Can Consistent but Different Axiomatic Systems Contradict Each Other?

Senia provided an insightful example involving simple axiom systems that illustrate the complexities within mathematical logic. Another similar example involves ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) and its independency from certain statements. In this article, we will delve into how consistent but different axiom systems can seemingly contradict each other, using Peano Arithmetic (PA) as a prime example. We will also explore the broader implications of these findings in the context of mathematical logic and the incompleteness theorem.

Introduction to Peano Arithmetic (PA)

Peano Arithmetic (PA) is a fundamental theory in the study of natural numbers. It is a consistent theory, and the consistency of PA can be proven within the framework of ordinary mathematics, which can be formalized in ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice) or simpler theories. A notable statement about PA is ConPA (Consistency of PA), which asserts that PA is consistent. This statement itself can be formalized in the language of arithmetic used in PA.

The Formalization of ConPA

The statement ConPA is formally written as "01 is not a theorem of PA". This is achieved by encoding the axioms of PA and the deduction rules of first-order logic into arithmetic manipulations. While this process is tedious, the resulting formula for ConPA can be constructed. This process demonstrates how logical statements can be translated into arithmetic expressions, providing a bridge between different branches of mathematics.

Consistency vs. Truth

Senia correctly noted that while ConPA is true (since 0 does not equal 1 and PA cannot prove this contradiction), it cannot be proven within PA itself. This is a direct consequence of G?del's Second Incompleteness Theorem. According to this theorem, any sufficiently powerful, consistent, and recursively enumerable axiomatic system (such as PA) cannot prove its own consistency. This is a profound result in mathematical logic, as it shows that consistency is a property that goes beyond the system's ability to prove it.

The Coexistence of Consistent but Inconsistent Statements

With this in mind, let us consider the following two axiomatic systems:

PA ConPA: This system includes PA and the statement ConPA, which asserts the consistency of PA. PA ?ConPA: This system includes PA and the negation of ConPA, which asserts the inconsistency of PA.

Despite their differences, both systems are consistent. However, only one of these systems can be considered sound, as the second system asserts an false claim (the inconsistency of PA), making it an unsound system. The second system can still be consistent, meaning it has models where the non-standard natural numbers appear to provide proofs of contradictions in PA.

Implications in Mathematical Logic

This example from Peano Arithmetic illustrates the intricate nature of consistent but different axiom systems. In particular, it highlights the following points:

The concept of consistency does not necessarily imply truth within the system. The burden of proving the consistency of a system is shifted to a stronger system, which is a non-trivial task. Theoretical consistency does not preclude the possibility of different models where the same system behaves differently.

Conclusion

In summary, different consistent but distinct axiomatic systems can coexist without contradiction, as long as they are formalized and studied within appropriate frameworks. The example of PA and the two systems PA ConPA and PA ?ConPA further underscores the importance of G?del's Incompleteness Theorems and the nuances of consistency in mathematical logic. These findings have implications in various fields of mathematics, including set theory, logic, and the foundations of mathematics.

Keywords

- Axiom systems

- Consistency

- Incompleteness theorem

- ZFC

- PA