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 those indicated by the expressions 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, traditionally we decide to identify them. If we do not record differences, we must thus assign distinct collections the same specification 'infinity': as a consequence, we are led to expressions like 'infinity - infinity', which are not reducible to any numerical specification in base ten.
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.
One simple, pleasant effect of this approach is that it turns many paradoxes of infinity, which essentially rely on the unavailability of determinations for infinite collections, into arithmetical exercises. Besides giving rise to much research work in applied mathematics (especially the fields of optimisation and linear and non-linear programming), the Arithmetic of Infinity has been the subject of a number of engaging workshops in schools across Italy, Lebanon and the UK.
Aim
The resources on this website are a way for teachers and students to find out more about the Arithmetic of Infinity, design their own enrichment session on it or ask for one to be given at their school.
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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