Welcome !
This website is a venue for students and teachers to learn about the Arithmetic of Infinity, a way of treating infinitely large and small numbers numerically whose employment in applied mathematics was established by the mathematician Yaroslav Sergeyev. The Arithmetic of Infinity discussed on this website can be formally developed within the axiomatisation of arithmetic presented in Montagna F., A. Sorbi and S. Simi (2015). Taking the Piraha seriously. Communications in Nonlinear Science and Numerical Simulation, 21, pp.52-69.
What is the Arithmetic of Infinity?
When we use base ten notation, we can specify numerical values to carry out calculations or to determine the size of certain collections. For instance, the specifications 3 and 4 allow us to carry out certain computations like 3 + 4 or 4 - 3. They also allow us to count the items in certain collections: thus, the collection {0, 1, 2, 3} is determined by the specification 4 and, deleting the item 0 from it, we obtain the collection {1, 2, 3}, determined by the specification 3.
As long as we stick to base ten, we cannot use any numerical specifications to determine an infinite collection like {1, 2, 3, 4, ...} as distinct from the infinite collection {2, 3, 4, ....}, obtained from the former by deleting the item 1. When we attempt to assign determinations to such collections, we often identify them. Without registering their differences, we must assign them the same specification 'infinity': this assignment leads us to expressions like 'infinity - infinity', which are not equivalent to any numerical specification.
The Arithmetic of Infinity is a way of avoiding this inconvenience. Its starting point is a different notation in an infinite base: the new base is a specification intended to determine the collection of all positive integers {1, 2, 3, ...}. Given the new base, called 'gross-one', we can determine infinite collections and use suitable numerical specifications to record the results of finitely many operations carried out on an infinite collection that we can determine numerically.
Many paradoxes of infinity, which essentially rely on the unavailability of determinations for infinite collections, are avoided by turning to arithmetic with gross-one. If, for instance, we can numerically specify the number of terms in an infinite series of the form 1 - 1 + 1 - 1 + ... we find that its sum depends on the numerical determination of the collection of added terms, and cannot be uniquely determined when we do not provide any specification of the number of terms occurring in it.
Explore the resources on this website to find out more about infinity, paradoxes, and gross-one!
Acknowledgments
The teaching resources available on this website and the related teaching activities carried out so far have been developed with the financial support of three Impact Grants from the University of East Anglia.
Contact
This website was created and is maintained by Davide Rizza, Associate Professor of Philosophy in the School of Politics, Philosophy, Language and Communication Studies, University of East Anglia, Norwich, UK.
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