Mathematical  Culture and Thought

Mathematical Culture and Thought

Is Arithmetic Determinate?

Document Type : Survey

Author
Faculty of Mathematical Sciences, Shahid Beheshti University, Iran
Abstract
Is arithmetic determinate and definitive? In other words, are there reasons
for the correctness or falsity of each arithmetic sentence? For example, is there evidence
that Goldbach’s conjecture is true or false, even if we don’t know about it? At
first glance, the answer seems to be clearly yes. But how can you be sure? What does
Gödel’s incompleteness theorem say about this? What is the relationship between independence
of some axioms of set theory, such as the axiom of choice and the continuum
hypothesis, with these questions? In this article, we will examine these questions. In
addition, we will examine the impact of the presence of unconventional facilities such
as computing machines that are able to perform an infinite number of instructions in a
finite time, as well as proof systems equipped with infinite rules on the answers to the
above questions.
Keywords

Subjects


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Volume 41, Issue 2 - Serial Number 71
December 2022
Pages 97-105

  • Receive Date 21 May 2022
  • Revise Date 25 June 2022
  • Accept Date 07 September 2022
  • Publish Date 21 January 2023