Re: infinity

David R Tribble said:
>> At each step, you add ten balls, then remove one. That is very
>> different from simply adding nine balls at each step.
>
Tony Orlow (aeo6) wrote:
> You mean, when you hand the cashier a ten and they give you a buck in change,
> you didn't just spend 9 bucks?

Yep, I did. But let's be clear about how this happens: I give her
ten dollars, which she puts into her drawer, let's say on the top of
the stack of one-dollar bills in her drawer. Then she gives me one
dollar from the bottom of the stack, a bill that I previously gave
her.

So, yeah, at each finite step prior to noon, I'm out a net nine bucks.
At each _finite_ step _prior_ to noon.


> Maybe you should try doing that an infinite
> number of times, and you could own the world without paying anything.

Which is the whole point of the problem: an infinite number of times.

If I did the transaction described above an infinite number of times,
at noon the cash drawer would be empty and I'd have every dollar I
gave her (an infinite number of dollars) back in my hand; she would
have pulled every dollar I ever gave her from the bottom of the stack
and handed it back to me.

But I do have to pay something at each step; in fact, the whole
procedure costs me an infinite number of dollars.


>> It's that "remove one ball" part that is critical to solving the
>> problem. You can't simply ignore it.
>
> I haven't. You do it as many times as you add ten, for a net gain of 9.

At each finite step, yes. But there are an infinite number of steps.
You're getting tired of hearing this, aren't you?

The important thing is that I'm getting a different bill back at each
step (one that I gave her in a previous step) than the ones I'm giving
her at that step (except for the very first bill I get back, which was
the very first bill I gave her).


>> If all we did was add nine balls an infinite number of times, sure,
>> we'd get an infinite number of balls. But we're not just adding
>> balls, we're also removing balls as we go. An infinite numer of
>> them, in fact.
>
> So, it's different if you remove the ball from your handful, rather
> than from the vase?

Of course it's different. If I add ten dollars to the drawer, then
take back one of dollars I just gave her, and she adds the nine bills
to the top of stack in her drawer, I'm not getting back any of the
dollars at the bottom of the stack which I previously gave her.
So of course she'll end up with all my money, because there are bills
left in the drawer (an infinite number of them) that I never take out.
But that's not the same procedure as it was originally described.

.

Relevant Pages

  • Re: infinity
    ... >>> different from simply adding nine balls at each step. ... > the stack of one-dollar bills in her drawer. ... > Which is the whole point of the problem: an infinite number of times. ... > The important thing is that I'm getting a different bill back at each ...
    (sci.math)
  • Re: infinity
    ... >>> never get the empty set at any finite step. ... > Which balls are in the vase at state E. ... > the vase at any finite step. ... A larger infinity than the infinite number that have been ...
    (sci.math)
  • Re: INFINITY Revisited
    ... Are you then saying that by noon an infinite number of balls have been ... I believe it follows that balls with an infinite number of digits must ... nonterminating decimals. ...
    (sci.math)
  • Re: infinity
    ... >> You seem to have pulled the infinite series ... > there are nine more balls in the vase that there were at the end of the ...
    (sci.math)
  • Re: infinity
    ... >> infinite but at the end the number of balls is infinite. ... >>> possible to empty the vase by removing a ball at a time, ... >> somehow imagining that this scenario is physically possible. ...
    (sci.math)