Re: The Dedekind Snap

MoeBlee wrote:
On Dec 8, 3:30 pm, John Jones <jonescard...@xxxxxxx> wrote:
MoeBlee wrote:
On Dec 8, 2:21 pm, John Jones <jonescard...@xxxxxxx> wrote:
But then it looks as if the rationals can express the irrationals.
SUBSETS of the set of rationals can be used to "encode" the reals.
If the reals are comprised of the rationals and the irrationals, then
how can the rationals and the irrationals be encoded in the rationals?

They're not encoded by rationals, they're encoded by SETS of
rationals.

How can numbers encode, or stand in for another number? Can 1 and 3 stand in for 2?

There are only countably many rationals but there are uncountably many
sets of rationals. There are enough SETS of rationals to accomodate
encoding reals as certain kinds of SETS of rationals.

....there are enough certain combinations of rationals to 'encode' reals (present reals) as certain combinations of rationals? Can you clear that up.


Recall that there are more subsets of the set of rationals than there
are rationals.

Yes, there are generally more permutations of things then there are things.

There are plenty of subsets of the set of rationals to
use to encode the reals.

You mean to encode the irrationals.

It may be common practice to paraphrase "there are more rationals than
rationals" as "there are more subsets of rationals than rationals".

No, please don't play me for a fool. I meant exactly what I said:

There are more SUBSETS of the set of rational numbers than there are
rational numbers.

Subsets meanst permutations of the digits. We assume of course that the permutations of a certain class of number are also in that class. It's not always true, however.

But
the term "subsets" refers to another mathematical construction. You
can't allow "subsets of rationals" to do business for "rationals" when
you say 'There are plenty of subsets of the set of rationals to
use to encode the reals.'

I don't say anything about "to do business for".

You are assuming of course that the subsets of the rationals can do business for, or are, rationals.

As I said, you may find a proof that the system of real numbers as
Dedekind cuts with the defined operatons is a complete ordered field.

I can say "there are more permuations of x than there are x, but
permutations of x isn't x.

So what? I DON'T say real numbers are encoded as rational numbers. I
say that, with the cut method, real numbers are encoded as certain
kinds of SETS or rational numbers.

I thought there was only one kind of rational number - the rational number.
.

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