Dear Henryk and dear Participants,

As promised (and also on behalf of KAR), I should like to introduce “zeration”, a subject considered as controversial by standard and non-standard mathematicians and that, despite the various debates (I must say: around ... the world), still remains fresh and new. Some of those who, based on correct existing mathematical standards, said “It’s just ... wrong”, are now coming back to their statement, thinking that perhaps, despite the new language and the lack of precision in expressing these new concepts, there might be something behind those ideas, still needing a careful attention.

An article on zeration was submitted sometime ago to Wikipedia and it was rejected, mainly because “Wikipedia is not the place for information on new scientific terminology and controversies and zeration is not present in the classical encyclopedias”. Moreover, due to the fact that the original research on it was covered by scientific literature in Russian (Kostantin Rubcov, 1989), the article was presented in two versions, in Russian and in English. This fact was considered as “spamming” by the distinguished Wikipedia Revisers.

Zeration, as part of the hyper-operation hierarchy was described in a paper cited in the bibliography of mentioned in the Stephen Wolfram’s book “A New Kind of Science”, under the title

“Ackermann’s Function and New Arithmetical Operation”, K. A. Rubtsov, G. F. Romerio.

(see: http://www.wolframscience.com/reference/...raphy.html ).

The same subject was also submitted twice to the International Congress of Mathematicians, during ICM-1994, Zurich, and ICM-2006, Madrid, where a poster on zeration and tetration was jointly presented by me and Konstantin Rubcov (Rubtsov). A thread on the subject was submitted by KAR and GFR to the Wolfram Research Institute NKS Forum, since Jan 6th, 2006, with a detailed report under the title: "Hyper-operations. Progress Report, Zeration".

(see: http://forum.wolframscience.com/showthre...eadid=1372)

Recently, zeration has also been discussed in the Republic of Korea, during a seminar organized by the National Institute of Mathematical Sciences (NIMS), on “Tetration and Zeration”.

(see: http://www.nims.re.kr/news/news_seminar_...600&seq=99

At the very simple common sense level, zeration is an attempt for finding an operation filling the gap in the following operations’ sequence:

a ^ a = a [3] a = a [4] 2 = a # 2 exponentiation <-> tetration

a . a = a [2] a = a [3] 2 = a ^ 2 multiplication <-> exponentiation

a + a = a [1] a = a [2] 2 = a . 2 addition <-> multiplication

which should be logically completed by a new operation that we may call zeration (indicating it by the infixed operation sign “°”) and that should at least have the following “strange” property:

a ° a = a [0] a = a [1] 2 = a + 2 zeration <-> addition.

The choice of the word “zeration” is justified by the fact that its rank is immediately lower than the rank of addition (normally considered at rank s=1) and by the common acceptance that iterative exponentiations produce the tetration operation, normally supposed to be at rank 4. The theoretical way for justifying such new operation is provided by the Ackermann Function. In fact, the definition of Ackermann Function (AF) can be summarised as follows:

A(0, n) = n + 1

A(s, 0) = A(s-1, 1)

A(s, n) = A(s-1, A(s, n-1))

By analysing the AF, it is known that we can find the following pattern:

in row s=0, by definition: A(0, n) = n + 1

in row s=1: A(1, n) = 2 + (n + 3) – 3 = n + 2;

in row s=2: A(2, n) = 2 . (n + 3) – 3 = 2n + 3;

in row s=3: A(3, n) = 2 ^ (n + 3) – 3 = 2n+3 – 3;

in row s=4: A(4, n) = 2 # (n + 3) – 3 = n+32 – 3;

With the provisional exception of row s=0, we could re-define Ackermann’s Function as follows:

A(s, n) = 2 [s] (n + 3) – 3, or:

2 [s] n = A(s, n-3) + 3

For s=0 we have:

