## The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity

#### Citation

Bagdasaryan, A. G. (2009). The arithmetic foundations of mathematics: constructing new mathematics with negative numbers beyond infinity. s. 112.#### Abstract

The basic and fundamental concept underlying the foundations of mathematics is the notion of natural number. Negative
numbers had been introduced to extend natural numbers to the set of all integers. Some properties of negative numbers
had long been remaining unclear, in particular, the order relation between positive and negative numbers. There existed at
least two approaches: (1) negative numbers are less than ”nothing” (zero), −1 < 0 (Descartes, Girard, Stifel), (2) negative
numbers are ”greater” than infinity, −1 > ∞ (Wallis, Euler, and probably Pascal) [1].
We present theoretical statements of a new mathematical conception underlying the construction of a new theory [2]
based on: (1) a new method for ordering the integers (first introduced, but in other form, in [3]): let a, b ∈ Z, then
a ≺ b ⇔ −1
a < −1
b
, thus getting Z = [0, 1, 2, ..., −2, −1]; the set Z can be geometrically represented as cyclically closed; (2)
a new class of real regular functions f(·) and the definition of Pb
a
f(·) that extends the classical definition to the case b < a:
let Za,b = [a, b] if a ¹ b and Za,b = Z \(b, a) if a Â b, Z \(b, a) = [a, −1]∪[0, b], then ∀a, b ∈ Z,Pb
k=a
f(k) = P
k∈Za,b
f(k);
(3) a set of conditions imposed on regular functions. From these we define a new regular method for infinite series summation
and find a unified approach to summation of divergent series, and to determination of limits of unbounded and oscillating
functions.
In this new setting we recently elementarily evaluate the zeta function and the zeta alternating function at integer
points [4-5]. We discover various surprising phenomena and unexpected results concerning some areas of mathematics,
obtained within the framework of this new theoretical background, which is being futher developed. We also discuss some
aspects of future research which will be based on the theory to be formulated as a paradigm.

#### Source

International Conference of Mathematical Sciences#### URI

https://www.maltepe.edu.tr/Content/Media/CkEditor/03012019014112056-AbstractBookICMS2009Istanbul.pdf#page=331https://hdl.handle.net/20.500.12415/5804

#### Collections

- Makale Koleksiyonu [586]

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