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Cantor has also developed an arithmetic for the infinite cardinal numbers. Some of the operations of this arithmetic including N0 and C are given below:

N0+1 = N0, N0+2 = N0, N0+N0 = N0,

(We remind that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘9’, ‘nine’, and ‘IX’ are different numerals, but they all represent the same number.)

C+1 =C, C+2 =C, C+C =C, C+N0 =C.

Again, it is possible to see a clear similarity with the arithmetic operations used in the numeral system of Piraha.

Advanced contemporary numeral systems enable us to distinguish within ‘many’ various large finite numbers. As a result, we can use large finite numbers in computations and construct mathematical models involving them. Analogously, if we were be able to distinguish more infinite numbers probably we could understand better the nature of the sequential automatic computations (remind the famous

phrase of Ludwig Wittgenstein: ‘The limits of my language are the limits of my

world.’) .

The goal of this paper is to study Turing machines using a new approach and allowing one to write down different finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. It is worthy to mention also that the new computational methodology has given a possibility to introduce the Infinity Computer working numerically with finite, infinite, and infinitesimal numbers (its software simulator has already been realized).

The rest of the paper is structured as follows. In Section 2, a brief introduction to the new methodology is given. Due to a rather unconventional character of the new methodology, the authors kindly recommend the reader to study the survey before approaching Sections 3 – 5.

Section 3 presents some preliminary results regarding description of infinite sequences by using a new numeral system. Section 4 shows that the introduced methodology applied together with a new numeral system allows one to have a fresh look at mathematical descriptions of Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by different mathematical languages (the instruments of investigation). Mathematical descriptions of automatic computations obtained by using the traditional language and the new one are compared and discussed. An example of the comparative usage of both languages is given in Section 5 where they are applied for descriptions of deterministic and non-deterministic Turing machines. After all, Section 6 concludes the paper.

N0+1 = N0, N0+2 = N0, N0+N0 = N0,

(We remind that numeral is a symbol or group of symbols that represents a number. The difference between numerals and numbers is the same as the difference between words and the things they refer to. A number is a concept that a numeral expresses. The same number can be represented by different numerals. For example, the symbols ‘9’, ‘nine’, and ‘IX’ are different numerals, but they all represent the same number.)

C+1 =C, C+2 =C, C+C =C, C+N0 =C.

Again, it is possible to see a clear similarity with the arithmetic operations used in the numeral system of Piraha.

Advanced contemporary numeral systems enable us to distinguish within ‘many’ various large finite numbers. As a result, we can use large finite numbers in computations and construct mathematical models involving them. Analogously, if we were be able to distinguish more infinite numbers probably we could understand better the nature of the sequential automatic computations (remind the famous

phrase of Ludwig Wittgenstein: ‘The limits of my language are the limits of my

world.’) .

The goal of this paper is to study Turing machines using a new approach and allowing one to write down different finite, infinite, and infinitesimal numbers by a finite number of symbols as particular cases of a unique framework. It is worthy to mention also that the new computational methodology has given a possibility to introduce the Infinity Computer working numerically with finite, infinite, and infinitesimal numbers (its software simulator has already been realized).

The rest of the paper is structured as follows. In Section 2, a brief introduction to the new methodology is given. Due to a rather unconventional character of the new methodology, the authors kindly recommend the reader to study the survey before approaching Sections 3 – 5.

Section 3 presents some preliminary results regarding description of infinite sequences by using a new numeral system. Section 4 shows that the introduced methodology applied together with a new numeral system allows one to have a fresh look at mathematical descriptions of Turing machines. A deep investigation is performed on the interrelations between mechanical computations and their mathematical descriptions emerging when a human (the researcher) starts to describe a Turing machine (the object of the study) by different mathematical languages (the instruments of investigation). Mathematical descriptions of automatic computations obtained by using the traditional language and the new one are compared and discussed. An example of the comparative usage of both languages is given in Section 5 where they are applied for descriptions of deterministic and non-deterministic Turing machines. After all, Section 6 concludes the paper.

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