Ultra Tangible f.o.m.

[Harvey Friedman]

On Sat, Jan 25, 2020 at 1:02 AM Joe Shipman <joeshipman at aol.com> wrote:
>
> There are some questions about integers which we are never going to know the answer to, and we’re ok with that.
>
> For example: either there is a sequence of a googolplex consecutive identical digits in the decimal expansion of pi that is both the first such sequence and involves an even digit, or not.
>
> It seems very plausible that there is just no way we could ever come to know the answer to that question. But does anyone think that it is a sentence whose meaning is unclear?
>
> I don’t understand why so few mathematicians are willing to say “of course CH is either true or false, but we don’t have access to the answer because it makes no difference to physics, and with no direct access to uncountable sets we can’t prove anything about it.”
>
> No one seems to think it’s a problem we can’t answer my question about pi. Why is it such a problem that we can’t answer CH?
>
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One can try to get into the same mood about such integer problems as
one fairly easily gets into about CH. Specifically, in the mood that

*the notion of being a natural number is nonsensical"

In this mood, we feel that the natural numbers are supposed to go on
forever, but that "forever" is just an imaginary fiction that we use
when we do not know what we are really talking about.

It is no longer interesting in the 21st century to bask in this mood,
or try to jolt people in this mood out of this mood.

It is much more interesting to see what mathematics survives under this mood.

Actually, in my humble (smile) opinion, almost all interesting
mathematics survives under this mood, including large cardinal theory.

I have thought about this in bits and pieces, for some time, and today
I would like to call it

ULTRA TANGIBLE FOUNDATIONS OF MATHEMATICS

Harvey Friedman