Re: Logarithm of transfinite numbers

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,...}.

At none of those additions of an incremented value does the
value reach an infinite point OR the set reach an infinite size,
because those two numbers are always exactly the same,
being incremented in tandem.

True, but again, this applies only to sets of the form {1,2,...,n}.
Your induction proof says nothing about unbounded sets.

But, if all of the elements in the set are finite, and in positions
within the set equal to their finite value, then there exists no
element in the set which marks anything but a finite set, ...
<snip>

Correct.

...and there exists nothing else in the set besides its elements
which contribute to the size of the set.

Set size may or may not have any relation to the value of its elements.
Occassionally, sets do have such a relationship, such as sets of the
form {1,2,...,n}. The vast majority though, such as { 17, 101, 10^10
}, do not.

You do not have an infinite set of naturals in the quantitiative
sense until you have infinite quantities in the set.

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

When you use "finite" in the definition of a set, that is a weak
definition, because finiteness imposes a restriction that is not
well defined. There is no clear point where the finite becomes
infinite, so you have what appears to be some kind of hazy
infinity at the end which doesn't quite go on forever.

Infiniteness is not a quality that happens at a "point". It happens at
the end of all points. What you call "points" is analogous to what set
theorists call "successor cardinals", that is, cardinals which have an
immediate predecessors. Limit cardinals, such as Aleph_0, have no such
immediate predecessor (nor can it). That doesn't make it a "weak
definition"; it merely means that it does not contain properties which
you assume it does.

Between any two finite naturals is a finite number of naturals...

Correct.

...so there is no infinite sequence within the set of finite naturals.

Incorrect. Your conclusion does not follow.

Regards,

Jonathan

.

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