How Many Constructible Points Exist on the X-Axis?

[kingwinner]

The points in the plane are pairs of constructible numbers, (x,y) where x and y are constructible. I don't see why you don't think this is the same thing as CxC where C is the constructibles. Yes, you would be hard put to find an explicit 1-1 function. But proving it exists is a different matter from actually writing it down in detail. And the proof it exists is all you need.

I can see that it's the same thing as CxC, but you said that:
"The constructible numbers in R have cardinality N. So the constructible points have cardinality NxN"

C={constructible numbers}
N={natural numbers}

|C|=|N| => |CxC|=|NxN| ? <----I can't follow this step...

 

[Dick]

If |C|=|N| then there is a bijective map f:C->N. Use it to construct a bijective map from CxC to NxN.

 

[kingwinner]

2) Find the cardinality of the set of all finite subsets of Q.

Attempt:
Let S={all finite[/color] subsets of Q}
For every k E N, let Ak = {all subsets of Q having EXACTLY k elements}

S is equal to
∞[/color]
U A_k U {empty set}
k=1
...
=======================
Something is funny here, S is set of all finite[/color] subsets of Q, but in the union of A_k, we are summing fom 1 up to infinity[/color] (so S may have infinitely many elements?) How can they be totally inconsistent (one is finite and the other is infinite)? Is there a mistake somewhere?

I am puzzled...could someone please explain?

 

[HallsofIvy]

2) Find the cardinality of the set of all finite subsets of Q.

Attempt:
Let S={all finite[/color] subsets of Q}
For every k E N, let Ak = {all subsets of Q having EXACTLY k elements}

S is equal to
∞[/color]
U A_k U {empty set}
k=1
...
=======================
Something is funny here, S is set of all finite[/color] subsets of Q, but in the union of A_k, we are summing fom 1 up to infinity[/color] (so S may have infinitely many elements?) How can they be totally inconsistent (one is finite and the other is infinite)? Is there a mistake somewhere?

I am puzzled...could someone please explain?

Why are you puzzled? Q is infinite itself so if S is set of all finite subsets of Q, there must be an infinite number of such sets! One can be finite and the other infinite, because they are different things: each set in S is finite but S itself is infinite because the number of sets in S is infinite.

Try the same thing for the natural numbers. Let S be the collection of all "singleton" sets: S= {{1}, {2}, {3}, ...}. Every set in S is finite but S itself is infinite.

 

[kingwinner]

Why are you puzzled? Q is infinite itself so if S is set of all finite subsets of Q, there must be an infinite number of such sets! One can be finite and the other infinite, because they are different things: each set in S is finite but S itself is infinite because the number of sets in S is infinite.

Try the same thing for the natural numbers. Let S be the collection of all "singleton" sets: S= {{1}, {2}, {3}, ...}. Every set in S is finite but S itself is infinite.

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[/color]), will S contain all subsets of Q with infinitely[/color] many elements? k is supposed to be finite, but from the union, k can be all the way from 1 to infinity. How come?

 

[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[/color]), will S contain all subsets of Q with infinitely[/color] 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.