Proof about size of a union of sets.

[cragar]

Lets say I have $\aleph_1$ numbers of sets that each have $\aleph_1$
number of elements and I want to show that the union of all of these sets has
$\aleph_1$ number of elements.
I start with the line segment [0,1] and shift this line segment up by all the reals from 0 to 1.
So now we have the unit square. Now we want to show that this unit square can be mapped to [0,1]. So can we use trick where you take the decimal form of a point and expand it to 2 dimensions. $(.x_1x_2x_3x_4...)\rightarrow (.x_1x_3...),(x_2x_4...)$
or another thought I had was to take the cantor set and move it around with a set of reals and map each set to a cantor set that was shifted across the real line.

 

[micromass]

You seem under the impression that $\aleph_1$ is the cardinality of the real numbers. This is not the case. The cardinality of the real numbers is $2^{\aleph_0}$.

 

[cragar]

I thought $2^{\aleph_0}=\aleph_1$
if i replace those 2 quantities, is my idea on the right track.

 

[micromass]

I thought $2^{\aleph_0}=\aleph_1$

Nope. The statement $2^{\aleph_0}=\aleph_1$ is known as the continuum hypothesis. It can be proven nor disproven. So under the usual ZFC axioms, we can't say that these two cardinals are equal.

 

[micromass]

But yes, your basic idea of doing

$$0.x_1x_2x_3x_4x_5x_6...\rightarrow (0.x_1x_3x_5...,0.x_2x_4x_6...)$$

is a good one.

 

[cragar]

thanks for the correction about the continuum

 

[Bacle2]

This is, I think, an interesting thing to think about re CH:

What happens when you assume ~CH : then you will find sets in your theory with

cardinality intermediate between Aleph_0 and Aleph_1. You may cut down on the

number of functions, say . Of course, I'm being a bit loose here.

 

[micromass]

This is, I think, an interesting thing to think about re CH:

What happens when you assume ~CH : then you will find sets in your theory with

cardinality intermediate between Aleph_0 and Aleph_1. You may cut down on the

number of functions, say . Of course, I'm being a bit loose here.

You mean: between $\aleph_0$ and $2^{\aleph_0}$?? There is nothing between $\aleph_0$ and $\aleph_1$, even without CH.

 

[Bacle2]

You mean: between $\aleph_0$ and $2^{\aleph_0}$?? There is nothing between $\aleph_0$ and $\aleph_1$, even without CH.

.

Yes, between $\aleph_0$ and $2^{\aleph_0}$.

 

[Bacle2]

And I hope this does not distract too much from the OP , but, if you think about it,

in a sense , there are infinitely-many mathematics: for every undecidable statement,

mathematics branches out into one system in which the undecidable statement holds,

and another system in which the statement does not hold. Something similar

happens with models of different cardinalities, say, the reals: which model is the

real model? We may choose the standard, the non-standard, or models of any

infinite cardinality per the compactness theorem.