Re: Logarithm of transfinite numbers

For example, you argue that it is impossible to have an
infinite number of finite numbers. This is simply untrue,
even using your definitions.

I never made that statement, and have repeatedly
corrected it.

Perhaps I have misunderstood you then. Let me ask you directly: Are
there, or are there not, an infinite number of finite natural numbers?

Let me drive the point home with the original bucket and balls
problem. Your claim is that at the end of the process, there
are an infinite (?) number of balls of undetermined size, do I
have that correct?

No, I specifically stated that given n iterations you will have 9n
balls in the vase, whether n is finite or infinite.

Okay, perhaps I have misunderstood you here as well. Let me again be
very specific:

We define Iteration #1 to perform two steps: first add Balls labelled
#1 through 10, and then remove Ball #1. We define Iteration #2 to add
Balls #11 through 20, and remove Ball #2. In general, Iteration #n
involves adding Balls #10n-9 through #10n into our bin, followed by the
removal of Ball #n, for each finite natural number n. We do nothing
(no adding, no subtracting) when n is anything other than a finite
natural number.

Furthermore, we add a filtering mechanism. By rule, every ball we add
to the bin must contain a label with a finite natural number on it, or
it is ignored. If we come across a ball containing no label, or a label
that has an infinite value, or empty label, or whatever is not a finite
natural number, we filter that ball out, and it never gets put into the
bin. This rule ensures that the bin always contains only balls with
finite natural labels on them.

We begin the clock at time T-1 seconds with an empty bin. We start by
performing Iteration #1 at time T-1/2 second, do Iteration #2 at time
T-1/4 second, and for each finite natural number n, we perform
Iteration #n at time T-1/2^n seconds. We stop the process at time T.
I reiterate again that by rule that any infinite n means the Iteration
does nothing.

Now at time T, with the process having been stopped, all iterations
have taken place for Iteration #n for every n which is a finite natural
number. Furthermore, no other n was involved.

So here we are finally. At the end of time T, Tony, is the bin empty
or not?

And if not empty, what is in there?

Jonathan Hoyle

.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... infinite number of finite numbers. ... balls in the vase, whether n is finite or infinite. ... We define Iteration #1 to perform two steps: ... We begin the clock at time T-1 seconds with an empty bin. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... infinite, because of an injection into a proper subset. ... Let me drive the point home with the original bucket and balls problem. ... fairly easy to determine if a ball has a stripe on it: ... removed at any iteration, it will be without a stripe.) ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... infinite number of finite numbers. ... balls in the vase, whether n is finite or infinite. ... We define Iteration #1 to perform two steps: ... Yes, and at any given finite n, you have 9n balls in your vase. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... Since, by rule, no ball with an non-finite label is allowed to be put ... finite natural numbered Iteration is run and so these balls would have ... no non-finite numbered ball is ever added to the bin. ... your claim that those balls are in the bin is in direct ...
    (sci.math)
  • Re: infinity
    ... >> balls with an integer written on each one, ... >> you will never get an infinite value. ... > I am not sure, if this is the same, as Tony is saying, try to think about it and tell me. ... iteration takes a constant finite time, the process will never end, becuas ethe ...
    (sci.math)