Re: Review of Mueckenheims book.

On Mar 17, 4:57 pm, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
cbr...@xxxxxxxxxxxxxxxxx wrote:
On Mar 17, 8:59 am, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
cbr...@xxxxxxxxxxxxxxxxx wrote:
On Mar 13, 9:22 am, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
cbr...@xxxxxxxxxxxxxxxxx wrote:
On Mar 12, 2:11 pm, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:

Where we can count in sequence from one element to any other, that
neighborhood is finite, even if unbounded. Where we can never count
between some pair of objects, such as between, say, ...1111 and
....2222, they are actually infinitely distant elements of a sequence,
since successor() exists.
Ermm... what th'!?!
Cheers - Chas
Was that a question?
One question might be "what is an example of a neighborhood that is
finite yet unbounded?"
The atmosphere. :)

In what sense is the atmosphere finite? In what sense is it unbounded?
If the atmosphere is finite, then there is a molecule of the
atmosphere which is the farthest from Earth. A distance greater than
that distance is a bound on the atmosphere, no?


No comment?

If you add 0 forever, you will never get anywhere, right? If you append
points to points, you can never make a line of any length, right? But,
you have lines of finite length, and they have an uncountable number of
points within them. No countable number of points can constitute a line
segment of any length, but an uncountable number can, in theory.

Consider this an analogy to the addition of finite units. No countable
number can achieve any infinite measure, where such a thing is properly
established. Aleph_0 is, the way I see it, the equivalent of the
smallest positive number, on the infinite scale. The smallest positive
number does not exist, finite or infinite.

So, "countably infinite" means "finite but unbounded" to me.

So in your terminology, the distinction between something which is
finite and something which is not finite is so vague that a
mathematical object can simultaneously have the property of being
finite and also /fail/ to have the property of being finite.


No comment?

That argues for your perhaps developing a new term that corresponds to
what is usually meant by "finite", so that the term actually /
describes/ the property of being "finite" in the usual sense.
Otherwise, I have no idea what you mean by "finite". In the usual
sense, it is not possible for a mathematical object to be both finite
and not finite.

In a countable set, there are only a finite number
of elements between any two specific elements.
Counterexample: The set of rationals in [0,1] is a countable set, and
there are an infinite number of elements between any two distinct
elements of that set.
Not in the order in which they are countable.

Is 1/2 between 1/4 and 3/4 "in the order in which they are countable"?

Starting at 1/1 and traversing the standard table diagonally to make a
sequence, no, 1/2 is reached before 1/4, which is reached before 3/4:

1/1 1/2 1/3 1/4 ...

2/1 2/2 2/3 2/4 ...

3/1 3/2 3/3 3/4 ...

4/1 4/2 4/3 4/4 ...

5/1 5/2 5/3 5/4 ...

You cannot state two rational numbers included in Cantor's diagonal
proof of their countability which are infinitely distant from each other
in that sequence.

Yes that is /an/ order in which 1/2 is not between 1/4 and 3/4. You
claimed that it was /the/ order in which they are are countable; and
no such /unique/ order exists.




Why don't you give me two
rational numbers, which are not finitely distant in the
pseudo-sequential non-quantitative (read, bogus) ordering of the
rationals through sparse diagonalization? Which comes infinitely beyond
any other, in that "countable" sequence? Hmmm....

So you yourself claim that your statement only makes sense if we use a
"bogus" ordering of the rationals? Hmmm indeed!

That's the ordering used by Cantor to prove their countability. What I
said was that no two elements in any countable set are infinitely
distant from each other in any linear ordering that makes them
countable.

Actually, what you said was:

In a countable set, there are only a finite number
of elements between any two specific elements.

And that was insufficient to make sense out of your statement.

If what you /intended/ to say was:

If a set S is countable, then there is a total order < of S such that
for all distinct x, y in S with x < y, there are at most a finite
number of elements z such that x < y < z.

then I would of course agree.

If this sequential order is valid for the rationals, then
this fact applies. If you have a problem with the fact that there are an
infinite number of rationals between any two in their natural
quantitative order...

Why would I have "a problem" with that statement? I'm the one who
mentioned it.

