fom
Posts:
1,033
Registered:
12/4/12
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Re: looking for example of closed set that is *not* complete in a
metric space
Posted:
Feb 3, 2013 6:40 AM
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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.
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