# No subject

Mon Jan 2 07:34:30 CET 2012

```of irrationals that can be represented at all no matter how. For a certai=
n=20
ontology useful to constructivism, it can be said that the uncountably=20
many irrationals that are inexpressible also don't exist.

This leaves you with countably many rational numbers and countably many=20
irrationals, that can't be represented in a finite format.

> We could also debate over whether infinity is exactly represented.
> When some math operation overflows (exceeds the range of floats), the
> result assigned is inf.

Every float represents a range of numbers. The difference with infinities=
=20
is that they represent half-intervals, that is, a line bounded only on on=
e=20
side.

> That's not the definition of infinity either: Take the set of real=20
> numbers R and the ordering operation <, then add an additional point=20
> "infinity" such that for any x belonging to R, x < infinity.

You should know that there are several competing definitions of infinity=20
for real numbers (not considering other number systems in which this=20
definition doesn't work).

There are three definitions of Real numbers (R) in common use=A0: one=20
without any infinite number, one with two infinite numbers as endpoints,=20
and one with a single infinite number without a sign. There are different=
=20
motivations for the use of each of those three sets. There's no definitio=
n=20
that fits all purposes, though the one without infinite numbers at all is=
=20
considered generally =ABcleaner=BB in the field of pure math.

> So, the inf in the float definition only represents "infinity" defined=20
> relative to the finitely countable set of numbers that can be=20
> represented as floats

Yes, except NaN.

You'll also find out that certain definitions of infinity that applies to=
=20
the whole set of Reals also are relative to just that set, and don't work=
=20
as-is for all possible extensions of Reals=A0; for example, Complex numbe=
rs=20
don't have a single coherent definition of less-than and greater-than=20
anymore, because all you can do is extract features of Complex numbers an=
d=20
compare those features as Reals... thus you need more specific=20
definitions (and there are more possibilities of them).