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The recursive definitions of addition, multiplication, and exponentiation usually stop after exponentiation ("${\small+}1$" to be read as "the successor of"):

$x \boldsymbol{+} (y\ {\small+}1) := (x \boldsymbol{+} y)\ {\small+}1$

$x \times (y\ {\small+}1) := (x \times y) \boldsymbol{+} x$

$x$ ^ $(y\ {\small+}1) := (x$ ^ $y) \times x$

Sometimes, this seems due to a lack of symbols, only. But it seems feasible to define a recursive sequence of operations $\circ_i$:

$x \circ_{i{\small+}1} (y\ {\small +}1) := (x \circ_{i{\small+}1} y) \circ_{i} x$

with

$x \circ_{0} y := x\ {\small+}1 $

Where resp. under which name is this sequence of operations investigated?

And:

Why is it - eventually - O.K. to stop after exponentiation?

Hans-Peter Stricker
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    http://en.wikipedia.org/wiki/Hyper_operator – Asaf Karagila Nov 26 '13 at 16:32
  • Here some interesting links of questions about Hyp. at MSC: http://math.stackexchange.com/questions/388217/are-hyperoperators-primitive-recursive/388265#388265 http://math.stackexchange.com/questions/170398/does-anything-precede-incrementation-in-the-operator-hierarchy/346004#346004 http://math.stackexchange.com/questions/390091/has-this-phenomenon-been-discovered-and-named/393521#393521 And About Commutative Hyperoperations: http://math.stackexchange.com/questions/383168/where-can-i-learn-more-about-commutative-hyperoperations/389646#389646 – MphLee Jan 10 '14 at 15:47
  • As Asaf say this sequenque is investigated under the name of Hyperoperations with other similar sequences like the commutative Hyperoperations. – MphLee Jan 10 '14 at 15:48

2 Answers2

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As Stephen points out, these operations are given by the three-argument Ackermann function, plugging in 0 (for addition), 1 (for multiplication), etc. to the third argument.

Another notation used for the same thing is Knuth's up-arrow notation: it starts with $a \mathbin{\uparrow} b$ to denote $a^b$ and continues by denoting the next functions in the sequence with multiple arrows: $a \mathbin{\uparrow\uparrow} b$, $a \mathbin{\uparrow\uparrow\uparrow} b$, etc.

Trevor Wilson
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It is called the Ackermann function (or some variant thereof).

http://en.wikipedia.org/wiki/Ackermann_function

Stephen Montgomery-Smith
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  • I did not ask for a specific function. (Do you mean $\circ_4$, eventually? I asked for the sequence $\circ_i, i = 0, 1,\dots$) – Hans-Peter Stricker Nov 26 '13 at 16:24
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    @HansStricker A function $\mathbb{N}^{n+1} \to \mathbb{N}$ naturally corresponds to a sequence of functions $\mathbb{N}^n \to \mathbb{N}$. – Trevor Wilson Nov 26 '13 at 16:30
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    I thought you were asking for the name under which this function is investigated. If you read the web page I posted, you will see that Ackermann's original function had three arguments. It also calls it Knuth's up-arrow notation. – Stephen Montgomery-Smith Nov 26 '13 at 16:30