Talk:Hyperoperation/Archive 1

Older discussions
AJRobbins (talk) 07:47, 22 April 2009 (UTC) Formatted by — Arthur Rubin (talk)

Ackerman definition
This definition is currently on the page:$$ H_n(a, b) = \begin{cases} 1 & \text{if } b = 0, \\ b + 1 & \text{if } n = 0, \\ H_{n-1}(a, H_n(a, b - 1)) & \text{otherwise.} \end{cases} $$ This has to be wrong; its value never depends on $$a$$! 71.137.5.190 (talk) 21:51, 14 May 2009 (UTC)


 * Titled by (me), and apparently fixed. — Arthur Rubin  (talk) 20:04, 23 June 2009 (UTC)

Exponentiation with a negative radix?
In the Examples section, we have:
 * a > 0, b real, or a non-zero, b an integer, with some multivalued extensions to complex numbers

Figure exponentiation is defined for negative bases, I changed this to:
 * b real, with some multivalued extensions to complex numbers

Arthur Rubin reverted my edit, citing:
 * nope... negative ^ real is undefined, in general

One of us here is completely and utterly confused, and I hope it's me. Exponentiation with a negative radix:
 * (−4)³ = −64

dm yers t urnbull  ⇒ talk 04:20, 5 September 2009 (UTC)


 * negative ^ real is, in general, undefined. negative ^ integer can be defined, but it may produce different results than the "principle value" of negative ^ real for reals near that integer.  — Arthur Rubin  (talk) 06:17, 5 September 2009 (UTC)
 * That is an interesting point. Although the operation produces radically different (complex) results for a^b when a < 0 and b has a fractional component, I think the accepted contemporary definition of a^b includes the described. Is the issue that such operations are no longer technically exponentiation? If so, could we provide an in-line citation?

dm yers t urnbull  ⇒ talk 06:14, 6 September 2009 (UTC)

Hyperoperation navbox
Proposal one (generally stable for a few months, until this month),

Proposal 2 (Robo37, as modified)

Perhaps Proposal 1 would be better as a table than as a navbar....

Comments? — Arthur Rubin (talk) 14:01, 30 March 2010 (UTC)


 * As this was moved from a template, perhaps a version of Navbox without the v-d-e would now be better, regardless of which version is selected. — Arthur Rubin  (talk) 14:05, 30 March 2010 (UTC)


 * Maybe, but before that gets discussed I think that as my navbox doesn’t link to the same articles twice and doesn't group addition and pentation together it should be in use for the time being. I see nothing that makes it any worse than the original. Robo37 (talk) 10:28, 5 April 2010 (UTC)


 * My version could be sensibly converted to a (Wiki)table of links, which might actually serve some purpose. Perhaps additional columns could be added to the table in #Examples.  Robo37's version only makes sense as a navigation template, which seems to me to be unnecessary.  — Arthur Rubin  (talk) 10:40, 5 April 2010 (UTC)

Article name
Shouldn't hyper operation be two words? aparently the article used to be hyper operator, is there a reason that it was squished into a compound word? Cheers, — sligocki (talk) 02:11, 24 October 2009 (UTC)
 * There was a discussion about this on the Tetration Forum, which discusses various terms including "hyperoperation" and its related term "hyper operator". If you use that forum as a whole (or this thread) as an example, the term "hyperoperation" is very popular, being used by most of the original members: andydude (myself), GFR, bo198214, Ivars, JamesKnight, and BaseAcidTetration. In short, Tetration Forum members have the tendency to use "hyperoperation" more frequently than "hyper operator". Also, to add further merit, if the term is to denote a generalization of addition, multiplication, exponentiation, it would be a poor choice linguistically if it did not end with the same suffix. AJRobbins (talk) 01:02, 9 January 2011 (UTC)

The define of zeration (Hyper(a,0,b))
$$H_0(a,b)$$ should be Max(a,b)+1, if a=/=b, or a+2=b+2, if a=b.

Not always b+1. Otherwise, a+3 = a(1)3 = a(0)(a(0)a) = a(0)(a+1) = (a+1)+1=a+2, this is false!!!

