A Semantic Situation without Syntax (Non- axiomatizibility of Theories)

TL;DR

This paper introduces a semantic paradox related to the Unexpected Hanging paradox, demonstrating that certain theories, including Computability Theory, cannot be fully axiomatized due to their non-axiomatizable semantic nature.

Contribution

It presents a new semantic paradox and argues that some theories inherently lack the possibility of complete axiomatization, challenging traditional views on formal systems.

Findings

Semantic paradoxes can demonstrate non-axiomatizability of theories

Computability Theory cannot be fully axiomatized

Many theories in physics and mathematics are non-axiomatizable

Abstract

Here, by introducing a version of Unexpected hanging paradox first we try to open a new way and a new explanation for paradoxes, similar to liar paradox. Also, we will show that we have a semantic situation which no syntactical logical system could support it. Finally, we propose a claim in the subject of axiomatizibility. Based on this claim, having an axiomatic system for Computability Theory is not possible. In fact, the same argument shows that many other theories are non-axiomatizable. (Dare to say: General Theories of Physics and Mathematics).