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Sewing-Machine comments on Exponentiation goes wrong first - Less Wrong
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Comments (81)
Very nice. It's interesting to think about how hard it would be to prove the proposition "3^^^3 is a counting number" using your trick but not using induction.
You'd have to meta my trick (apply it to itself) at least once. Probably several times.
ponders
Let's go with 4s instead of 2s.
"Since it's true for gaps of one, and 1+1+1+1 equals four, it's true for gaps of four" "....4....16...." "....16....64..." So that's 4^N
"Since it's true for gaps 4 times larger than it's true for, it's true for gaps 444*4=256 times larger" "...256...4 294 967 296..." so that's 4^(4^N)
"Since it's true for gaps the fourth power of gaps it's true for, it's true for the 256th power of gaps it's true for" "...256th power....4 294 967 296th power..."
That's 4^(4^(4^N), or 4^^4 if N=4.
But I'm really struggling to get from that to 4^^^4