2,...,n], thereby getting a Pi00 sentence, the sentence is "independent"
of ZFC in the sense that it can be proved with large cardinals, but has no
proof in ZFC with fewer than 2^1000 symbols. Numbers like 10^(10^100) may
give me enough wiggle room to accomplish this.

Harvey Friedman