A(0, n) = 2 [0] (n + 3) – 3 = n + 1 (zeration)

which gives: 2 ° (n+3) = n + 4

therefore: 2 ° n = n + 1 , (for: n ≥ 3)

to which we may add: 2 ° 2 = 2 + 2

and: n ° n = n + 2

We can start using these expressions in order to find out the first properties of the “zeration” operation, which can be described 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

Zeration can therefore be defined as a new binary arithmetical operation belonging to the Grzegorczyk hierarchy and coinciding, in some particular cases, with the “successor” unary operation. It is not associative, but it is commutative and, therefore, it has a unique inverse operation, called “deltation”. In fact, zeration follows the general hyperoperation rules:

a [s] a = a [s+1] 2 , s = 0, 1, 2, 3, 4, ....

and, in particular:

a [0] x = a ° x = x ° a = x [1] 1 = x + 1 for x > a

x [0] x = x ° x= x [1] 2 = x + 2 for any x

and, also, for a = 2:

2 ° 2 = 2 + 2 = 2 * 2 = 2 ^ 2 = 2 # 2 = .... = 4

Zeration follows its own algebraic rules and generates, with its inverse, a new set of numbers, called the “delta numbers”, which can be put in bijection with the logarithms of negative numbers. Functions built with the help of zeration are discontinuous and singular. For these reasons, zeration can be used for defining Boolean operators and fundamental discontinuous functions, such as the Dirac function and the Heaviside function. Zeration and Delta numbers are usable for the construction of approximated algorithms concerning tetration (e.g. in y = b[4]a, with b > 1 and a < -2)

Thanking you very much in advance for your possible interest and expected ... patience , please find attached the detailed progress report on zeration, presented as an attachment to the NKS Forum thread of 2006-01-06.

The authors encourage the readers of the attachment to freely use the terminology and symbols that they have proposed. They will welcome any suggestion and comment and, in case of partial or total text citations, they would appreciate the reference to: “C. (or K.) A. Rubtsov – G. F. Romerio; Hyper-operations. Progress Report. Zeration. The Wolfram Research Institute NKS Forum, Jan 6th, 2006”

GFR

As promised (and also on behalf of KAR), I should like to introduce “zeration”, a subject considered as controversial by standard and non-standard mathematicians and that, despite the various debates (I must say: around ... the world), still remains fresh and new. Some of those who, based on correct existing mathematical standards, said “It’s just ... wrong”, are now coming back to their statement, thinking that perhaps, despite the new language and the lack of precision in expressing these new concepts, there might be something behind those ideas, still needing a careful attention.

An article on zeration was submitted sometime ago to Wikipedia and it was rejected, mainly because “Wikipedia is not the place for information on new scientific terminology and controversies and zeration is not present in the classical encyclopedias”. Moreover, due to the fact that the original research on it was covered by scientific literature in Russian (Kostantin Rubcov, 1989), the article was presented in two versions, in Russian and in English. This fact was considered as “spamming” by the distinguished Wikipedia Revisers.

Zeration, as part of the hyper-operation hierarchy was described in a paper cited in the bibliography of mentioned in the Stephen Wolfram’s book “A New Kind of Science”, under the title

“Ackermann’s Function and New Arithmetical Operation”, K. A. Rubtsov, G. F. Romerio.

(see: http://www.wolframscience.com/reference/...raphy.html ).

The same subject was also submitted twice to the International Congress of Mathematicians, during ICM-1994, Zurich, and ICM-2006, Madrid, where a poster on zeration and tetration was jointly presented by me and Konstantin Rubcov (Rubtsov). A thread on the subject was submitted by KAR and GFR to the Wolfram Research Institute NKS Forum, since Jan 6th, 2006, with a detailed report under the title: "Hyper-operations. Progress Report, Zeration".

(see: http://forum.wolframscience.com/showthre...eadid=1372)

Recently, zeration has also been discussed in the Republic of Korea, during a seminar organized by the National Institute of Mathematical Sciences (NIMS), on “Tetration and Zeration”.

