What comes after tetration ? And after ? And after ? etc.

The power is to the multiplication what the multiplication is to the addition.

We can put those this way:

• Addition
• Multiplication
• Exponentiation
• Tetration

What comes after tetration? What comes whatever comes after tetration? Is this generalized and easily understandale for non-mathematician?

Note: I'm aware of this question, but it doesn't go deeper. I'm asking about the generalization of this process.

Side question:

We know that

2 + 2 = 4
2 * 2 = 4
2 ^ 2 = 4

Is 2 <sign> 2 always 4 ?

What comes after tetration ?

Pentation.

And after ?

Hexation.

And after ?

Heptation.

etc.

Take the Greek numerals in order. Tetra means four, penta means five, hexa means six, etc.

Is 2 <sign> 2 always 4 ?

Yes.

Is this generalized ?

Yes. $a\uparrow^nb$ is the consecrated notation.

P.S.: The operation of order $0$, coming right before addition, is incrementation.

Knuth's up-arrow notation is the usual generalized notation of this. It is used the following way:

• If there are only two numbers with only one arrow between them, then the arrow means "Raised to the power of", e.g. $3\uparrow 4 = 3^4 = 81$.
• If there are only two numbers with arrows between them (like $3\uparrow\uparrow\uparrow 4$), then you take the first of the two numbers (in this case $3$), you repeat it a number of times signified by the latter number (in this case $4$), and then between them all you put arrows, one less than what you had. So we have $3\uparrow\uparrow\uparrow 4 = 3\uparrow\uparrow 3 \uparrow\uparrow 3 \uparrow\uparrow 3$
• Lastly, if there are more than two numbers, you read it from right to left. Continuing on our example, that means $$ 3\uparrow\uparrow\uparrow 4 = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3 \uparrow\uparrow 3}\\ = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3 \uparrow 3 \uparrow 3} = 3\uparrow\uparrow 3 \uparrow\uparrow \color{red}{3^{3^3}}\\ = 3\uparrow \uparrow \color{blue}{3\uparrow \uparrow 3^{3^3}} = 3\uparrow\uparrow \color{blue}{3\uparrow 3 \uparrow 3 \cdots \uparrow 3} = 3\uparrow \uparrow 3^{3^{\cdots^3}} $$ which gets large. That is, it's a "power tower" of threes so tall you'd need a power tower of threes that's seven trillion tall to describe how tall it is ($3^{3^3} \approx 7\vphantom{\dfrac{1}{2}}$ trillion).
• When there are too many arrows to practically write down, you use exponentiation. So $3\uparrow\uparrow\uparrow4 = 3\uparrow^34$.

I'll answer the question about $2 ? 2=4$. It is true that it always gives $4$, and you can prove it by induction. We define $+_1=\cdot$, $+_2$ is the power, etc. For two natural numbers $a$ and $b$ we define, $a+_{n+1}b=\overbrace{a+_na+_na+_n\cdots+_na}^{b\text{ times}}$.

So $2+_{n+1}2=2+_n2=4$.