Re: The rational number 5 and the real number 5

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?
.

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