Re: Logarithm of transfinite numbers

In article <442d78c2$0$20703$6d36acad@xxxxxxxxxxxxxxxxxxxxx>, Matt Gutting <tchrmatt@xxxxxxxxx> writes:
Tony Orlow wrote:
Jonathan Hoyle said:
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?

There are not a quantitatively infinite number of finite natural numbers. There
CAN be an infinite number of finite numbers in a set, IF the set is dense at at
least one point. The naturals are everywhere sparse.

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.

Yes, and at any given finite n, you have 9n balls in your vase. Are any of
those n equal to 0?

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.

Uh, okay.

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.

Okay.

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?

9*aleph_0 balls, numbered aleph_0 through 10*aleph_0, if the gedanken makes any
sense at all. This analysis suffers from the same issues that your normal
bijections do. There is no consideration of the mapping function. It just goes
off forever into the distance, and you declare the input and output the same.
If you pour ten gallons of water into a lake for every gallon that flows out,
it will never empty, ever.

If you think that this process could ever result in an empty vase, then please
explain how it gets empty. You remove only one ball at a time, so at some point
there must be 1 ball in the vase for it to empty. But, before removing that
ball, you had to have just added 10 balls to get to that 1. Were there ever -9
balls in the vase? No, that's impossible, and the answer according to set
theory is, pardon me, simply ludicrous. It's conclusions like this that
motivate me to replace this theory with something sane. I simply cannot accept
that this artificial notion of labeling the balls makes any difference in what
is obviously a sum of an infinite number of 9's. I mean, set theorists admit
that the same process of adding and removing leaves 9/10 of the balls, if the
labels are just rearranged. Can you rearrange the labels afterward, and have
those 9/10 of the balls disappear? If that's not hoocus pocus, I don't know
what is. Sorry.

Jonathan Hoyle



I can't think of a ball that goes in that doesn't come out; by my lights that
means every ball comes out.

When I think of this problem, I tend to think in terms of a state machine.

We have a vase. And its state is characterized by the set of balls
contained therein.

We have a set of transitions where the state of the vase changes as balls
are inserted and removed.

And then we are asked a question that relates to the final state of the
vase.

The problem is that, when looked at in this way, the final state of the
vase is undefined. The final state is not an immediate successor
to any prior state.


Most of the mathematicians in this thread tend to skip past that
intuitive picture and look at the problem as if it were presented (as it
may indeed have been if the presenter was careful) in terms of the
limiting position of each individual ball.


The resulting paradox is nothing more than a commutation of limits
prettied up to make a false equivalence between the two approaches
intuitively appealing:

sum lim (if n <= i < 10*n then 1 else 0)
i = 1, oo n->oo

[This sum is zero. This is the "where is each ball when we're done"
approach]

is not equal to

lim sum (if n <= i < 10*n then 1 else 0)
n->oo i = 1, oo

[This limit is infinite. This is the "but the number of balls in the
vase goes up and never goes down" approach]

Folks with quantifier dislexia will naturally feel troubled by this and
want to fix things up so that both approaches give the same result.
.

Relevant Pages

  • Re: infinity
    ... >> I is the union of a bunch of sets. ... >> Define I_n to be the set of balls added at step n. ... by definition the vase is empty at state E. ... > are an infinite number of sets I_n", ...
    (sci.math)
  • Re: infinity
    ... >>> Which axioms allow completion of an infinite ... That's what a sequence is, by the way: ... > If you do not interrupt the process, the vase never "reaches" noon. ... > where xis the number of balls labeled i. ...
    (sci.math)
  • Re: An uncountable countable set
    ... The number of balls approaches infinity as time ... So, David, you think the fact that balls leave the vase only by being ... from infinite series, ... Very basic logic would hold that, if the vase is not empty at any time t ...
    (sci.math)
  • Re: infinity
    ... the argument that the vase is empty does not rely on any ... >> axioms that complete infinite sequences. ... > adding balls to it. ...
    (sci.math)
  • Re: infinity
    ... >> is to show that each ball is removed from the vase before noon and is not ... > is not empty at noon. ... # of balls in the vase can be ... there are an infinite number of steps and this statement ...
    (sci.math)
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