(See Robert Munafo (November 1999). "Inventing New Operators and Functions" (Referance 10) and G. F. Romerio (2008-01-21). Hyperoperations Terminology (Referance 14))


 * I don't agree. Perhaps it should be a+1 in that context, rather than b+1.  — Arthur Rubin  (talk) 02:47, 26 April 2014 (UTC)
 * Or the counter should be adjusted, or the induction step defining H1 should be adjusted. H0 was never really a good choice.   — Arthur Rubin  (talk) 06:25, 29 April 2014 (UTC)

However, if it is always a+1, than a+3 = a(1)3 = a(0)(a(0)a) = a(0)(a+1) =a+1, this is also false, so the definition isn't true either. — Preceding unsigned comment added by 140.113.136.218 (talk) 08:08, 13 May 2014 (UTC)

-- The above concerns are misguided -- Goodstein's original definition (the one presently in the article) is not inconsistent. Rather, the examples given above are using incorrect formulas; in particular, there is no basis for writing a[1]3 = a[0](a[0]a), as was done to arrive at contradictions. Correct formulas are a+3 = a[1]3 = a[0](a[1]2) = a[0](a[0](a[1]1)) = a[0](a[0](a[0](a[1]0))) = 1+(1+(1+(a+0))). NB: The definition gives (omitting parentheses for convenience) a[n]b = a[n-1]a[n](b-1) = a[n-1]a[n-1]...a[n-1]a[n]1 (with b a's), but a[n]1 = a holds generally only for n >= 2. — r.e.s. (talk) 02:34, 14 May 2014 (UTC)

Zeration in table
Could someone help me get the "a+1" in the second row 0 left-justified? I don't know why it isn't. (This change in the table relates to  above.)  — Arthur Rubin  (talk) 06:22, 29 April 2014 (UTC)


 * I've removed that from the table, because it is contrary to the article's definition of the 0th hyperoperation. The article presently uses Goodstein's original definition -- as it should, in my opinion. — r.e.s. (talk)  03:11, 14 May 2014 (UTC)

Grzegorczyk hierarchy
This page states that "as a three-argument function, e.g., G(n,a,b) = H_n(a,b)\,\!, the hyperoperation sequence as a whole is seen to be a version of the original Ackermann function \phi(a,b,n)\,\! — recursive but not primitive recursive".

And the footnote on this page states that the Grzegorczyk hierarchy is "more general" than hyperoperation.

And the page Grzegorczyk hierarchy states that the Grzegorczyk hierarchy is primitive recursive.

I don't think these can all be true unless what is meant is that the Grzegorczyk hierarchy is more general than particular 2-argument hyperoperations, but not more general than "the hyperoperation sequence as a whole" "as a three-argument function".

Is this correct? If so, the "more general" in the footnote next to the Grzegorczyk hierarchy should be clarified.

Bayle Shanks (talk) 00:11, 29 August 2013 (UTC)


 * The Grzegorczyk hierarchy is "more general" in the sense that it contains all primitive recursive functions, whereas the H_n hierarchy obviously contains only certain special functions (which are also prim.rec.). The terminology may be tricky, but each function in the hierarchy is pr.rec., while the hierarchy itself is not pr.rec. This is true of both the Grzegorczyk hierarchy and the hierarchy of functions H_n, and other Ackermann hierarchy variants. Thus, each function H_n: NxN -> N is pr.rec., but the function G: NxNxN -> N, G(n,a,b) = H_n(a,b), is not pr.rec. — r.e.s. (talk) 05:10, 14 May 2014 (UTC)

Domain table
Under Examples, the domains are vague and in some cases inaccurate. Firstly, this appears to describe the codomain, not the domain. Secondly, "arbitrary" is not a set of values. Perhaps, "real numbers, often extended to complex numbers," would be a better choice here. Thirdly, the codomain for exponentiation is confusing and just plain wrong, and it seems to approach describing the range rather than just the codomain. Even when restricted to natural numbers alone, 0^1 is well-defined in every algebra commonly used, for one example. TricksterWolf (talk) 17:44, 21 May 2012 (UTC)
 * No, it's not the codomain. For exponentiation, possible domains are (non-zero real)^real, complex^(positive integer), and (non-zero complex)^integer, with (non-zero complex)^complex being a possible domain for a multivalued function.  — Arthur Rubin  (talk) 17:02, 22 May 2012 (UTC)
 * I made a mistake again. Domains within C^C include (nonnegative real)^real, real^(Q_2 that is, rationals with odd denominator), and complex^(integer), all excluding 0^(non-positive).  — Arthur Rubin  (talk) 10:56, 22 May 2014 (UTC)

Standard arrow notation
(I'm not talking about Conway chained arrow notation.)

