Thanks, Ramana! I don't quite understand. In each theorem I should
make an extra hypothesis which will look something like assume
Hilbert_plane H (Between) (===) Line {HP};
And I'll add the label HP to each "by" list in which I refer to one of
my earlier theorems? In your earlier post, I think you said you
preferred shallow embeddings. Do you call this a shallow embedding?
Back to my earlier question, my miz3 code begins
new_type("point",0);;
new_type_abbrev("point_set",`:point->bool`);;
new_constant("Line",`:point_set->bool`);;
I still don't know what the HOL, or HOL4 meaning of this is, but I have a new
conjecture:
My new_type command defines a set called "point", And later when I write
let A B C be point;
that means that the variables A, B & C refer to elements of the set "point"
Does that sound right? In particular, a type is the name of a set?
I'm having trouble with the term HOL. If someone asked me to describe what HOL
Light was doing, I'd say that we were working in a model of ZFC with some extra
features. E.g. it's quite difficult in ZFC to construct the real line R (see
Halmos's book Naive Set Theory), but HOL Light hands us a nice copy of R with
all its properties. So if one did not have a model of ZFC, one would be doing
FOL, just manipulating the f.o. ZFC axioms, but it's a lot nice to actually
have a model of ZFC, i.e. a collection of sets. I don't mean that we have all
the sets that a model of ZFC would contain. There's a set UNIV, the universe
we work in, and a ZFC model would not contain that. All this is great, but I
don't know why we're using the term HOL to describe it. I understand that HOL
means 2nd order logic, 3rd order logic etc, but that just means you can
quantify over more and more complicated sets, and I guess we get all those sets
in a model of ZFC.
I'm also confused about the HOL Logic manual saying we don't get the empty set.
I use the empty set all the time, it's called EMPTY or [ {}, defined in
sets.ml. Maybe there's a technicality which going like this. In sets.ml, sets
are "constructed" as boolean functions. So (assuming that "point" is actually
a set, as I conjectured above) we can define the empty subset of point as the
function
{}: point -> bool
which takes every element of point to False. So is that it, we can get the
empty set by confusing sets with boolean functions?
--
Best,
Bill
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