Re: The rational number 5 and the real number 5

On Mar 7, 10:25 am, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
djr...@xxxxxxxxxx wrote:
On Mar 7, 5:52=A0am, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
djr...@xxxxxxxxxx wrote:
On Mar 6, 12:38 pm, David W. Cantrell <DWCantr...@xxxxxxxxxxx> wrote:
djr...@xxxxxxxxxx wrote:
On Mar 6, 10:48 am, William Elliot <ma...@xxxxxxxxxxxxxxxxxx>
wrote:=

On Thu, 6 Mar 2008 djr...@xxxxxxxxxx wrote:

The real number 5 is the name we give to [(5,5,5,...)] if
real numbers are defined to be equivalence classes of Cauchy
sequence=
s
of rationals. Maybe we would like to say that the real number
5 is equal to the rational number 5. In what sense does it
make sense to say this, and is there anything wise to say
here?

What makes sense is that there is an isomorphism between the
rational numbers and the subfield of the real rationals of the
reals.

In otherword, the rationals embed into the reals.

Of course 5 the integer, 5/1 the rational and 5.0000....
the real are not equal until you embed the integers
into the rationals and the rationals into the reals.

Then it makes sense to define 5/1 as 5.000... and 5 as 5/1.

So to be technical, the rationals Q and the rational reals Q',
the set of equivalence classes of the form [(q,q,q,...)] where q
is rational, are not equal, but are isomorphic?

Yes. [But the rational reals _could_ be precisely the same as the
rationals. See "N as a subset of C" (sci.math, 2000 Mar. 16) by Dan
Hoey at
<http://groups.google.com/group/sci.math/msg/d6e1ff510d0b0568>=
.
But I should note that I don't like the idea of doing what Hoey
describes; I prefer, as is most commonly done, that all reals be
the same sort of mathematical object.]

Concerning common constructions in which, as you say, Q and Q' are
not=

equal, but isomorphic:

Note that this merely means that the rational 5 and the real 5
behave the same under _field_ operations. It doesn't tell us that
they should=

behave the same otherwise. For example, the floor function is
computable on the rationals, but not on the reals. Thus, I can
imagine=

someone arguing that, although floor(rational 5) =3D 5, we should
leav=
e
floor(real 5) undefined.

David

The only thing I don't get is what you said about the floor function.
You are saying there is no general algorithm that, given a real
number, outputs the floor of that real number?

Yes.

I suppose there could
be a real number none of whose digits can be computed by the
(arbitrary) candidate algorithm, then the floor of that real number
could certainly not be computed by that algorithm. But surely there
is a general algorithm for computing the floor of rational reals?
Simply return the floor of the relevant rational.

But, assuming that I understand things correctly, we can't determine
the relevant rational.

If it's the real number 5, the relevant rational is the rational
number 5.

Indeed. But we don't have an algorithm which can determine whether we have
the real number 5 or not.

David

Well, I still don't understand how computing the floor of a rational
is different from computing the floor of a rational real in the way
you claim. It's not the main point of the thread though, I guess.
.

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