In many places in this article, we have lines such as:
 * $$F_4(a, b) = (x \to a \uparrow\uparrow (x + 1))^b(a)$$

or, probably better,
 * $$F_4(a, b) = (x \mapsto a \uparrow\uparrow (x + 1))^b(a)$$

I think, perhaps, we should use λ-calculus, as in
 * $$F_4(a,b)=\left(\lambda x . a\uparrow\uparrow (x + 1)\right)^b(a)$$

It has the advantage of being correct, and shouldn't be much more confusing if we define it the first time. Comments? — Arthur Rubin (talk) 11:25, 22 May 2014 (UTC)


 * The notations $$x \mapsto E$$ and $$\lambda x.E$$ have exactly the same meaning in the present context, but I think that in general mathematics this usage of the barred arrow is much more widespread than lambda notation. It's also practically self-explanatory. Rather than introducing lambda notation, in my opinion it would be better to use the barred arrow notation throughout, including a link to the function page where it is explained. — r.e.s. (talk) 04:17, 25 May 2014 (UTC)

Balanced hyperoperations? Lower hyperoperations?
The Balanced hyperoperations section appears to be unsupported by its claimed source. It says this source considers "balanced hyperoperations" using the recursion

$$F_{n+1}(a, b) = (x \to F_n(x, x))^{\log_2(b)}(a).$$

However, rather than the above, the cited paper discusses the sequence of operations defined by

$$\begin{cases} \alpha_0(a,b)&=b \\ \alpha_{n+1}(a,b)&={\alpha_n(a,b)}^{{\alpha_n(a,b)}^a}\ \ (n = 0, 1, 2, \dots) \end{cases}$$

where $$a,b$$ are complex numbers. This generates the sequence

$$(\alpha_0(a,b), \alpha_1(a,b), \alpha_2(a,b), \dots) \ = \ (b,\ b^{b^{a}},\ b^{b^{b^{a}\cdot a + a}},\ \dots)$$

which omits defining the basic arithmetic operations (successor, addition, multiplication, exponentiation); i.e., it is not a hyperoperation sequence.

On the other hand, the sequence actually described in this section (and the term "balanced hyperoperation") is presented without sourcing in a tetration forum thread and appears to be invented by the OP. Finally, the Lower hyperoperations section contains no sourcing at all. Until these sections are appropriately sourced, it seems to me that they should be removed per No_original_research. Or have I overlooked something? — r.e.s. (talk) 15:37, 25 May 2014 (UTC)

Recent additions
An IP cluster has been adding huge tables of entries, most of the tables being too large to view, and most of the entries being too large to understand; and selflinks (zeration, hexation) to this article. This is inappropriate. — Arthur Rubin (talk) 05:59, 29 April 2014 (UTC)
 * Still doing it. — Arthur Rubin  (talk) 10:41, 22 May 2014 (UTC)
 * By the way, by "too large", I mean the expressions are physically too large to understand, rather than the numbers being too large to understand, which is also true. — Arthur Rubin  (talk) 10:43, 22 May 2014 (UTC)
 * Still doing it, and other things (see below.)  — Arthur Rubin  (talk) 16:34, 31 October 2014 (UTC)

&minus;1
The anon has, I believe properly, noted that
 * $$a[n](-1)=0$$

for n ≥ 4 (and a ≠ 0). I would like some confirmation that this leads to proper calculation of $$(-1)[n]b$$, for b a positive integer. — Arthur Rubin (talk) 16:46, 31 October 2014 (UTC)

