Re: Review of Mueckenheims book.

mueckenh@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.

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. So, uncountable sequences exist. When we apply infinitude to points in "real" space, it behooves us to apply the finite unit, such that within each such unit of real space lie an infinite count of points, so as to wed count and measure with the introduction of an infinite unit. This is how I see the future of math...

Thoughts, Professor?

:)

Tony
.

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