[FOM] Harrison Advocates ZFC (tweaked)

[Harvey Friedman]

I am proposing ZFC with urelements and free logic as the NEUTRAL
UMBRELLA. Various factions can easily work smoothly within this
Neutral Umbrella, and even probably almost compatibly.

I agree that having urelements is not *necessary* but it seems
essential philosophically. It is just what is needed to neutrally
accommodate various positions concerning "what really is an ordered
pair", "what really is a natural number", etcetera. So urelements to
my mind are good to have in the neutral Umbrella I am proposing.

I'm a little surprised to hear that John doesn't like partial terms.
E.g., terms like 1/0. Again, all practicing mathematicians do this
without wanting to be bothered with any set theoretic or other
workaround to 1/0. Once again, 1/0 as a partial term like in free
logic seems to be a very good thing to have in the neutral Umbrella.

I need to wrap my head around the issues Larry is talking about
concerning n*x. If * is real multiplication, then n*x is automatically
covered if every integer is a real number. But maybe one is going to
have integers never real numbers. Then there is the canonical
embedding j:Z into R, and one instead will write j(n)*x. Perhaps Larry
is trying to avoid getting into a situation where you write j(n)*x.
However, I don't see why we can't thrive in a situation where we
simply make a choice:

1. Every integer is a real. Then n*x is not problematic.
2. Integers are not reals. Then we write j(n)*x. One does see working
mathematicians do stuff like this when they want to be a bit careful
and are not doing 1.

Of course, there are overloading problems galore. Are we going to
reuse j problematically? Sure, this is a problem, because we would
rather not clutter up stuff with putting modifiers on j. Things get
ugly quickly.

But I think that there is a separate subject about how to reduce the
number of symbols and keep everything super readable and totally
unambiguous.

I think that there is such a thing as "thorough design of a system"
that is nowhere near enough for actual working implementation, but
enough to flesh out the principal ideas and give critical examples -
more or less according to the usual standards for doing mathematical
research. I know from Jeremy that people in CS regard such a thing,
without actual working implementation, as worthless - or simply
elements of a grant proposal, and nobody will pay any attention to it
in CS - being essentially valued at zero. I view this as an indication
of a disastrously poor state of affairs intellectually in CS, and I
and many people on this little list can easily afford to completely
ignore this. I have worked for absolutely no money since soon after
2012, and https://u.osu.edu/friedman.8/foundational-adventures/downloadable-lecture-notes-2/

#69 is incomparable to anything I did before.

Harvey Friedman