Bracket notation
An anon has been replacing most of $$a \uparrow ^n b$$ and $$H_{n+2}(a,b)$$ with $$a [n+2] b$$. I see no real reason to emphasize Conway's uparrow notation, but I want to make sure this has consensus, especially the latter, as it replaces a definition using standard function notation with "standard" hyperoperation notation. — Arthur Rubin (talk) 16:37, 31 October 2014 (UTC)
 * I'm going to change it back. According to the table of notations, Conway's notation is the one overwhelming used in reliable sources, and the only one used by sources not affiliated with the source of the notation.  The alternative notation is Hn+2(a, b).  [n] is used only in ASCII sources to simulate $$a {\,\begin{array}{|c|}\hline{\!n\!}\\\hline\end{array}\,} b\,\!$$ which, in turn, very difficult to draw, even in LaTeX, and is used in few sources.  — Arthur Rubin  (talk) 00:49, 6 November 2014 (UTC)

Variant starting from 0
Please check me on this. In this variant,
 * $$F_4(a,0) = 0$$ with the possible exception of a = 0
 * $$F_5(a,0) = 0$$
 * $$F_5(a,n+1) = F_4(a,F_5(a,n)) = 0$$ by induction
 * — Arthur Rubin (talk) 18:08, 6 November 2014 (UTC)

Lower hyperoperations
I restored the "Lower hyperoperations" section, giving "Donner-Tarski" as a source for the extension to ordinal numbers, and noting that, not only is
 * $$a _{(4)} b = a^{a^{b-1}}$$, but also
 * $$a _{(2n)} b \leq a _{(2n-1)} (3 a b + 1)$$ for n &ge; 3.

We might add:
 * $$ a^{(4)}b \leq a_{(5)}b \leq a^{(4)}(b+2),$$

but, as far as I know, the result is unpublished. — Arthur Rubin (talk) 13:23, 19 January 2015 (UTC)

Note on the "original research" tag I added;

For a &ge; 2, b &ge; 1, and n &ge; 0, where appropriate:

$$\begin{array}{ll|l} a O_0 b = a+b & \text{Definition 1(i)} & a _{(1)} b = a + b \\ a O_{n+1} 1 = a & \text{Corollary 2(ii)} & a _{(n+2)} 1 = b \\ a O_{n+1} (b+1) = (a O_{n+1} b) O_n a & \text{Theorem 9} & a _{(n+2)} (b+1) = (a _{(n+2)} b) _{(n+1)} a \end{array}$$

Also note that $$ a _{(0)} b$$ must be defined as $$\boldsymbol a + 1$$ for $$ a _{(1)} b = a + b. $$ — Arthur Rubin (talk) 17:28, 19 January 2015 (UTC)

Number set required by each operation level
Note that addition is complete with integer numbers, and no higher set (as real or complex) is necessary to solve an equation with integers and addition, but once multiplication is introduced, it requires real numbers to solve any equation, but not complex ones. Complex are required once exponentiation is introduced.

So, for each operation, a new superset of numbers is required to solve any equation, and this should be mentioned on this article.

What new set of numbers are required by tetration? — Preceding unsigned comment added by 181.20.129.191 (talk) 13:33, 28 February 2015 (UTC)

From this article: Tetration However, if the linear approximation above is used, then $ ^y x = y + 1$ if &minus;1 < y ≤ 0, so $ ^y \sqrt{y + 1}_s $ cannot exist.

That is like saying that square roots of negative numbers do not exist. But is possible to define imaginary numbers that are solutions, by extending the set of real to the complex. So, if some equations in tetration do not have solutions, that should be used to extend the concept of number. — Preceding unsigned comment added by 186.59.42.207 (talk • contribs) 08:42, March 1, 2015‎
 * If the concept of "number" could be so extended to have reasonable propeties, (and it were done in a reliable source), that could be a good idea. I doubt the former, and categorically deny that the latter (a reliable source) has been identified.  — Arthur Rubin  (talk) 17:00, 1 March 2015 (UTC)

Disputed names
Are tetration, pentation, hexation, etc., the commonly used names? I'm not convinced. — Arthur Rubin (talk) 19:27, 23 June 2009 (UTC)
 * As I think you have gathered, tetration is fairly widely used, the others aren't but are obvious extrapolations from that one. &mdash; Charles Stewart (talk) 12:37, 30 June 2009 (UTC)

-- SMBC --

109.156.20.243 tried to include the SMBC names for these operations, penetration and sexation. These names are far superior, and at least deserve an honourable mention. Since the edit has been reverted, I didn't change the actual page, but I propose listing them as "alternative nomenclature" or some such thing. 118.208.134.60 (talk) 01:35, 20 May 2012 (UTC)


