Re: Logarithm of transfinite numbers

In article <MPG.1e8cb5b56f21361b98ab85@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Jonathan Hoyle said:
No, it inductively proves it for all set sizes which are
finite natural numbers. It is not proven (nor is it even
true) for sets of infinite size.

No, it proves for every natural number in the set that the set
up to and including that natural number cannot be infinite. So,
the set never is anything but finite, as long as you are adding
elements which are never anything but finite. See?

You are failing to see your own proof here. You have indeed proven
that sets of the form {1,2,...,n} are finite, for each and every
natural number n. That is what your induction proof demonstrated.
It does not however say anything about the set {1,2,...}.

It says soemthign about each and every n in that set, which are all
the points in the space of that set, that that space is not infinite
at any one of those points.

That "argument" manages to show only that certain Dedekind finite sets
are Dedekind finite.



Acutally, there is a universal rule about this. If a set has a
minumum finite difference between any two elements, or more
generally, has a finite average difference between successive
elements, then it can only have a finite number of elements in any
finite range.
But the set of von Neumann naturals as a whole does not have a "finite
range". It does not have a range at all that fits TO' definition of
range.




This is the basic assumption you are circularly making. You have
not proven it (nor can you since it is false).

It is only false in standard set theory,

And we do not have anyh toher kind to work with until TO completes his
system.



Infiniteness is not a quality that happens at a "point". It
happens at the end of all points.

Wording like, "at the end of all points" would get me excoriated. I
am not so particular about that wording, as it's more a matter of
political convention than anything. Still, I will disagree, and say
that infinity occurs, not at the end of all points, but within the
progression.

Wrong! "At the end of all points" is a possibly poor way of saying that
in determining infintieness one cannot count on being able to do it
without looking at the set as a whole and NOT merely ata parts of it.

TO seems to be saying that if you can't tell it is infinite by looking
only a a part of it the it can't be infinite. That is only valid for
dense sets like the rationals or reals. For discrete sets like the
naturals, you have to look at the whole, not merely at some part.

But TO is unwilling to do this, because it shows him to be wrong.
.

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