Cardinality of infinity

[Dick]

OK, I can now see that S has infinitely many elements.

But if I define the S = union of A_k (k summing fom 1 up to infinity), will S contain all subsets of Q with infinitely many elements? k is supposed to be finite, but from the union, k can be all the way from 1 to infinity. How come?

I don't think you are giving these questions much thought before you post them. Answer this one yourself. WHY doesn't S contain an infinite subset? Try answering it instead of asking it.

 

[kingwinner]

I don't think you are giving these questions much thought before you post them. Answer this one yourself. WHY doesn't S contain an infinite subset? Try answering it instead of asking it.

S certainly doesn't contain an infinite subset by definition, or by the statement of the problem.

But when I try to write this as S = union of A_k (k summing fom 1 up to infinity) U {empty set}
Then the right side would contain the infinite subset since k is summing from 1 to infinity.

On the other hand, if I write it as S = union of A_k (k summing fom 1 up to n) U {empty set}, then the right side would not contain A_n+1, A_n+2, etc...

So either of them seem to be an incorrect description of S, but what else can I do?

 

[Dick]

The notation S = union of Ak (k from 1 up to infinity) does not mean that you include A_infinity, you just union over all finite k. That fits with your notion of what S should be as you described in the first sentence, right?

 

[kingwinner]

The notation S = union of Ak (k from 1 up to infinity) does not mean that you include A_infinity, you just union over all finite k. That fits with your notion of what S should be as you described in the first sentence, right?

Is this simply a matter of notation/convention? Although it is an infinite union, it can still never go up to infinity?

However, I think the following is true:
union of A_k (k from 1 up to ∞) = A₁ U A₂ U...UA_∞
And by definition, A_k = {all subsets of Q having k elements}
Now put k=∞, UA_∞ contains infinite subsets of Q

I think I am missing something...

 

[Dick]

I already told you that most people wouldn't include A_infinity. If you insist on reading it that way, then you'll have to find a different way of expressing the union you want.