 * Seems absurd. Nonetheless, it might be listed if a reliable source commented on those names.  (Reliable sources, IMHO, in this context, would only include published peer-reviewed papers or published news articles, not "expert" opinions, as an expert in the field would not necessarily be an expert on the names, and WP:SPS requires that the item be written by an expert in the field.)  — Arthur Rubin  (talk) 09:47, 20 May 2012 (UTC)


 * Jokes are fine and well, but they aren't nomenclature until someone starts to use them.--Prosfilaes (talk) 20:39, 20 May 2012 (UTC)

zeration?
While "pentation" etc. may be the natural extension of "tetration" which is "commonly accepted" (say), this is IMHO far from being the case for "zeration" (its tetra, penta,... but zero and not zera! Note also: addition, not addation...); I can't find a serious reference for this (and at least, the article is lacking such). &mdash; MFH:Talk 20:05, 8 April 2015 (UTC)
 * One of your 's was a "smart quote". — Arthur Rubin  (talk) 00:25, 9 April 2015 (UTC)

Pentation
In the unlikely event that there is something in the anon-created Pentation which should be kept, it should be merged to this article somewhere. I don't see any information about the real-"exponent" extention which is possibly appropriate, as it (1) is sourced only to an unmoderated forum, and (2) depends on a particular version of tetration/ultra-exponentiation to real "exponent"s, which is almost certainly not the standard one. — Arthur Rubin (talk) 19:44, 23 June 2009 (UTC)


 * Actually, tetration, pentation and hexation are the standard names. This definition is standard, just not very well-known. --220.255.7.151 (talk) 02:42, 24 June 2009 (UTC)


 * Although I still have doubts about the names and definitions for natural number "exponent"s, it's the extension to real "exponent"s that I'm certain is non-standard. Why x^^y should be linear in y for y in (-1,0) is beyond me, and that's required to evaluate x^^^-infinity.  — Arthur Rubin  (talk) 03:03, 24 June 2009 (UTC)


 * To do that you calculate the limit as x approaches -infinity. --116.14.26.124 (talk) 03:09, 24 June 2009 (UTC)


 * P.S. Also see tetration. --116.14.26.124 (talk) 03:15, 24 June 2009 (UTC)


 * If $$f(x)=e{\uparrow}{\uparrow}x$$ were well-defined for real x, then $$e{\uparrow}{\uparrow}{\uparrow}-\infty$$ (if it were well-defined) would be a fixed point of $$f$$. However, neither is well-defined.  — Arthur Rubin  (talk) 22:41, 24 June 2009 (UTC)
 * There's a second substantial cite for pentation, besides Goodstein (1947): K.K. Nambiar, 1995, Ackermann functions and transfinite ordinals, Appl. Math. Lett. 8 (6), 5153. That should justify a redirect. I don't see a case for a separate article.  As an aside, there's a fun attempt to generalise teration, pentation and hexation to continuous functions, but its unpublished so doesn't count for notability: Markus Müller, Reihenalgebra &mdash; What comes beyond exponentiation?&mdash; Charles Stewart (talk) 11:39, 1 July 2009 (UTC)


 * I do not think Pentation should have its own page. Researchers are just starting to figure out Tetration, so maybe when we are competent with that, then we can move on to the next. Until then, Pentation is by definition a function from NxN -> N, as are all hyperoperations. Since there isn't much beyond that, it seems fitting to put it in this hyperoperation article. AJRobbins (talk) 14:06, 8 September 2009 (UTC)


 * "All" hyperoperations? So add, mul, exp are only NxN -> N functions? Hmm? mike4ty4 (talk) 01:12, 4 October 2009 (UTC)


 * I went through this on the hyperoperations page. Tetration is (approximately) from R+xN, so pentation is only defined on NxN.  — Arthur Rubin  (talk) 07:12, 4 October 2009 (UTC)

I don't particularly support a merge. While I share Arthur Rubin's doubts about the choice of real extension for tetration, I think that an article on this topic could be useful and can segregate some particulars away from this page. Of course notability is borderline here. I don't think anyone would suggest that hexation should have its own article, or that tetration should not. CRGreathouse (t | c) 16:23, 29 December 2009 (UTC)


