User blog:SuperSpruce/Aleph BEAF!

Let N stand for Aleph (א).

This is a variant of BEAF that sizes up cardinals. It uses the aleph system.

Rules:

The first entry of an array has to be infinite! ({N} or bigger)

1. Base rule: {N} is Aleph-null.

2. Base rule 2: {N,a} is Aleph-a.

3. Tailing rule: {#,0}={#}, where # is any non-empty string.

4. Prime rule: {a,0,#}={a}.

5. Catastrophic rule: If Rules 1-4 don’t apply,

A. The first non-zero entry after the prime (second entry) is the pilot, and the entry immediately before that is the copilot.

B. The pilot decreases by 1.

C. The copilot becomes the original array with the pilot decreased by 1.

D. All entries before that become the value of the first entry.

{N} is aleph-null, which is basically countable infinity.

{N,a}=aleph-a.

{N,1,1}={N,N}=aleph-aleph-null.

{N,2,1}={N,{N,N}}=aleph-aleph-aleph-null.

{N,N,1}= the first aleph fixed point.

But that’s just the beginning…

{N,1,2}={N,N,1}

{N,2,2}={N,{N,N,1},1}

{N,1,3}={N,N,2}= the second aleph fixed point?

{N,1,4}={N,N,3}= the third aleph fixed point?

{N,N,N}= the “aleph-null”th aleph fixed point?

{N,N,0,1}

{N,N,N,1}

{N,4(1)2}={N,N,N,N}

{N,5(1)2}={N,N,N,N,N}

…

{N,N(1)2}

{N,N(1)3}

{N,N(1)N}

{N,N(1)(1)2}

{N,N(2)2}

{N,N(N)2}

And we could just keep going...