Is Arithmetic Determinate?

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.