 * I don't particularly have an opinion. In a Wikipedia with an article for every 25-person-population village in India, I suppose it's reasonable to have an article on pentation. I'm not convinced that adequate reliable sources can support that article. Anyways, since there appears to be consensus, I'm removing the suggested-merge template. If someone really objects to that, they're welcome to re-add it. dm yers t urnbull   ⇒ talk 03:40, 2 May 2010 (UTC)


 * I don't think there is enough known about pentation to merit an entire article. AJRobbins (talk) 17:13, 10 June 2015 (UTC)

Steinhaus–Moser notation
I'm not a maths expert but shouldn't Steinhaus–Moser notation be listed among other Notiations? P4z (talk) 16:50, 26 May 2015 (UTC)
 * Interesting. We don't seem to have an article connecting them in general, although it can easily be shown that, since:
 * n ^^ (n + 1) <= (triangle) n,
 * and it seems likely that, for sufficiently large n,
 * (triangle) n < n ^^ (2 n)
 * but I don't think this helps bounds for higher M-S operations. — Arthur Rubin  (talk) 09:36, 16 July 2015 (UTC)

a+1 instead of b+1
a+1 for n=0 is more intuitive. b+1 produces problems... >> Perhaps it should be a+1 in that context, rather than b+1. — Arthur Rubin

MrFrety (talk) 20:37, 7 November 2015 (UTC)

Knuth's up-arrow notation
I propose that Knuth's up-arrow notation be merged here to hyperoperation. With the exception that Knuth only uses positive integers, and, here, we extend as far as possible (including &minus;1, but possibly failing for some combinations where the left-hand side is &minus;1 or 0, it's a notation for the same concept. — Arthur Rubin  (talk) 09:39, 1 November 2014 (UTC)

In view of the shagginess of the subject matter, it doesn't seem especially desirable to consolidate these articles. The advantage of retaining a separate article for the Knuth notation is that it can be allowed to sprawl without confusing (or reducing the compactness of) the article on hyperoperation. Gottlob Frege (talk) 13:53, 1 January 2015 (UTC)


 * Oppose. Knuth's notation is a special case and is significant enough to warrant its own article. Wqwt (talk) 02:33, 16 July 2015 (UTC)

In my opinion, Knuth's arrow up notation, as said above, has enough specific uses and characteristics to justify an article of its own. Idanbhk (talk) 15:56, 17 October 2015 (UTC)


 * Oppose. I'm not sure what the editor-speak is, but I feel that the page on hyperoperation should have a short blurb-like section about up-arrow notation with a link to saying that up-arrow-notation is the "main article". lol md4 U&#124;T 21:05, 9 March 2016 (UTC)

History: Hyperoperations before Goodstein
In 1926, Hilbert described Ackermann's result in Über das Unendliche (before Ackermann published it in 1928) with a slightly different function: φn(a, b) (Hilbert 1926) is really the same as a(n)b ((n) being the hypern operator), so hyperoperations were considered before Goodstein did it. -- IvanP (talk) 13:50, 19 March 2016 (UTC)

Hyperoperation with reals
In the Chinese page you can see what is between the operation addition, multiplication, exponentation and tetration ... etc. : https://zh.wikipedia.org/wiki/超运算#.E4.B8.80.E8.88.AC.E5.8C.96 But how can I evaluete Hn(a;b) = H(a;n;b) ? E.g. why H(3;1.5;3)≈7.34? — Preceding unsigned comment added by Xorter (talk • contribs) 15:33, 1 May 2016 (UTC)

Derivatives?
Do these operations have meaningful derivatives? — Preceding unsigned comment added by 70.247.163.191 (talk) 11:37, 15 June 2016 (UTC)

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Another definition of zeration
can be found at http://math.eretrandre.org/tetrationforum/showthread.php?tid=122

