Adapting the proof of Cantor's theorem

[Mikeezy]

Hi everyone.

I'm trying to show that the following inequality holds in ZFC:

w(w) < [w(w)]^w

where w(w) means omega-subscript-omega, and [w(w)]^w means omega-subscript-omega raised to the power omega.

I don't have a problem showing that the non-strict inequality holds, but I am having a hard time showing that the two are not equal. I know that doing so involves adapting the proof of Cantor's Theorem (i.e. using a diagonalization argument) but I'm having a hard time figuring out how to do that.

Thanks in advance for any tips.
Mike

 

[Guido]

My Reply

Cantor's thm sets a cardinality to infinity, e.g. all infinities are not equal. As an example consider integers, they go from a negative infinity integer to a positive integer infinity so the set has an infinite number of elements.

But is there a 1 to 1 relation between the elements in the set of integers and the elements in the set of rational numbers? Cantor argued that an infinite number of rational elements existed between any two integer set elements. So the answer was no, the infinity of the set of integers was smaller (had a smaller cardinality) than the set of rational numbers. The set of rational numbers had a higher cardinality (aelph) than the set of integers.

The same can be said of the set of irrationals over rationals, and the set of imaginarys over irrationals.
Now to your problem---I really don't understand the fixation with Cantor
I see the following condition
if w(w)>1
then
w(w)<w(w)^n if n<1
w(w)=w(w)^n if n=1
w(w)>w(w)^n if n>1

if w(w)=1
then
w(w)=w(w)^n for all n

if w(w)<1
then
w(w)>w(w)^n if n<1
w(w)=w(w)^n if n=1
w(w)<w(w)^n if n>1

 

[pka]

Mikezy,
Because you wrote “I don't have a problem showing that the non-strict inequality holds…”,
I assume that you realize that just about everything in the previous reply is wrong!

Having said that, I can make a few comments that may help.
First, there is no standard notation for these ideas.
If I knew the author or the text I may be able to give you more guidance.
I am not sure how your text defines (ω<SUB>ω</SUB>).
I assume you are doing cardinal exponents, so (ω<SUB>ω</SUB>)<SUP>ω</SUP> is the set of all functions from (ω) to (ω<SUB>ω</SUB>).
If that correct, then the required Cantor Theorem may not be the one using a diagonalization argument.
Another theorem of Cantor states that for any set A, card(A)<card[℘(A)] , ℘(A) is the power set of A.
We also know that card[℘(A)]= card[2<SUP>A</SUP>], 2 is a cardinal <ω.
I hope some of this may help. But again it depends upon your definitions and notation.