Re: The rational number 5 and the real number 5

djrt20@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 sequences
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) = 5, we should leave
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.

David
.

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