FOM: Re: Reals and reality

[Harvey Friedman]

This is in response to Shipman, 10:06AM 12/17/97.

Let me remind you my posting of 2:28AM 12/8/97 on the continuum hypothesis.
There I talk of "complete pictures" and conjecture that this has a clear
formalization, that many systems of FOM demonstrably represent "complete
pictures" and that there should be a theorem that no "complete picture"
settles the continuum hypothesis.

Our conception of the ordered real number line, say with the 4-ary relation
R(a,b,c,d) if and only if a < b and c < d and the distance from a to b is
the same as the distance from c to d may well not correspond to anything in
physical reality, but does reperesent a "complete picture." Of course, the
language can be expanded, but that is another matter. And in this complete
picture, one has what technically amounts to a completely ordered group,
which determines this complete picture up to isomorphism.

Further "complete pictures" lead to various axiomatizations of 2nd and even
3rd order arithmetic.

Of course, the whole matter has to be handled with great finesse to avoid
collapsing into a sea of blather.