Re: Review of Mueckenheims book.

cbrown@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:
mueck...@xxxxxxxxxxxxxxxxx wrote:
On 8 Mrz., 22:46, Tony Orlow <t...@xxxxxxxxxxxxx> wrote:
mueck...@xxxxxxxxxxxxxxxxx wrote:
WM, you don't disagree that there are infinite sets containing just
finite values, such as the reals in [0,1], are you? I certainly agree
that an infinite set of naturals must contain infinite values, but
that's only because they are spaced apart by a unit in value. Isn't tat
your thinking?
If you disregad physical restrictions, then there are infinitely many
real numbers in the interval. Their cardinality, however, is not
larger than "infinite" for any set. Therefore we need no alephs etc.
The binary tree shows that different alephs are self contradictive.
If you take into account the physical restrictions, then there is no
infinite set. And that is the only correct approach.
Regards, WM
Well, since numbers are not physical entities, they don't actually
occupy space on the number line - they are true points. So, between any
two finitely distant points are indeed some infinite number of points.
You say that the only correct approach is to take into account
"physical" restrictions, but where the subject is non-physical, those
restrictions don't exist, though relations do, even if between infinite
nonphysical concepts called numbers.
Wow! That actually made sense for an /entire/ /paragraph/.
:)

DM said, "Wow! That makes even less sense than WM's posts. Although, it
doesn't quite reach the heights of Ross's nonsense."


I think he was referring to the content of the following paragraph.


Perhaps, but he didn't comment on the first. :(


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. :)

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.

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. 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....

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.

Smiles,

Tony
.

Relevant Pages

  • Re: Review of Mueckenheims book.
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  • Re: Review of Mueckenheims book.
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  • Re: Review of Mueckenheims book.
    ... you don't disagree that there are infinite sets containing just ... finite values, such as the reals in, are you? ... that an infinite set of naturals must contain infinite values, ... If you disregad physical restrictions, ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... you don't disagree that there are infinite sets containing just ... finite values, such as the reals in, are you? ... If you disregad physical restrictions, ... And the adics each ...
    (sci.math)
  • Re: Review of Mueckenheims book.
    ... you don't disagree that there are infinite sets containing just ... finite values, such as the reals in, are you? ... that an infinite set of naturals must contain infinite values, ... If you disregad physical restrictions, ...
    (sci.math)