It is as follows:

a ° b = a + 1, if a > b

a ° b = b + 1, if a < b

a ° b = a + 2 = b + 2, if a = b

2600:1702:29C0:12E0:DDB4:FDB5:1F8D:2971 (talk) 05:32, 6 June 2018 (UTC)

how about this notation
3⑤3 = 3④(3④3) = 3④(3③(3③3)) = 3④(33 3 ) = 3④(327) = 3④7625597484987 = $$\begin{matrix}\underbrace{3^{3^{3^{3^{\cdot^{\cdot^{\cdot^{\cdot^{3}}}}}}}}}\mbox{7625597484987} \end{matrix}$$ — Preceding unsigned comment added by 49.215.49.24 (talk) 06:06, 21 August 2018 (UTC)

a op 0
As Hn(a, 1) = a for n >= 2, there seems no harm in defining Hn(a , 0) = 1 for n >= 3. The only crosscheck required is:

$$a = H_n(a, 1) = H_{n-1}(a, H_n(a, 0))) = H_{n-1}(a, 1) = a.$$

— Arthur Rubin (talk) 01:49, 22 November 2018 (UTC)

Commutative exponentiation (?)
Could an expert page-watcher please review Commutative exponentiation, and (if appropriate) link it into Exponentiation and Hyperoperation. Thanks ~Hydronium~Hydroxide~(Talk)~ 03:29, 4 March 2020 (UTC)

Removed claim; more OR-removal may be needed
I have removed the claim that a certain operation is "[a] commutative form of exponentiation." The operation in question is certainly commutative, but there is no source given that it is a form of exponentiation, and it's not clear what that would even mean.

I think what it means is that this operation is in some sense an answer to the analogy question addition : multiplication :: multiplication : ? and exponentiation is also an answer to that. I can agree that this is true, but I don't think it justifies calling the operation a "form of exponentiation", and even if it did we'd still need a source.

Along those lines, has anyone looked at the sources to see how much of this article really comes from them? It's not even clear to me that the term "hyperoperation" is standardly used in this way in the wild. --Trovatore (talk) 22:04, 4 March 2020 (UTC)


 * I think there isn't a standard term for this sequence of operations: 'hyperoperation' then is a neologism for the purposes of Wikipedia, but not one that bothers me, since it is a perfectly good term for a useful concept. McBeth (Combinatorial Number Theory, 1994, p.259) calls these operations 'generalisations of exponentiation'; in that book he generalises the notion of hyperoperation to ordinals, so we have G_omega, G_{omega+1), ... G(epsilon_0), ... etc.
 * Nothing in the article, based on a quick scan, struck me as surprising enough to count as original research --I was not familiar with the 'commutative series', but that seems adequately sourced and the other claims seem clear enough-- but the article as a whole could be better sourced. I might work on that. &mdash; Charles Stewart (talk) 10:12, 6 March 2020 (UTC)
 * Hmm. "Neologisms for the purposes of Wikipedia" pretty much always bother me.  Maybe I'm too sensitive on this point, but I think Wikipedia has to rein itself in from being an instrument of linguistic change, because it would be too easy.  If you want to make up a word, you ought to have to get it published, like everyone else. --Trovatore (talk) 04:38, 7 March 2020 (UTC)
 * If it makes you feel happier, I did find a journal article that used essentially this word in this sense: Hooshmand, M. H. (2018); Ultra power of higher orders and ultra exponential functional sequences; Journal of Difference Equations and Applications, 24(5), pp. 675-684; 10.1080/10236198.2017.1307350. It's not a very good source, because it is not written by an authority (AFAICS, this is only his 2nd work on fast-growing functions), but it crosses the rubicon of a number theory work in a Scopus-listed journal. The thing is, I don't know of a better term: despite the frequency with which the concept occurs in the literature, authors seem to go out of their way to avoid giving the family a surname. &mdash; Charles Stewart (talk) 19:35, 7 March 2020 (UTC)
 * That's the thing &mdash; if it's not standard in the wild, we shouldn't write as though it is, because that could promote the development of a standard, which is not our role. I would prefer a messier title that avoided putting WP in that position. --Trovatore (talk) 20:52, 7 March 2020 (UTC)
 * Would rephrasing the lead sentence help? Something along the lines of "Hyperoperation is a term used occasionally in scholarship to describe the members of a hierarchy of operations growing faster than exponentiation that are defined using a simple, nested-recursive scheme." &mdash; Charles Stewart (talk) 16:45, 8 March 2020 (UTC)
 * Honestly I would prefer to move the whole article to something like generalizations of exponentiation, and mention "hyperoperation" among other terms that have been used. --Trovatore (talk) 17:36, 8 March 2020 (UTC)

"Heptation" listed at Redirects for discussion
An editor has asked for a discussion to address the redirect Heptation. Please participate in the redirect discussion if you wish to do so. 1234qwer1234qwer4 (talk) 13:24, 17 April 2020 (UTC)

Lack of generality
Addition, multiplication and exponentiation are particular operations. Then on we use $$a[n](a[n](a[n]...$$ without any indication whatsoever what ( means, is it concatenation or exponentiation or a totally new operation.

