[OT] Number theory [Was: A use for integer quotients]

[David C. Ullrich]

On Thu, 26 Jul 2001 00:01:15 +1000, "michael" <serrano at ozemail.com.au>
wrote:

>> But strictly
>> speaking the Integer 2 and the Real Number 2 are different entities.
>
>So strictly speaking, Z (the set of integers) is not a subset of R (the set
>of reals)?

Strictly speaking yes, but nobody _ever_ speaks this strictly
(except in contexts like the present).

We math guys have reduced more or less everything to sets these
days - fewer things at the bottom to keep track of. So when we
want to define a "type" with certain "properties" we figure out
how to "code" such a thing as a certain sort of set. But almost
always what set something actually is makes no difference, all
that matters is how it works.

Strictly speaking the natural number 2 is the set {{},{{}}}
(where {} denotes the empty set) and with one typical definition
the integer 2 is a certain infinite set of ordered pairs of
natural numbers {(2,0), (3,1), (4,2),...}. With some other
definition the integer 2 would be a different set. There
are several definitions of real numbers - with one
definition the real number 2 is a certain set of rationals,
with another popular definition the real number 2 is a
certain set of sequences of rationals.

But those are irrelvant implementation details - when someone
says something about the real number 2 people don't ask whether
he's talking about a Dedkind cut or an equivalence class of
Cauchy sequences, because it makes no difference; in fact
pretending it made a difference would be a bad thing. Both
definitions give reals that work the way we want reals to
work.

So yes, strictly speaking the integers are not a subset
of the reals. But nonetheless if someone asks "Are the
integers a subset of the reals?" the _right_ answer
is "yes"; the answer becomes "no" only if we're focussing
on the irrelevant aspects of things.




David C. Ullrich