09/07/2011, 03:33 PM

So essentially [$\aleph_{\aleph_{0}}+1=\aleph_{\aleph_{0}}$],

[$2*\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$],

[$2^\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$], and

2^^[$\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$].

However I do not agree that [$\aleph_{\aleph_{1}}] does not exist.

My heuristic reasoning is:

1 (the first integer past the addition identity) + 0 = 1 (the first integer past 0)

(assuming the Continuum Hypothesis)

2 (the first integer past the exponentiation identity) ^ [$\aleph_{0}$] = [$\aleph_{1}$] (1 being the first integer past 0)

if these are true then

2 (the first integer past the pentation identity) ^^^ [$\aleph_{\aleph_{0}}$] = [$\aleph_{\aleph_{1}}$] (1 being the first integer past 0)

and you could extend the pattern.

Of course I have no other reasons to believe that the third statement is true, as one would have to prove that there does not exist a bijection from [$\aleph_{\aleph_{0}}$] to 2^^^[$\aleph_{aleph_{0}}$].

Also, where would be a place I could go to on the internet to find more discussion on this topic?

Thanks,

Hassler Thurston

[$2*\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$],

[$2^\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$], and

2^^[$\aleph_{\aleph_{0}}=\aleph_{\aleph_{0}}$].

However I do not agree that [$\aleph_{\aleph_{1}}] does not exist.

My heuristic reasoning is:

1 (the first integer past the addition identity) + 0 = 1 (the first integer past 0)

(assuming the Continuum Hypothesis)

2 (the first integer past the exponentiation identity) ^ [$\aleph_{0}$] = [$\aleph_{1}$] (1 being the first integer past 0)

if these are true then

2 (the first integer past the pentation identity) ^^^ [$\aleph_{\aleph_{0}}$] = [$\aleph_{\aleph_{1}}$] (1 being the first integer past 0)

and you could extend the pattern.

Of course I have no other reasons to believe that the third statement is true, as one would have to prove that there does not exist a bijection from [$\aleph_{\aleph_{0}}$] to 2^^^[$\aleph_{aleph_{0}}$].

Also, where would be a place I could go to on the internet to find more discussion on this topic?

Thanks,

Hassler Thurston