Re: Two results of set geometry

WM wrote:
On 12 Sep., 14:11, William Hughes <wpihug...@xxxxxxxxxxx> wrote:
On Sep 12, 7:31 am, WM <mueck...@xxxxxxxxxxxxxxxxx> wrote:

But a stronger result is that
lim{n --> oo} |{2, 4, 6, ..., 2n}| / 2n < 1.
So there is always, i.e. for any n, a greater natural number than
|{2, 4, 6, ..., 2n}|,
Yes. Note that every sequence of this form has a last element.

and hence, there is a greater natural number
than
|{2, 4, 6, ...}|
No. Note that this sequence does not have a last element.

Note, however, that it has only natural numbers as elements - and only
that is important.

You cannot assume that something that is true for sequence
with a last element must be true for sequences without
a last element.

But you can assume that every element of N is a natural number and
nothing else. For any set consisting only of even natural numbers I
proved that there is a nunber contained which is greater than the
cardinal number of the set.

Regards, WM


Hi Wolfgang -

After being compelled to respond to William Hughes' comment on Orlovia, I am inspired to respond to this post of yours. I have made the argument numerous times, and been accused to no avail of quantifier dyslexia, that the fact that NO element within the set can possibly be in any infinite position within the set makes the set NOT actually infinite to me. That there is NO location, NO element even possible within the set, where it attains "infinity", makes it at best potentially infinite, which is apparently what "countably" infinite means. To me, that's fine. The set is unbounded, but not infinite in the sense of being greater than any finite value. It is greater than OR EQUAL to any finite value. None of that gives us any actual measure for the set. It's all in the undefined outer limits of some mathematical atmosphere of the planet Zero.

So, you do not believe in infinities... Do you believe in infinitesimals? Probably not, but I'd like to hear your thoughts on such matters, and promise as always to respect what I hear, and respond accordingly. In the meantime, do take care.

Peace,

Tony
.

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