Document Type : Brief Research
Author
Associate Professor of Mathematics Department of Shahid Beheshti University
Abstract
First, in the light of Feferman’s views, we will examine Gödel’s dichotomy that either the capabilities of the human mind are beyond any finite machine, or there are Diophantine-type mathematical equations that are absolutely unsolvable. Then we examine Putnam’s argument that if scientific competence of the mind can be simulated by a Turing machine with the ability to prepare a list of scientific propositions, this machine will not print out the sentence that expresses this ability. In an effort to better understand this proof, we restate it in the language of modal logic. Then, we discuss the possibility of supertask computations to perform infinite basic operations in finite time. This is a possibility that has recently been proposed based on new physical theories. We argue that, assuming that such a possibility is realized, arithmetic will be determinate, meaning that the truth or falsity of each arithmetic sentence will be explainable.
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