Infinite computations and the generic finite

This paper introduces the concept of a generic finite set and points out that a consistent and significant interpretation of the grossone, ? notation of Sergeyev is that ? takes the role of a generic natural number. This means that ? is not itself a natural number, yet it can be treated as one and used in the generic expression of finite sets and finite formulas, giving a new power to algebra and algorithms that embody this usage. In this view, N = { 1 , 2 , 3 , ? , ? - 2 , ? - 1 , ? } is not an infinite set, it is a symbolic structure representing a generic finite set. We further consider the concept of infinity in categories. An object A in a given category C is infinite relative to that category if and only if there is a injection J : A ?s? A in C that is not a surjection. In the category of sets this recovers the usual notion of infinity. In other categories, an object may be non-infinite (finite) while its underlying set (if it has one) is infinite. The computational methodology due to Sergeyev for executing numerical calculations with infinities and infinitesimals is considered from this categorical point of view.

[1]  Yaroslav D. Sergeyev,et al.  Numerical computations and mathematical modelling with infinite and infinitesimal numbers , 2012, ArXiv.

[2]  Yaroslav D. Sergeyev,et al.  Blinking fractals and their quantitative analysis using infinite and infinitesimal numbers , 2007 .

[3]  Alfredo Garro,et al.  Observability of Turing Machines: A Refinement of the Theory of Computation , 2010, Informatica.

[4]  Anatoly A. Zhigljavsky,et al.  Computing sums of conditionally convergent and divergent series using the concept of grossone , 2012, Appl. Math. Comput..

[5]  Yaroslav D. Sergeyev,et al.  Solving ordinary differential equations on the Infinity Computer by working with infinitesimals numerically , 2013, Appl. Math. Comput..

[6]  Alfredo Garro,et al.  Single-tape and multi-tape Turing machines through the lens of the Grossone methodology , 2013, The Journal of Supercomputing.

[7]  Yaroslav D. Sergeyev,et al.  Numerical point of view on Calculus for functions assuming finite, infinite, and infinitesimal values over finite, infinite, and infinitesimal domains , 2009, 1203.4140.

[8]  Yaroslav D. Sergeyev,et al.  On accuracy of mathematical languages used to deal with the Riemann zeta function and the Dirichlet eta function , 2011, 1203.4142.

[9]  W. Luxemburg.  Non-Standard Analysis , 1977 .

[10]  Yaroslav D. Sergeyev,et al.  Solving ordinary differential equations by working with infinitesimals numerically on the Infinity Computer , 2013 .

[11]  Yaroslav D. Sergeyev,et al.  Counting systems and the First Hilbert problem , 2010, 1203.4141.

[12]  Yaroslav D. Sergeyev,et al.  Evaluating the exact infinitesimal values of area of Sierpinski’s carpet and volume of Menger’s sponge ☆ , 2009, 1203.3150.

[13]  Maurice Margenstern.  Using grossone to count the number of elements of infinite sets and the connection with bijections , 2011, ArXiv.

[14]  J. M. Child.  The Early Mathematical Manuscripts Of Leibniz , 1921, The Mathematical Gazette.

[15]  Yaroslav D. Sergeyev,et al.  A New Applied Approach for Executing Computations with Infinite and Infinitesimal Quantities , 2008, Informatica.

[16]  Yaroslav D. Sergeyev,et al.  Interpretation of percolation in terms of infinity computations , 2012, Appl. Math. Comput..