Question on cardinality of sequences.

[MathematicalPhysicist]

i need to show that there exists a class of sets A which is a subset of P(Q) such that it satisfies:
1) |A|=c (c is the cardinality of the reals)
2) for every A1,A2 which are different their intersection is finite (or empty).

basically i think that i need to use something else iv'e proven: which for every a in R, let s(a) in Q^N is an increasing sequence which converges to a, then |{s(a)|a in R}|=c

i think that such a class could be: A={P(A')||A'|<alephnull} where A' is a subset of Q, A is the union of all P(A') where A' is Q.
im not sure if A's cardinality is c, but other than this example i don't see how to show it.
i think it's related to what i typed in the second paragraph, perhaps i need to find a subset to {s(a)|a in R} which is still uncountable, but i don't see how.

 

[matt grime]

The question states 'show there exists', right? That suggests to me that the first thing you shuold do is forget about constructing one explicitly.

 

[MathematicalPhysicist]

so how should i prove it?
i mean do i have to assume that A is a subset of P(Q) and then prove that |A|=c and for every 2 sets in A, their intersection is finite or empty.
then one way to do so is, bacause A is a subset of P(Q) then |A|<=c
but we also have c<=|A| cause |A|=c... well I am stuck.
i don't think i can prove it without showing explicitly.

 

[matt grime]

IMPORTANT: I haven't proved this myself, so do what you will with this advice.

ARGUMENT: how do you demonstrate that there are transcendental numbers? Cardinality of real numbers=c. Cardinality of Algebraic numbers=/=c. Hence there are transcendental numbers.

ANALOGY: take the set of all possible As with the finite intersection property. Count them. Count those where |A|=/=c. Are they the same? I suspect the answer is no.

 

[MathematicalPhysicist]

what i did so far is:
A is a subset of P(Q), so i defined A as the union of the sets {Q_n} where Q_n are finite subsets of Q, which have cardinality |Q_n|=n, i think that bacuase there are c sets in P(Q) A equals alephnull*c=c so its cardinality is c.
and we have that every two sets which are in A are either their intersection is finite or empty, is this good definition?

 

[matt grime]

There are countalby many Q_n for each n. Thus A is a countable union of countable sets, hence countable.

the fact that |P(Q)|=c doesn't have anything effect.

 

[AKG]

There's a simple constructive solution. HINT: think about decimal expansions.

 

[Hurkyl]

i think that such a class could be: A={P(A')||A'|<alephnull} where A' is a subset of Q, A is the union of all P(A') where A' is Q.

Incidentally, the "finite power set" of an infinite set is the same cardinality as that set:

$$|S| = | \{ \, A \subseteq S \, | \, |A| < |\mathbb{N}| \, \} |$$

 

[MathematicalPhysicist]

my mistake, perhaps this appraoch will do:
iv'e defined Q_n to be finite subsets of Q, and A={Q_n|Q_n subset of Q} such that |Q_k|=k for every k nonnegative integer.
|A|=|UQ_n|=c
Q_n a subset of Q
and every two elements of A their intersection is finite or empty.
will that be enough?

 

[matt grime]

In what way is that different from what you just did? Why is A uncountable? It is a countable union of countable sets.

 

[AKG]

To add to my hint, think about decimal expansions of all real numbers. And think about how R is constructed as the completion of Q.

 

[HallsofIvy]

To add to my hint, think about decimal expansions of all real numbers. And think about how R is constructed as the completion of Q.

Second this but note that there are several ways to construct R from Q. Since AKG uses the phrase "completion of Q", I assume he means defining R as equivalence classes of the set of Cauchy sequences of rational numbers.
One could also use the definition in terms of equivalence classes of the set of non-decreasing sequences of rational numbers with upper bound but not the definition as Dedeking Cuts.

 

[MathematicalPhysicist]

the decimal expansion of all real numbers is infinite, but the decimal expansion of Q can also be finite.
perhaps i can construct A to be a family of sets of the finite decimal expansions of a particular real number, for example pi would have the set {3,3.1,3.14,...} and so on.
but i don't see how the intersection of two different sets in A WOULD BE FINITE OR EMPTY.

 

[Hurkyl]

Well, what happens if two of these sequences agree infinitely often?

 

[MathematicalPhysicist]

then they are the same.
anyway, i saw the solution in class, i tried to do something similar beore i handed the assignment, i hope i did well.