10-01-2026, 04:21 PM
The article “What Stronger Mathematical Language Can Reveal About Computation, Sets, and Infinity”, by Louis D’Alotto and Dmitri E. Kvasov, examines the idea that some apparent mathematical limitations may arise not from the objects themselves, but from the language and notation used to describe them. The authors begin with examples such as Roman numerals, where the absence of zero makes certain calculations awkward, and the Pirahã numeral system, which reportedly distinguishes only a few exact quantities before using the general concept of “many.” They use these examples to argue that richer mathematical languages can reveal distinctions that weaker systems conceal. This perspective is then extended to the mathematical treatment of infinity, where the authors introduce Yaroslav Sergeyev’s grossone numeral, written as ①, as an alternative framework for representing infinite and infinitesimal quantities.
The central mathematical example is Turing’s diagonal argument, traditionally used to demonstrate the existence of uncomputable functions. In the standard formulation, all computable binary functions can be arranged in an infinite square table, and changing each diagonal element produces a new function that differs from every function in the table. The article argues that this construction depends crucially on how infinity is represented. Sergeyev’s grossone system treats ① as the number of elements of the natural numbers and permits expressions such as $①-1$, $①$, and $①+1$. In this framework, the natural numbers can be represented as a finite-looking sequence extending to ①, while the number of binary functions is represented as $2^{①}$.
From this perspective, the table used in Turing’s argument is interpreted not as a square table but as a rectangular table with ① columns and $2^{①}$ rows. The function constructed by diagonalization can therefore appear explicitly as another row of the table rather than as an uncomputable object outside it. The authors emphasize that this resembles the finite case, where diagonalization over a finite collection of binary functions does not produce an uncomputable function because the resulting function is simply another member of the complete finite table. The grossone framework is also presented as allowing finer distinctions among infinite quantities—for example, distinguishing $①$, $①+1$, $2①$, and $2^{①}$—where conventional cardinal arithmetic often identifies quantities such as $\aleph_0$, $\aleph_0+1$, and $\aleph_0-1$.
The article ultimately presents Infinity Computing based on grossone as a proposed extension of mathematical language that could affect both pure mathematics and computer science. The authors describe possible applications involving infinite sets, Turing machines, computation, optimization, and numerical analysis, arguing that richer notation may make previously invisible distinctions observable and computationally expressible. Importantly, the article presents this as a new mathematical paradigm and research program, rather than as an established replacement for classical computability theory. Its central philosophical message is that changing the mathematical language can change what distinctions we are able to formulate and investigate, while the grossone approach challenges readers to reconsider whether some conclusions traditionally associated with infinity reflect properties of infinity itself or limitations of the formal systems used to represent it.
ARTICLE
The central mathematical example is Turing’s diagonal argument, traditionally used to demonstrate the existence of uncomputable functions. In the standard formulation, all computable binary functions can be arranged in an infinite square table, and changing each diagonal element produces a new function that differs from every function in the table. The article argues that this construction depends crucially on how infinity is represented. Sergeyev’s grossone system treats ① as the number of elements of the natural numbers and permits expressions such as $①-1$, $①$, and $①+1$. In this framework, the natural numbers can be represented as a finite-looking sequence extending to ①, while the number of binary functions is represented as $2^{①}$.
From this perspective, the table used in Turing’s argument is interpreted not as a square table but as a rectangular table with ① columns and $2^{①}$ rows. The function constructed by diagonalization can therefore appear explicitly as another row of the table rather than as an uncomputable object outside it. The authors emphasize that this resembles the finite case, where diagonalization over a finite collection of binary functions does not produce an uncomputable function because the resulting function is simply another member of the complete finite table. The grossone framework is also presented as allowing finer distinctions among infinite quantities—for example, distinguishing $①$, $①+1$, $2①$, and $2^{①}$—where conventional cardinal arithmetic often identifies quantities such as $\aleph_0$, $\aleph_0+1$, and $\aleph_0-1$.
The article ultimately presents Infinity Computing based on grossone as a proposed extension of mathematical language that could affect both pure mathematics and computer science. The authors describe possible applications involving infinite sets, Turing machines, computation, optimization, and numerical analysis, arguing that richer notation may make previously invisible distinctions observable and computationally expressible. Importantly, the article presents this as a new mathematical paradigm and research program, rather than as an established replacement for classical computability theory. Its central philosophical message is that changing the mathematical language can change what distinctions we are able to formulate and investigate, while the grossone approach challenges readers to reconsider whether some conclusions traditionally associated with infinity reflect properties of infinity itself or limitations of the formal systems used to represent it.
ARTICLE
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