In fact one can equally say:

If a set S is countable, then there is a total order < of S such that
for all distinct x, y in S with x < y there are an infinite number of
elements z such x < z < y.

But that isn't the same as saying: In a countable set, there are an
infinite number of elements between any two specific elements.

, then you should consider the possibility that
non-quantitative orderings of subsets of the reals are not suitable for
relative measure of those subsets.


Erm? What th'?!?





There are an infinite
number of adic numbers between ...111 and ...222, no? And the adics each
have a distinct successor, yes? What was the question?
Another question might be "are you aware of the difference between the
definitions of a list of elements from a set, a total order on the
elements of that set, and a well-ordering of the elements of that
set?"
Cheers - Chas
Make it relevant to the topic, and we'll discuss that.

You said:

"Where we can never count between some pair of objects, such as
between, say, ...1111 and ...2222, they are actually infinitely
distant elements of a sequence, since successor() exists."

By "since successor() exists", you seem to imply that:

"/Because/ successor() exists, they are actually infinitely distant
elements of a sequence".

Since successor(x) is defined for every such string, it constitutes a
sequence.

Are you aware of the definition of the term "sequence"? It appears
not.

Since there exist elements more than any finite number of
successions from each other, it is uncountable.


Let succ(x) = x + 2. Using this definition, is not succ() a successor
function on the naturals, with two elements (0 and 1) having no
predecessor? What "number of successions" is it from 0 to 1? Is N
therefore uncountable? I.e., is N is a set which has the property of
being countable, and which also /fails/ to have the property of being
countable?



But the existence of a successor function does /not/ imply a set is
(or can be made into) a sequence or list, in the usual definition. It
doesn't even imply a set is infinite, in the usual definition. Nor
does it necessarily impose a total order; it may be at best a partial
order.

If xeS -> s(x)eS, and ExeS AyeS s(y)<>x, then you have a countably
infinite sequence at the very least.

S = {0,1,2}; s(0) = 1, s(1) = 2, s(2) = 1 is a finite set satisfying
the above premises.

But you get 10 points for actually producing a clear yet false
statement, as opposed to gibberish! Seriously, that is progress!

The question is whether one can
have an uncountable sequence, which the adics clearly are.


No, the question is "what do you mean by an uncountable sequence, of
which you assert the adics are an example?" A sequence (by any
definition I know) is countable, because it is always basically
equivalent to some function f : N -> S, and N is countable. You
obviously don't mean "an uncountable sequence is a sequence (which is
countable by definition) which is not countable", because that would
be silly. But what /do/ you mean?

You /might/ mean "suppose C is an uncountable set with a total order,
then an uncountable sequence on S is a function f : C -> S", but there
are /many/ different types of sets which are uncountable, and have a
total order, but whose order types are "as different" as the order
types of N and Q.



Let T be the set of all triangles in the plane. If t is a triangle,
define successor(t) to be t after being rotated around the origin by
pi/sqrt(2) radians. Every t has a unique successor. Every triangle t
has a unique triangle for which t is its successor.

Indeed, and it constitutes an uncountably long sequence which is
actually circular.


Erm? What th?'?



Despite the existence of successor(), T is not a sequence; nor can it
be made into a sequence, because T is uncountable

Uncountable sequences aren't a problem for me.

Well, "uncountable sequences" may or may not be "a problem" for anyone
else either; of course, depending on what the heck you're talking
about.

The only thing that
distinguishes this from a sequence for me is the fact that every point
is a successor as well as having one. But, like the integers, where this
may also be considered true, it's not surprising that an overall
circular nature may manifest itself.


What is "circular" about the ordering of integers?

Yes, poetically speaking, a circle has no beginning or end; and the
integers have no begininng or end. Also, my love for my niece has no
beginning or end. Is that all you mean to say?

Also, we can certainly define "t1 < t2" as "there is a natural number
n such that t2 is the result of applying the successor function n
times to t1". But there are triangles t and u such that it is not the
case that t < u or u < t or t = u.

Right. Some will be uncountably many successions beyond any other.

No.

Firstly, and most obviously, there is /no/ number of applications of
rotating an equilateral triangle by /any/ amount which will yield a
non-equilateral triangle. So we cannot say (purely from the ordering
imposed by succ() on T) that an equilateral triangle is <, >, or = a
triangle which is not equilateral.

