A cardinal is a number, not a set.

Cardinals: one, two, three, ..., aleph₀, ...
Ordinals: first, second, third, ..., (often) ù, ...

>2) Let A1, A2, ... be some infinte cardinal subsets of A
>which does not contain commonn elements with each other
>(they do contain common elements with A however). For
>exemple think of A as the positive natural numbers, of A1 as
>positive even natural numbers and of A2 as positive uneven
>natural numbers
>3) Since A1 is an infinte subset, then there is one-on-one
>correspondence between A and A1. So # elements in A = #
>elements in A1. Same is true for A2. So, if we take out A1
>from A there will be still an infinite number of elements of
>A (Something like Hilbert's hotel paradox?
>https://en.wikipedia.org/wiki/Hilbert%27s_hotel)

OK so far.

>4) Dynamically, at least wihtin the finite realm (can we
>talk of the passage finite - infinite here? can we talk of
>dynamics for the infinite cas? I think we can't...)

Do not even know what this means.

>Gennie's
>and Jenny's strategies seem similar: for both # balls in the
>urn rises steadily. There must be some point however from
>which Jenny's balls in the urn should start to decrease...

Do not know about that. You appear to apply "finite" reasoning to infinite sets.

>Maybe this point is inside the infinite set? How fast is
>this decrease? Maybe it is instant...

Truly do not know. It's hard to imagine how things actually (whatever this may mean) happen. This is why the whole story is referred to as a paradox.

>
>5) The original article mentions a much stronger reusult
>which lead to zero balls as did Jenny's strategy
>(https://www.cut-the-knot.org/Probability/infinity.shtml).
>Actually, we can allow Jenny to choose randomly the balls at
>each period (accepting that she attributes equal probability
>for the balls in the urn). Probability to end up with zero
>balls is still 1.

Yes. For each ball, the probability to stay in is 0.

>On the other hand, we can also generalise Gennie's strategy
>as well. What if Genny chose, at each period to take out
>randomly one of the last ten balls to enter the urn? I think
>that her urn will still be infinitely laden.

Yes, of course.

>6) One last thing (for alexb) Please reexplain your argument
>about cardinals and sets. I'm afraid I've stopped following
>you from the line where you wrote:
>a-b = a if a>b

A cardinal number is a common attribute of sets in 1-1 correspondence. "Infinity", as a cardinal number, is undefined because there are many infinities.

aleph₀ is the cardinal number of the set of integers (and also the set of even integers, the set of whole numbers, the set of irrational numbers, the set of algebraic numbers, and many others.)

This is the smallest infinite cardinal. There are other "alephs". The rule for the addition of infinite cardinals says

cardinal₁ + cardinal₂ = max(cardinal₁, cardinal₂).

It is sort of "many to one" operation. This is why the reverse operation of subtraction is said to be "undertermined."

But note that you can always subtract one set from another. Their difference will be a set of certain cardinality. For two sets say C₁ and C₂ with cardinalities c₁ and c₂, the cardinality of the difference may be any cardinal number from 0 to max(cardinal₁, cardinal₂).

>I'm aware, that we should reject the axiom that "the whole
>has more terms thant the part", as B. Russell put it.

For finite sets this is true. So this is in fact how you define an infinite set: a set is ininite if it's in a 1-1 correspondence with its proper part.

We can do some basic operations on finite numbers, like addition and subtraction. These are intended to represent operations we can do with finite sets, like taking the union of sets (if they have no elements in common) or removing subsets. The same operations on sets may of course be extended to operations on infinite sets. But, and this is important, this does not mean that the operations on finite numbers extend to operations on infinite cardinal numbers. With some care, you can show that addition does extend in this way. But subtraction does not. That is part of the paradox.

We are not talking about an infinite cardinal set A, but rather about an infinite set A. To put it another way, we are talking about a set A whose cardinal number is infinite.

