Turing Machine Halting Problem, Russell’s Paradox and Gödel Incompleteness Theorem

Abstract

The Turing Machine Halting Problem is a major problem in computer theory, Russell’s Paradox is the root of the Third Mathematical Crisis, and the Gödel Incompleteness Theorem is a major discovery in modern logic. The three have had a profound impact on the development of science and have attracted the attention of scientific and philosophical circles. However, since the Gödel Incompleteness Theorem was put forward, the scientific and philosophical significance of its proof has been questioned; in particular, Wittgenstein regards it as a certain logical paradox, and Russell’s Paradox has not yet been settled. This paper makes a detailed analysis of the three based on the view of dialectical infinity. The author notes that the Principle of Comprehension based on the view of actual infinity is the root of Russell’s Paradox. The Turing Machine Halting Problem shows that it is impossible to make an actual-infinite ultimate judgment of the constantly generated infinite world, but the philosophical significance of the Gödel Incompleteness Theorem is that our understanding of the world is essentially potentially infinite. At the end of the article, the author raises several questions about the proof of the Gödel Incompleteness Theorem, finds out the specific paradox form in the proof, points out the high consistency of its proof method and Russell’s Paradox, which strongly supports Wittgenstein’s view. The author points out that the philosophical basis of the proof of the Gödel Incompleteness Theorem is the idea of actual infinity, the proof of the theorem is based on a logically invalid circular formula, the contradiction of the proof originates from the Gödel formula itself, and cannot be attributed to the incompleteness of the system, so the proof is wrong. Therefore, the conclusion of this paper is that the world is constantly developing and changing, and our human understanding of the world is essentially a potential infinite, that is, the world is Aristotelian, not Platonic.