Re: Logarithm of transfinite numbers

In article <MPG.1e9b09dd9411d2cf98abd0@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

Jonathan Hoyle said:
Do you ever remove more than one at a time? No. Does the vase
become empty? According to you.

So far we understand each other.

So, how many balls were in the vase right before it became
empty?

Please define "right before". You can answer that with an
Iteration # for some n, or if you prefer, give me a time T. Once I
know what you mean by "right before", I can happily give you
answer.

Before the last ball is removed, at noon.

There goes TO again, claiming the existence of a last natural. What
about its necessary successor?


Yes, you have a contradiction here. According to the gedanken,
you must have had -9 balls in order to empty the vase.

The gendanken says nothing of the kind. The only contradiction
here is with the false assumption that there is a last iteration
before the process ends. And this is a false assumption that you
are making, not I. The gendanken, once freed from your erroneous
proposition, quite contently and consistently empties all the balls
from the bin, one at at time, Ball #n at Iteration #n.

And yet, at each of those iterations, you are adding 10 more balls.
The gedanken states specifically that each iteration consists of the
adding of 10 balls and the removal of 1. I am not making any
assumption about that. That is the way the problem is stated. So,
logically speaking, if you wish to claim that the vase becomes empty,
it behooves you to address the contradiction that creates, wherein
the last ball removed has to have been preceded immediately by 10
balls added, and the resulting reuqirement that the vase previously
contain a negative number of balls.

Let us change the problem slightly by adding each ball enough earlier
than previously so that all balls have been added before any are
removed.

Adding balls earlier certainly cannot diminish the number left in at the
end. But it blows TO's argument away entirely. In this new scheme
clearly every ball is removed.


I didn't label the balls. I don't need labels.

No of course you don't! The labels are entirely your undoing here!
You want to change the problem and ignore the labels, because you
know (whether you wish to admit it publically or not), these labels
entirely throw out your whole theory.

No, the labels simply aren't necessary, and the introduction of the
transfinite treatment of the set of naturals into this problem only
serves to confound the issue with hocus pocus, while the answer is so
simple any child could tell you without coaching.

Then why does adding the balls earlier but removing them as before
earlier leave fewer at the end? Any child will tell you that with the
longer period between insertion and removal all the balls will get
removed.

The fact that the labels make a difference should be enough to
convince you the answer is misguided. If you change the labels
afterwards, do the 9n balls disappear from the vase?

With the increased time between insertion and removal, labelling does
become irrelevant. It is only relevant when a time of insertion and a
time of removal cooincide. But when labelling is irrelevant, TO is wrong.


You ask about the labels. I say they don't matter. The balls are
all labeled '1', okay. It doesn't matter. This is one of many
reasons I cannot ascribe to transfinite set theory.

So instead of honestly answering the question as given, with the
labels, you choose instead to pretend they don't matter. But if
they didn't matter, you wouldn't have a problem answering the
questions, would you? They matter all right. They matter so
deeply that your house of cards crumbles with their admission.
You so desperately want to avoid the labels, but I'm not letting
you. If they truly don't matter, then you should be able to answer
the damn question.

I have answered why the paradox arises. You make no distinction
between your mapping functions. The set of balls going in is 10 times
the size of the set coming out, but you have no means of
distinguishing between aleph_0 and 10 *aleph_0, so you erroneously
conclude those are the same value, and essentially do a subtraction
to get zero.

But inserting an interval between insertions and removals removes both
the need for labels and TO's arguments.

You are working within what SOUNDS like a distinct set,
and yet it has no distinct boundary

Endless ordered sets are like that, they do not contian upper bounds.
That is what makes them endless.



so you can play these games by
not treating the boundary as a variable

What is not there at all is not a variable, it is a "not there at all".



and looking at the infinity
formulaically. There are many such games played with transfinite set
theory, and I reject them as unsound.

And mathematics rejects TO as unsound.


If you stop ignoring the
functions that map your sets to each other, then you'll start getting
more sensible results.

So, what's your question? "Which" balls are in the vase? There is no
well defined end to your set, or I could answer that.

If there were any "end" to the set, there could be no such thing as the
set of natural numbers.





But you can't, can you? All these many months of posting and hours
upon hours of investing yourself into this group, you are finally
seeing a flaw, a crack in your armor. You would rather throw out
the thought experiment than face the possibility that this past
year's postings have all been in vain.

Relax. Your theory is to blame.

What "is to blame" is that our intuitions about finite sets and finite
procedures tend to mislead us when the sets and processes are no longer
finite.

It has numbers that aren't numbers.

It only uses the standard naturals, but TO has long been unhappy with
them.

There is no well defined end to the set

That is the whole point!

I keep telling myself, "this has to be a joke." I mean, when I
think about it, it makes me laugh, when it doesn't piss me off.

Don't piss your pants about what you can't understand, TO.
.

Relevant Pages

  • Re: Logarithm of transfinite numbers
    ... you must have had -9 balls in order to empty the vase. ... with the false assumption that there is a last iteration before the ... You want to change the problem and ignore the labels, ... It doesn't matter. ...
    (sci.math)
  • Re: infinity
    ... William above said the labels are ultimately unimportant. ... > easily demonstrate that there are certainly balls left in the vase at noon, ... > that your labelling IS the problem. ... Scheme A: At each step remove the ball that has been in the ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... you must have had -9 balls in order to empty the vase. ... with the false assumption that there is a last iteration before the ... You want to change the problem and ignore the labels, ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... then please explain how it gets empty. ... At *NO* point is there ever one ball in the bin. ... you must have had -9 balls in order to empty the vase. ... I don't need labels. ...
    (sci.math)
  • Re: infinity
    ... >>> without labels, labels are redundant, but labels are a useful, though ... >> easily demonstrate that there are certainly balls left in the vase at noon, ... >> for the same problem indicates that your labelling IS the problem. ... Virgil says, and it's true, the labels are only a handy guide to keep track of ...
    (sci.math)
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