Date: Feb 3, 2013 6:40 AM Author: fom Subject: Re: looking for example of closed set that is *not* complete in a<br> metric space On 2/2/2013 8:41 PM, Shmuel (Seymour J.) Metz wrote:

> In <GbydnfG1Xq3Nu5HMnZ2dnUVZ_jqdnZ2d@giganews.com>, on 02/01/2013

> at 02:32 PM, fom <fomJUNK@nyms.net> said:

>

>> So, for example, there are "gaps" in the system of rational

>> numbers. One can, assuming completed infinities, define infinite

>> sets of rational numbers corresponding to the elements of a

>> Cauchy sequence.

>

> What would be the point? You're introducing extra machinery into what

> is a very simple construction.

That is not me. I would be happy to understand

the real numbers in the simple manner in

which it had been taught to me.

The construction of the reals from the natural numbers

is a sequence of logical types for which the order relation

of the natural numbers grounds the order relation of the

derived type.

You can find an excellent construction of the integers in

Jacobson:

Lectures in Abstract Algebra

Van Nostrand Co., Inc.

Princeton NJ, c. 1951

Each integer is an infinite class of pairs.

From this, the similar construction for defining

the rationals should be apparent using quotients

rather than differences. Some particular

may be needed for handling 0.

Once again, each rational is an infinite class of pairs.

From this, one constructs the reals. The following

is from Cantor's Grundlagen concerning the logic

of definition for a real number:

==============================

"I come now to the third definition

of real numbers. Here too an *infinite*

set of rational numbers of the first

power is taken as a basis, but it has

properties other than the Weierstrassian

definition. [...]

"Every such set (a_v), which also can

be characterized by the requirement:

lim_(v=oo) (a_(v+u) - a_v) = 0 (for arbitrary u)

I call a fundamental sequence and correlate

with it a number b, *TO BE DEFINED THROUGH

IT*, [...]

"Care must be taken on this cardinal point,

whose significance can easily be overlooked:

in the third definition the number b, say,

is *NOT DEFINED AS THE 'LIMIT' OF THE TERMS

a_v OF A FUNDAMENTAL SEQUENCE (a_v); FOR

THIS WOULD BE A LOGICAL ERROR SIMILAR TO

THE ONE DISCUSSED FOR THE FIRST DEFINITION,

i.e., WE WOULD BE PRESUMING THE EXISTENCE

OF THE LIMIT lim_(v=oo) a_v. BUT THE

SITUATION IS RATHER THE REVERSE.

and earlier...

"These definitions agree that an

irrational real number is *GIVEN BY

A WELL-DEFINED INFINITE SET OF RATIONAL

NUMBERS* of the first power. But they

differ over the way in which the set

is linked with the number it defines,

and in the conditions which the set

has to fulfill in order to qualify as

a foundation for the definition in

question.

"In the first definition a set of

positive rational numbers a_v is

taken as a basis, is designated by

(a_v), and satisfies the condition

that, whatever and however many of

the a_v are summed (so long as the

count is finite) this sum always

remains less than a specifiable

limit. [...]

"One sees here that the creative

element which binds the set with

the number defined through it lies

in the formation of sums; but it must

be emphasized that only the summation

of an always finite count of rational

elements is used *AND THAT THE NUMBER

b TO BE DEFINED IS NOT SET AT THE

BEGINNING AS EQUAL TO THE SUM Sum_v(a_v)

OF THE INFINITE SERIES (a_v); THIS

WOULD BE A LOGICAL ERROR,[...]

=================

By this account, the real numbers are a

logical type wherein each individual real

number is an equivalence class of fundamental

sequences.

I have no doubt that I am "wrong" in the

eyes of many. But it would be nice to

have that error explained.

At what point did we start ignoring the "foundations"

of mathematics because the logical rigor was

excessive? Believe me. I get the usual axiomatization

of the real numbers as presented in any decent calculus

text. But strict logical form that places the

foundations of mathematics within set theory using

only the axioms of set theory obtains each real number

through a sequence of definitions describing classes

of distinct logical types.

I am fully aware that much research on the

real numbers in set theory involves the isomorphic

set omega^(omega) which is closely related

to the description of real numbers relative to

infinite continued fractions (Baire space).

But, the sets involved in this isomorphism do

not have members identifiable as real numbers

relative to construction using continued

fractions. Such a definition would also

involve equivalence classes of infinite

collections of rationals.

Now, I certainly did make the mistake

of thinking that the original poster

had less knowledge than subsequent posts

indicated. But, to the best of my

knowledge, my remarks have been accurate

and to the point with regard to what

modern mathematicians claim to believe

when they say the things they say concerning

sets -- especially ZFC.

>

>> When the limit of the sequence is, itself, a rational number,

>> that infinite set becomes a representation of that rational number

>> in the complete space whose "numbers" are equivalence classes of

>> Cauchy sequences sharing the same limit.

>

> No; the set of values taken by the sequence is irrelevant. If a Cauchy

> sequence in the rationals converges then the *sequence* is a

> representative of its limit.

>

You are making precisely the logical error spoken of by Cantor.

>> When the limit of a Cauchy sequence does not exist as a rational

>> number, that Cauchy sequence becomes a representative of the

>> equivalence class of Cauchy sequences that cannot be

>> differentiated from that representative using the order relation

>> between the rational numbers of the underlying set.

>

> What are you trying to say? The definition of the equivalence relation

> is the same whether the Cauchy sequences converge or not; two

> sequences are equivalent if their difference converges to zero. Any

> Cauchy sequence is a representative of its equivalence class, by

> definition.

This is the point at which the order relation of the underlying

set of rationals is used to ground the identity relation for the

equivalence classes of objects defined individually as infinite

collections of rationals.

>

>> These "numbers" have no corresponding rational number as a limit

>> and are, therefore, distinguished as a different logical type in

>> the *new*, completed space.

>

> Non sequitor, and false. There is nothing logically special about

> equivalence classes of Cauchy sequences that converge.

Sadly I only speak English. But the false part I take

issue with. The entire problem since the discovery of

incommensurables has been to find a satisfactory logical

explanation for them. In Dedekind's definition using cuts,

there certainly is a logical difference between cuts containing

a maximum value (or minimum depending on which is used) and

those that do not.

And Cantor's careful distinctions have been made precisely

to accommodate the *DEFINITION* of real numbers, both rational

and irrational.

>

>> To call a subset of a complete space a dense subset is to say that

>> such a logical type construction could be made from that subset

>> to recover the original space.

>

> Completeness only applies to metric spaces[1], while denseness applies

> to subsets of arbitrary topological spaces. You can't in general

> reconstruct a topological space from only a dense subset, not even if

> the space is compact.

>

> [1] Well, slightly more general.

>

As I said when I concluded before, there are certainly

more knowledgeable topologists than I.

Yet,

"A uniform space is complete if and only

if every Cauchy net in the space converges

to a point of the space"

"A uniform space is complete if and only

if each family of closed sets which has

the finite intersection property and

contains small sets has a non-void

intersection"

"A pseudo-metrizable uniform space is

complete if and only if every Cauchy

sequence in the space converges to a

point"

And,

"A set A is dense in a topological

space X iff the closure of A is X"

All of these statements are from General

Topology by Kelley. The first and fourth

are definitions. Both contradict what

you have just stated. What you are thinking

of as a dense subset is a set dense in a

subset with the subspace topology.

As I said, I am probably wrong. But I try

to be careful.