(see: http://www.nims.re.kr/news/news_seminar_...600&seq=99

At the very simple common sense level, zeration is an attempt for finding an operation filling the gap in the following operations’ sequence:

a ^ a = a [3] a = a [4] 2 = a # 2 exponentiation <-> tetration

a . a = a [2] a = a [3] 2 = a ^ 2 multiplication <-> exponentiation

a + a = a [1] a = a [2] 2 = a . 2 addition <-> multiplication

which should be logically completed by a new operation that we may call zeration (indicating it by the infixed operation sign “°”) and that should at least have the following “strange” property:

a ° a = a [0] a = a [1] 2 = a + 2 zeration <-> addition.

The choice of the word “zeration” is justified by the fact that its rank is immediately lower than the rank of addition (normally considered at rank s=1) and by the common acceptance that iterative exponentiations produce the tetration operation, normally supposed to be at rank 4. The theoretical way for justifying such new operation is provided by the Ackermann Function. In fact, the definition of Ackermann Function (AF) can be summarised as follows:

A(0, n) = n + 1

A(s, 0) = A(s-1, 1)

A(s, n) = A(s-1, A(s, n-1))

By analysing the AF, it is known that we can find the following pattern:

in row s=0, by definition: A(0, n) = n + 1

in row s=1: A(1, n) = 2 + (n + 3) – 3 = n + 2;

in row s=2: A(2, n) = 2 . (n + 3) – 3 = 2n + 3;

in row s=3: A(3, n) = 2 ^ (n + 3) – 3 = 2n+3 – 3;

in row s=4: A(4, n) = 2 # (n + 3) – 3 = n+32 – 3;

With the provisional exception of row s=0, we could re-define Ackermann’s Function as follows:

A(s, n) = 2 [s] (n + 3) – 3, or:

2 [s] n = A(s, n-3) + 3

For s=0 we have:

A(0, n) = 2 [0] (n + 3) – 3 = n + 1 (zeration)

which gives: 2 ° (n+3) = n + 4

therefore: 2 ° n = n + 1 , (for: n ≥ 3)

to which we may add: 2 ° 2 = 2 + 2

and: n ° n = n + 2

We can start using these expressions in order to find out the first properties of the “zeration” operation, which can be described 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

Zeration can therefore be defined as a new binary arithmetical operation belonging to the Grzegorczyk hierarchy and coinciding, in some particular cases, with the “successor” unary operation. It is not associative, but it is commutative and, therefore, it has a unique inverse operation, called “deltation”. In fact, zeration follows the general hyperoperation rules:

a [s] a = a [s+1] 2 , s = 0, 1, 2, 3, 4, ....

and, in particular:

a [0] x = a ° x = x ° a = x [1] 1 = x + 1 for x > a

x [0] x = x ° x= x [1] 2 = x + 2 for any x

and, also, for a = 2:

2 ° 2 = 2 + 2 = 2 * 2 = 2 ^ 2 = 2 # 2 = .... = 4

Zeration follows its own algebraic rules and generates, with its inverse, a new set of numbers, called the “delta numbers”, which can be put in bijection with the logarithms of negative numbers. Functions built with the help of zeration are discontinuous and singular. For these reasons, zeration can be used for defining Boolean operators and fundamental discontinuous functions, such as the Dirac function and the Heaviside function. Zeration and Delta numbers are usable for the construction of approximated algorithms concerning tetration (e.g. in y = b[4]a, with b > 1 and a < -2)

Thanking you very much in advance for your possible interest and expected ... patience , please find attached the detailed progress report on zeration, presented as an attachment to the NKS Forum thread of 2006-01-06.

The authors encourage the readers of the attachment to freely use the terminology and symbols that they have proposed. They will welcome any suggestion and comment and, in case of partial or total text citations, they would appreciate the reference to: “C. (or K.) A. Rubtsov – G. F. Romerio; Hyper-operations. Progress Report. Zeration. The Wolfram Research Institute NKS Forum, Jan 6th, 2006”

GFR