Re: Dedekind Cuts, Fundamental Sequences: why?

Dave Seaman wrote:

On Fri, 08 Jun 2007 09:19:22 -0400, Hatto von Aquitanien wrote:
Dave Seaman wrote:

We also have the possibility that x \in \Q and
y !\in \Q, which implies x!=y, and thus x-y!=0. Now, suppose I have
some
series which converges in finite time to a real value v !\in \Q. For
every member of any Cauchy series in the equivalence class representing
v, we can find z \Q such that |v-z| > 0.

This is the kind of thing which makes me feel as though logic alone is
insufficient to prove that a sequence of rational numbers can converge
to an irrational real number.

Try as I might, I cannot find any connection between that last paragraph
and the one preceding it.

Well, I didn't provide the qualification explaining what I meant by z.
That should have been |v-a_n| > |v-z| > 0 where a_n is a member of the
sequence. I don't know if you gleaned that from the context or not.

That was not the paragraph that I had trouble with.

Since the latter paragraph references the previous one, I am at a loss
regarding your failure to see a connection between them. If you do not see
the logical connection between the ideas expressed in the two paragraphs, I
suggest you read it with close attention to semantics.

Yes, I know I am introducing a condition which may be superfluous.
Nonetheless, it shows that there is a difference between the equivalence
class representation of a rational number and the equivalence class
representation of an irrational number.

Remember, the point of using equivalence classes is that we don't want to
talk about "convergence" while we are defining the reals, because that
would be circular.

I don't think I can "remember" any such motivation since I don't believe I
ever perceived that there was such. I'm also confused by your statement
about convergence. I can define convergence strictly in terms of rational
numbers. What I cannot, in general, talk about are limits.

After the reals are defined, we can then talk about a
sequence of rationals converging to an irrational.

A simpler way of saying what I think you are getting at is this: an
equivalence class in C/I represents a rational number iff the equivalence
class has a constant sequence as one of its members. Note that an
equivalence class has infinitely many (uncountably many) members, and at
most one of them can possibly be a constant sequence.

That would suffice.

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