What comes after tetration ? And after ? And after ? etc.
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> 2always4?
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 $2+_{n+1}2=2+_n2=4$.