On the Grossone and the Infinity Computer

S. S. Kutateladze

The mass media announced on November 3, 2010 that Yaroslav Sergeyev, a professor of Calabria University and Lobachevsky State University of Nizhny Novgorod, has received the Pythagoras Award. It was mentioned that “the professor constructed and patented a new `Infinity Compute'” and “he suggested a new mathematical language that enables one to record various infinitely large and infinitely small numbers.” This information requires commenting.
Sergeyev’s idea is to introduce into arithmetic some infinitely large number—grossone, consider only the numbers that are less than the grossone, and operate exclusively on these numbers using the grossone as the radix. Sergeyev embellishes his idea with metaphysical arguments, emphasizing that he does not use Cantor’s approach and returns to Ancient Greeks.
Elliot Mendelson remarked in his review of Sergeyev’s book [1] that “the systems he deals with consist of objects which are called extended real numbers, but the descriptions of these objects and their properties are not clear enough to permit any warranted judgments about the assertions made by the author about these systems.”
Sergeyev confronts his ideas with the nonstandard analysis of Abraham Robinson, defining his grossone as “the number of elements of the set of natural numbers.” In fact, the role of this would-be mysterious entity can happily be performed by the factorial of an arbitrary infinite number which are galore in nonstandard analysis. The principal shortcomings of Sergeyev’s approach and attempts at implementing calculations with a grossone on a computer were given in [2]. Unfortunately, the series of Sergeyev’s publications continues in the various international journals having little if any in common with foundations of analysis. Miraculously, there are no Sergeyev’s publications on his grossone in  Russian.
Ancient Italian grossones are linguistically close to Sergeyev’s grossone but differ in value.

References

[1]
Sergeyev Ya. D., Arithmetic of Infinity. Edizioni Orizzonti Meridionali, Cosenza (2003).
[2]
Gutman A. E. and Kutateladze S. S. "On the theory of grossone." Siberian Math. J., 49:5, 835-841 (2008).

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November 7, 2010


Newsletter of the European Mathematical Society, No. 79, March, 2011, p. 60.
File translated from TEX by TTHgold, version 4.00.
On 05 Mar 2011, 22:19.

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