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ArisKatsaris comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong
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...I don't understand how this can be. 3^^3 is just (3^(3^3))= 3^27 = 7625597484987 And 3^^^3 is just (3^(3^(3^(... 7625597484987 times ...))))
Superexponentiation is just made of exponentiation many times. And exponentiation is made of multiplication, and multiplication is made of addition.
How can superexponentiation be made invalid without making invalid even normal addition?
I can say more if you want, but maybe see here and here for an explanation.
Any finite calculation of superexponentiation will be valid. But as I understand it you can't in general in Nelson's formulation prove that superexponentation is well-defined in general.