Secondly, because sqrt(2) is irrational, there is no number of
successive applications of succ() to a triangle t such that we arrive
at the triangle t rotated by pi/(10^(10^1000)) radians. So these two
triangles t, and t rotated by pi/(10^(10^1000)), while in some sense
being "very, very close" to being "the same" by "distance", are still
incomparable as regards my given definition of <, >, = as derived from
succ().

You
can define '<' in this way, or you can define "t1 < t2" as "t2-t1>0".

Erm? What th'?!? We can?

Let t1 be the unit equilateral triangle with 1 vertex at the origin
and base on the x-axis. Let t2 be the right triangle with sides
1,1,sqrt(2)/2 having its right angle at the origin. Is t1 - t2 > 0? Or
is t1 - t2 <= 0? How are we to know for arbitrary triangles s, t,
except by directly asking you?

And how does the ordering you claim we can define relate to your claim
that "/because/ there is a successor function, blah blah..."? The
question isn't, "/can/ we somehow totally order T?", because that is
fairly obviously true given AofChoice for any set (and thus trivial).
The question is "is there a specific total ordering on T, /since/
succ() exists?"

If
t2 is infinitely past t1, then t1<t2. Like I said, the problem in this
case is the circularity of the uncountable system.


If t1 is some equilateral triangle, and t2 is t1 rotated about the
origin by pi radians, then is t1 "infinitely past" t2, or is it the
other way round? t1 is not a member of the sequence (t2, succ(t2),
succ(succ(t2)), ...); nor is t2 a member of the sequence (t1,
succ(t1), succ(succ(t1)), ...). So how are we to know, except by
directly asking you?

Again, the question isn't "/can/ we somehow totally order T?", because
the answer is trivially "yes". The question is how do you suggest we /
define/ this total order using /only/ succ(), when you aren't around
to help us?



These observations arise from the definitions of "sequence" and "total
order"; which is why I asked: ""are you aware of the difference
between the definitions of a list of elements from a set, a total
order on the elements of that set, and a well-ordering of the elements
of that set?"

Cheers - Chas

So, what do you call what I am calling an uncountable sequence? A
nonexistent concept?

No, I would instead call it a poorly-defined concept.

Given your fixation on "succ()", the best I can understand what you
are thrashing about for by "uncountable sequence" is a function which
is indexed by an ordinal A whose /ordinality/ is greater than omega.
By extension of the example that f : N -> S is a sequence of S, you
want "uncountable sequence" to be a function f : A -> S, where A is an
ordinal and omega in A (i.e., omega < A where < is defined by set
inclusion); and that gives us a function which is "like" a sequence,
only "longer".

But there are ordinals greater than omega in the /ordinality/ sense
(e.g., the ordinal succ(succ(omega)) > omega) which have the same /
cardinality/ as omega; and that is where you begin getting mixed up.
Such ordinals are /not/ uncountable, they are /countable/. There are
indeed uncountable ordinals; but countable/uncountable does /not/
refer to comparison by /ordinality/, rather by /cardinality/.

This becomes further confused when you then mix in other order types,
such as the orderings of Q and R and the adics, which are not orders
which can be compared with ordinals by their /ordinality/ because Q, R
and the adics are /not ordinals/ (nor are they order-isomporphic to an
ordinal).

And your concept of "succ()" is vague enough that it ends up including
order types where succ() does not even impose a total order, as in the
examples S and T I gave above (which are pre-orders and partial
orders, resp.).

Thus my question, which I will expand slightly: "are you aware there
is a difference between the definitions of a sequence, a cardinal, an
ordinal, a well order, a total order, a partial order, a pre-order?".

Every sequence is indexed by a cardinal (omega). Every cardinal is an
ordinal. Every ordinal imposes a well order. Every well-order is a
total order. Every total order is a partial order. Every partial order
is a pre-order.

But:

Not every pre-order is a partial order. Not every partial order is a
total order. Not every total order is a well order. Not every well-
order is order-isomporphic to an ordinal. Not every ordinal is a
cardinal. Not every cardinal is the index of a sequence: only omega
fits that bill.

Cheers - Chas

.

Relevant Pages

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