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The new interpretation of arithmetic operation symbols
viXra, 2014
We introduce the permutation group of arithmetic operations symbols by getting the permutations of all the common arithmetic operations symbols, with keeping the brackets out of ordering. We find 6 ways of doing the arithmetic operations. Therefore the output of any mathematical formulas depends on which one element of the arithmetical permutation group we work on. We find invariants by the reordering of the arithmetic operation x+y, xy. Working with the irreducible representation of the permutation arithmetic symbols group we define new arithmetic structures called arithmetic particles symbols.
Annals of Pure and Applied Logic, 1985
We establish by elementary proof-theoretic means the conservativeness of two subsystems of analysis over primitive recursive arithmetic. The one subsystem was introduced by Friedman [6], the other is a strengthened version of a theory of Mint ; each has been shown to be of considerable interest for both mathematical practice and me&mathematical investigations. The foundational significance of such conservation results is clear: they provide a direct finitist justification of the part of mathematical practice formalizable in these subsystems.
2022
The purpose of this article is to introduce \textit{the geometry of operations}. Let psi\psipsi be a bijection from a nonempty set GGG into another set G′G^{'}G′, we prove that for any fixed operation on GGG, there exists a \textit{unique} operation on G′G^{'}G′, where psi\psipsi is an \textit{isomorphism}. This finding allow us to show that we can make the set of natural numbers a \textit{commutative field}. Therefore we establish that the symmetric group mathcalSG\mathcal{S}_{G}mathcalSG of a given nonempty set GGG \textit{acts} on the set of all operations on GGG. Then we generalize the notions of ttt-norms, ttt-conorms and we give the equivalent of the ttt-conorm for \textit{the triangle function}.
Applied Mathematics, 2014
New operators are presented to introduce "arithmetic calculus", where 1) the operators are just obvious mathematical facts, and 2) arithmetic calculus refers to summing and subtracting operations without solving equations. The sole aim of this paper is to make a case for arithmetic calculus, which is lurking in conventional mathematics and science but has no identity of its own. The underlying thinking is: 1) to shift the focus from the whole sequence to any of its single elements; and 2) to factorise each element to building blocks and rules. One outcome of this emerging calculus is to understand the interconnectivity in a family of sequences, without which they are seen as discrete entities with no interconnectivity. Arithmetic calculus is a step closer towards deriving a "Tree of Numbers" reminiscent of the Tree of Life. Another windfall outcome is to show that the deconvolution problem is explicitly well-posed but at the same time implicitly ill-conditioned; and this challenges a misconception that this problem is ill-posed. If the thinking in this paper is not new, this paper forges it through a mathematical spin by presenting new terms, definitions, notations and operators. The return for these out of the blue new aspects is far reaching.
The Tractatus System of Arithmetic
Synthese, 1997
The philosophy of arithmetic of Wittgenstein's Tractatus is outlined and the central role played in it by the general notion of operation is pointed out. Following which, the language, the axioms and the rules of a formal theory of operations, extracted from the Tractatus, are presented and a theorem of interpretability of the equational fragment of Peano's Arithmetic into such a formal theory is proven.
On arithmetical first-order theories allowing encoding and decoding of lists
Theoretical Computer Science, 1999
In Computer Science, n-tuples and lists are usual tools ; we investigate both notions in the framework of first-order logic within the set of nonnegative integers. Gödel had firstly shown that the objects which can be defined by primitive recursion schema, also can be defined at first-order, using natural order and some coding devices for lists. Secondly he had proved that this encoding can be defined from addition and multiplication. We show this can be also done with addition and a weaker predicate, namely the coprimeness predicate. The theory of integers equipped with a pairing function can be decidable or not. The theory of decoding of lists (under some natural condition) is always undecidable. We distinguish the notions encoding of n-tuples and encoding of lists via some properties of decidabilityundecidability.
On Certain Extentions of the Arithmetic of Addition of Natural Numbers
Mathematics of the USSR – Izvestia. 15:2, 1980
In this paper the problems of expressibility and decidability are studied for elementary theories obtained by extending the arithmetic of order and the arithmetic of addition of natural numbers. Results are obtained on the decidability and undecidability of elementary theories of concrete structures of the form ⟨N;+,P⟩, where P is a fixed monadic predicate, as well as results on the class of sets definable in the theory T⟨N;+,λx,∃y(x=dy)⟩.