How are the hyperoperations defined

Question

For the first few operations such as addition and multiplication they follow rules such as
$$A(x,y+1)=A(x,y)+1$$
$$A(x,0)=x$$
For multiplication
$$M(x,y+1)=M(x,y)+x$$
$$M(x,1)=x$$
So for a general hyper operation you would think that
$$\Pi_n(x,y+1)=\Pi_{n-1}(\Pi_n(x,y),x)$$
But then this doesnt work for tetration because
$$\Pi_4(x,y+1)=\Pi_{3}(\Pi_4(x,y),x)$$
$$^{y+1}x =(^{y}x)^x$$
But this implies
$$^{y}x =(x)^{x^y}$$
Because exponents multiply
When it should be
$$^{y}x =x^{x^{x^{x^{x^{{…}}}}}}$$
What should be the correct formula for the nth hyperoperation.
Thanks.

Answer 1

What should be the correct formula for the nth hyperoperation?

Well, there is no answer because both of them are currently called Hyperoperations. Your definition gives what are called lower-hyperoperations.

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The recursive relation for lower h.os is the one you give: $$h_{n+1}(x,y+1)=h_n(h_{n+1}(x,y),x)$$

As you can read on Wikipedia, this is only one of the possible definitions and probably not the most natural one.

The "natural hyperoperations", as I call them, or Goodstein's hyperoperations satisfy instead this relation: $$H_{n+1}(x,y+1)=H_n(x,H_{n+1}(x,y))$$

Goodstein himself, who proposed the term tetration, pentation , ecc, defines them in this way in R. L. Goodstein (Dec 1947). "Transfinite Ordinals in Recursive Number Theory". Journal of Symbolic Logic. 12 (4): 123–129.

It's easy to prove that both definitions agree on the first 3 operations, and diverges at the fourth.

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Is this the correct definition? We can't know unless we specify what we mean with correct. It's not the oldest definition. For sure it's the most natural and, I claim, the most fundamental one.