Can Mathematical Theorems Exist Without Proofs if Their Axioms Are Not True?

At the heart of mathematics lies the concept of axioms, which are postulated as self-evident truths that serve as the foundational building blocks of any logical system. However, the question of whether mathematical theorems can exist without proofs when their axioms are not true is a complex and intriguing one. This piece explores this enigma through the lens of mathematical logic, particularly focusing on G?del's incompleteness theorems and the limitations of formal proof systems.

Understanding Axioms and Their Role

Axioms are defined as self-evident truths that form the basis of a logical system. By their very nature, axioms are considered to be true without the need for further proof. However, in the realm of logic, if a set of axioms is inconsistent (i.e., contains contradictory statements), it can lead to the derivation of any statement as a true theorem. This inherent flaw in the system presents a significant challenge to the concept of mathematical truth and proof.

The Impact of G?del's Incompleteness Theorems

In 1930, Kurt G?del published his famous incompleteness theorems, which fundamentally altered our understanding of formal axiomatic systems. G?del's first incompleteness theorem states that any consistent formal system rich enough to express the basic arithmetic of the natural numbers is incomplete; that is, there exist statements within the system that cannot be proven either true or false. This theorem challenges the idea that every mathematical statement can be definitively resolved within a system.

G?del’s second incompleteness theorem adds an extra layer of complexity: it asserts that for any consistent system capable of expressing basic arithmetic, the system cannot prove its own consistency. This means that even if we have a set of axioms that are consistent, we cannot prove that they are consistent using those axioms alone. This profound insight highlights the limitations of formal proof systems and the inherent uncertainty in mathematical reasoning.

Examples of Unprovable and Independent Statements

One notable example of a true but unprovable statement within a system is known as the G?del sentence. Consider a system based on the Zermelo-Fraenkel set theory with the axiom of choice (ZFC). The G?del sentence for a given system asserts its own unprovability. While the system cannot prove this statement, it is true because the system cannot prove its own consistency. This example demonstrates that there can exist true statements that are unprovable within the system itself.

The Nature of Axiom Systems and Proofs

Axiom systems are typically structured such that each axiom is considered self-evident and true. The proofs of these axioms are often trivial, consisting of a single line stating the validity of the axiom. In more complex systems, axioms can come in schemas, with each instance of the schema having a one-line proof. The concept of an axiom does not reflect the idea of proof being informative in the sense that the proof should be lengthy and detailed.

Many mathematicians and logicians reject the idea that informal proofs must necessarily be more informative than formal proofs. The distinction between formal and informal proofs is often blurred by the vagueness and subjective nature of the informal proofs. An informal proof can be true but not considered a valid proof if it relies on assumptions that are not part of the formal system.

The Diophantine Equation and Algorithmic Complexity

A concrete example illustrating the limitations of formal proof systems can be found in the realm of Diophantine equations. A Diophantine equation is a polynomial equation with integer coefficients, and the problem is to find integer solutions. While it is theoretically possible to find solutions by calculation, there is no known algorithm that can determine whether a Diophantine equation has a solution in general. This is equivalent to the famous halting problem in theoretical computer science, highlighting the inherent computational limitations.

Given a formal axiom system, there are well-known limitations. For instance, there is no axiom system that can correctly decide all Diophantine equations. If such a system existed, it would provide a slow algorithm to determine the existence of solutions by exhaustively searching through all possible proofs in a controlled order of increasing complexity. However, the absence of such a system is not a point of concern, as it reflects the inherent complexity and limitations of mathematical proof systems.

Mathematicians generally accept proofs based on the axioms of Zermelo-Fraenkel set theory with the axiom of choice (ZFC). While ZFC is presumed to be consistent, it cannot prove its own consistency. This inconsistency is flagged by mathematicians, but a proof from the assumption of ZFC's consistency is still considered a valid informal proof, provided that the extra assumption is made clear.

Conclusion

In summary, the question of whether mathematical theorems can exist without proofs if their axioms are not true is addressed through the lens of G?del's incompleteness theorems and the inherent limitations of formal axiomatic systems. The G?del sentence exemplifies true but unprovable statements, and the Diophantine equation problem demonstrates the computational boundaries of formal proof systems. While there are limits to what can be proven in a formal system, this does not negate the truth of certain mathematical statements, as highlighted by the G?del sentence and the concept of algorithmic undecidability.