Re: infinity

snapdragon31 wrote:
> You are still at a very early stage of discussion on this topic. If
> you want to talk about it, please show me the proof. I want a concrete
> proof with no guessing.

You have a strange notion of "proof" incompatible with what is called a
proof in mathematics. In mathematics we do the following: We want to
show: "the vase is empty at noon". So let V(n) be true if ball n is in
the vase at noon, let R(n) be true if ball n was removed before noon.
Now the statement that the vase is empty at noon is the statement that
"for all n in N, not V(n)". So let's negate this: "the vase is not
empty at noon" then becomes: "there exists n in N such that V(n)". Now
either the vase is empty at noon or it is not. If you claim that the
vase is not empty, then there must exist a specific n such that V(n).
That is precise logical statement. But that's a sidenote, let's show
that the vase IS in fact empty. Note that "R(n) implies not V(n)",
because no ball that is removed is ever put back. Now we know by the
process that "for all n, R(n)," because we go though all the n before
noon and for each n we remove the ball. Now "for all n, R(n)" and
"R(n) implies not V(n)" simply implies that "for all n, not V(n),"
which is precisely the logical statement that "the vase is empty."

Now I can't imagine there being a more rigorous, elementary, direct
proof of this. It is not a guess, it is a proof. All the people
arguing against such a proof have at some point invoked machinery they
don't quite understand such as infinite series and limits, and most
important, they have invoked their finitistic real world intuition and
assumed it works even for infinite cases.

You and Tony and many others often mistake "intuition" for "logic."
Just because something seems like it "ought to be" in some particular
way, doesn't mean that it logically "is" that way.

> Do worry, I would not force you to answer the second question. I just
> want to let you know that there is a strong agrument to support that if
> you do not know which ball is the last ball then the vase is not empty.
> I am not looking for strong agrument for question 1 and question 2. I
> want concrete proof.

You got a concrete proof. Your question 2, which asks for the last
ball, is a nonsensical question since it assumes the existence of such
a thing as last ball, so it is two questions at once, the second one
making no sense if the existence one is answered with a negative. You
and Tony assume that if all balls have been removed, then there has to
have been a last ball that was removed. Again finitistic intuition,
not logic. Your question two is akin to "Is the dog's escape hatch
painted with strawberries?" It is hard to answer once you realize that
the dog, not being a submarine, has no escape hatch. You have
conjectured "if all balls have been removed, then there is a last ball
removed." Then if in fact that would be true, you would be home free,
but that statement would require proof, and there is no proof since
it's a false statement.

You have concrete proof above that the vase is empty. It is also very
easy to see that there is no last ball removed since you (and tony and
others) have argued this point several times. Obviously I don't expect
you to accept this proof because you already have a preconcieved notion
of what the answer should be, and so I expect you to find some
rediculous "flaw" with the proof and reiterate that "if you keep adding
9 balls at every step you can't have an empty vase" or some other vague
argument like "there would have to be a last ball removed."

Jiri

.

Relevant Pages

  • Re: An uncountable countable set
    ... the vase, is consistent with the fact that no balls are removed at noon? ... The only relevant question is "According to the rules set up in the ... is each ball inserted before noon also removed before noon?" ... An affirmative answer confirms that the vase is empty at noon. ...
    (sci.math)
  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... But in order for the vase to transition from not-empty ... If the vase ever became empty, ...
    (sci.math)
  • Re: An uncountable countable set
    ... Does Han claim that there is any ball put in that is not taken out? ... the vase empties". ... If the vase ever became empty, ...
    (sci.math)
  • Re: infinity
    ... Number of balls in the vase at noon is f= OO. ... Unfortunately, if infinity gets involved, this statement alone is not sufficient to claim the vase is empty before noon. ... ball) is in the vase. ... You also keep ignoring my question. ...
    (sci.math)
  • Re: An uncountable countable set
    ... The only critical time dependency is that each ball to be inserted shall ... the vase is empty at noon of anything of any balls ... An affirmative answer confirms that the vase is empty at noon. ... given the times of insertions and removals. ...
    (sci.math)