Question about a measure of a set

[cragar]

Could we use the fact that all countable sets have zero measure
to prove that their must be a larger infinity.
we know countable sets have measure zero because I could just start by making
boxes around each number and then add up their widths.
For the first number I will make a box that has width \frac{\epsilon}{2}
and then each box will have half the width of the previous box.
so the sum will be \epsilon(1/2+1/4+1/8...)
and i can make \epsilon as small as I want.
This proof comes from Gregory Chaitin.
If the reals were countable they would have measure zero, but we know this isn't true
because the reals have positive width. Can i do this to prove there is a larger infninty than countable.

 

[mathman]

Could we use the fact that all countable sets have zero measure
to prove that their must be a larger infinity.
we know countable sets have measure zero because I could just start by making
boxes around each number and then add up their widths.
For the first number I will make a box that has width \frac{\epsilon}{2}
and then each box will have half the width of the previous box.
so the sum will be \epsilon(1/2+1/4+1/8...)
and i can make \epsilon as small as I want.
This proof comes from Gregory Chaitin.
If the reals were countable they would have measure zero, but we know this isn't true
because the reals have positive width. Can i do this to prove there is a larger infninty than countable.

Short answer - yes. The only objection, compared to Cantor proof, is that it is necessary to develop measure theory first.

 

[cragar]

ok thanks for your answer. Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.

 

[mathman]

ok thanks for your answer. Instead of using measure theory could I just talk about lengths and use convergence of this infinite series.

Yes - although if you look at it closely you will find you are using some elementary facts from measure theory.

 

[cragar]

ok thanks