>>4) Dynamically, at least wihtin the finite realm (can we
>>talk of the passage finite - infinite here? can we talk of
>>dynamics for the infinite cas? I think we can't...)
It is important to remember also that infinite sets are objects in their own right. That is, a set should not be confused with an infinite process by which it could be built. The paradox deals with both sets and processes: Processes have dynamics but sets do not. Nor do numbers. So we must be careful not to confuse any of the three closely related ideas mentioned; sets, processes and numbers.

>>Gennie's
>>and Jenny's strategies seem similar: for both # balls in the
>>urn rises steadily. There must be some point however from
>>which Jenny's balls in the urn should start to decrease...
This is not true.

If we analyse the process taking place here, it is clear that throughout the process the number of balls in Jenny's urn is increasing, just as is true for Genie. This process does not include a point at which the number of balls in Jenny's urn begins to decrease.

If, on the other hand, we analyse the set of balls finally in Jenny's urn, we find that it does not contain any of the balls. A good way to see this is to think about it from the point of view of an individual ball. Any particular ball is put in, and then, a while later, is taken out again. So it ends up not being in the urn. The set of all balls that are left in the urn forever is empty. But there is no point at which the number of balls in this set decreases: It is a set, and not a process.

Finally, this shows that the number of balls remaining in the urn is 0. This number is directly related to the set, not the process, and presents a striking contrast to the increasingly large numbers associated with the process. Hence the paradox.

>>On the other hand, we can also generalise Gennie's strategy
>>as well. What if Genny chose, at each period to take out
>>randomly one of the last ten balls to enter the urn? I think
>>that her urn will still be infinitely laden.
You are right, newscam9, as alexb has mentioned. Can you think of a way to prove this?

>aleph₀ is the cardinal number of the set of
>integers (and also the set of even integers, the set of
>whole numbers, the set of irrational numbers, the set of
>algebraic numbers, and many others.)
Of course, you mean the set of rational numbers.

>But note that you can always subtract one set from another.
>Their difference will be a set of certain cardinality. For
>two sets say C₁ and C₂ with
>cardinalities c₁ and c₂, the
>cardinality of the difference may be any cardinal number
>from 0 to max(cardinal₁, cardinal₂).
By 'difference' here, do you mean the symmetric difference? With the more usual meaning, if c₁ is strictly bigger than c₂ then the cardinality of their difference is c₁. Otherwise the cardinality of the difference could be anything in the range from 0 to c₁. This seems to fit a little better with what you wrote in your original post.

Thankyou

sfwc
<><

>Of course, you mean the set of rational numbers.

Yes, of course. Sorry and thank you.

>>But note that you can always subtract one set from another.
>>Their difference will be a set of certain cardinality. For
>>two sets say C₁ and C₂ with
>>cardinalities c₁ and c₂, the
>>cardinality of the difference may be any cardinal number
>>from 0 to max(cardinal₁, cardinal₂).
>By 'difference' here, do you mean the symmetric difference?

Does it really matter? Ah, because of the symmetry in max(.,.), here it does does. Originally I meant the common difference.

>With the more usual meaning, if c₁ is strictly
>bigger than c₂ then the cardinality of their
>difference is c₁. Otherwise the cardinality of
>the difference could be anything in the range from 0 to
>c₁.

Right. The point to be made is that the cardinality of the difference could be anything in an infinite range, which makes the difference of the corresponding cardinals indeterminate.

I think a way to do this is to prove that

Prob<#"balls that are NEVER taken out of Gennie's urn"<M>=0
for M any positive integer

Since M is finite, we can be sure that after periods there will be at least M balls that will stay permanently in the urn.

As the procedure of piling balls in the urn goes to infinity, number of "permanent" balls will be greater than any finite number, which is another way to say that Gennie's urn should be infinitely laden with balls.

Thank you guys for your insights on this paradox and on infinite sets.

I tested some people with this nut, have to say it'surely provokes an eyeopener :-) People have trouble to accept it, especially those with some (superficial) knowledge and fondness to maths. Only one person which is quite away from maths gave away the correct anwser within seconds... I figure that's the way with paradoxes...

Finally, here's a link for those of you fascinanted with paradoxes that confront some erroneous perceptions concerning infinity.

https://www.c3.lanl.gov/mega-math/workbk/infinity/infinity.html