Notice that first formula has the condition $$n \geq 2$$ which is utterly confusing on all levels, because further on it says that the function is universal. The article is speaking about representing addition, multiplication and exponentiation, which we know already and then even ask question what is beyond exponentiation and tetration... Since we do not know those higher functions as well as addition and multiplication we are confused over what notation means in the first place.

So let me fill in a serious gap by a few examples, for $$n=1$$ the chain is longer by 1:

$$4[1]3=4[0](4[0](4[0]4))$$

$$4[0]4=\text{next in line on 0 level}=4+1=5$$

$$4[1]3=4[0](4[0](5))=4[0](4[0]5)$$

$$4[0]5=\text{next in line on 0 level of 5 (4 has no role)} = 5+1 = 6$$

$$4[1]3=\text{4[0](4[0](5))}=4[0](6)$$

$$4[0]6=\text{next in line on 0 level}=6+1=7$$

$$4[1]3=7$$

Very similar for multiplication, but here the chain is not longer.

$$4[2]3=4[1](4[1]4)$$

$$4[1]4=\text{apply previous operation 4 times}=4+1+1+1+1=8$$

$$4[1]8=4+1+1+1+1+1+1+1+1=12$$

So technically all higher operations can be represented through addition if we want to. Since it is exponentiation that is the last operation that we have far more experience than with tetration, we do express higher operations using exponentiation. We do not express exponentiation through addition, although in order to understand the higher operations we should do it as an exercise.

$$4[3]3=4[2](4[2]4)=4[2](4[1](4[1](4[1]4)))=4 \cdot (4+4+4+4)=(4+4+4+4)+(4+4+4+4)+(4+4+4+4)+(4+4+4+4)$$

This last is showing how it goes on the upper levels. So where is $$3$$ hidden. It is on the grouping. Addition is on the level $$1$$, so we have one group of $$4$$ elements that is raising the level $$(4+4+4+4)$$ and another group of $$4$$ elements $$+++$$

However, exponentiation is not associative so it creates some trouble when we try to explain tetration as we have to keep parenthesis.

$$4[4]3=4[3](4[3]4)=4^{(4^4)}$$

This is to say:

$$4[3]4=4 \cdot 4 \cdot 4 \cdot 4$$

And now

$$4[4]3=4^{4 \cdot 4 \cdot 4 \cdot 4}$$

which means that previous operation (multiplication) is applied $$4 \cdot 4 \cdot 4 \cdot 4$$ times. Which is giving $$a[n]b=a[n-1](a[n](b-1))$$

So let us try to create a multiplicative structure as with addition and multiplication previously

$$4[4]3=4^{4 \cdot 4 \cdot 4 \cdot 4}=(((4^4)^4)^4)^4$$

So first we have $$4 \cdot 4 \cdot 4 \cdot 4$$

Next this is repeated four times

$$(4 \cdot 4\cdot 4\cdot 4)\cdot(4 \cdot 4\cdot 4\cdot 4)\cdot(4 \cdot 4\cdot 4\cdot 4)\cdot(4 \cdot 4\cdot 4\cdot 4)=\alpha $$

Next again four times

So first we have $$\alpha \cdot \alpha \cdot \alpha \cdot \alpha$$

and again four times

$$(\alpha \cdot \alpha \cdot \alpha \cdot \alpha) \cdot (\alpha \cdot \alpha \cdot \alpha \cdot \alpha) \cdot (\alpha \cdot \alpha \cdot \alpha \cdot \alpha) \cdot (\alpha \cdot \alpha \cdot \alpha \cdot \alpha)$$

This last example is clearly showing what is happening with higher levels beyond exponentiation. So in some sense addition and multiplication are too simple to explain the complexity. — Preceding unsigned comment added by 158.248.76.166 (talk)
 * Please see our core content policy WP:OR, and our behavioral policy WP:NOTFORUM. --JBL (talk) 13:50, 24 July 2021 (UTC)