Justin T. Moore: Why Every Size \aleph_1 Has Measure Zero

[cragar]

I have attached part of 2 pages from Justin T. Moores dissertation.
I am wondering why he says every set of size $\aleph_1$ has measure zero.
He is probably using some axioms that i am not familiar with. And I am not sure
what the $k_2$ is.
He says this towards the bottom of the page.
any help will be much appreciated.

 

[AKG]

It definitely uses extra axioms, indeed it can't follow from ZFC alone since it would contradict CH, which is consistent with ZFC. Whenever you see something like:

Theorem: <Statement>

It means the <Statement> is a theorem of ZFC. Whenever you see:

Theorem: (<Axiom(s)>) <Statement>

It means the <Statement> is a theorem of ZFC + the additional <Axiom(s)>. So in this case, the theorem about $\aleph_1$-sized sets being null assumes $\mathcal{K}_2$. A quick Google search yields some relevant results. In particular, look at definitions 4.1 and 2.1 here:

http://ir.lib.shizuoka.ac.jp/bitstream/10297/2406/1/080701001.pdf

 

[cragar]

ok thanks for your help