Can You Use a Theorem in Its Own Proof?

When proving a theorem, it is generally not advisable to use the theorem itself in the proof. This would create a circular argument, which defeats the purpose of a rigorous proof. A proof should rely on previously established results, axioms, or definitions that are independent of the theorem being proved.

Standard Mathematical Practice

In standard mathematical practice, a theorem must be proved without assuming its own truth. For example, if you were to attempt to prove the statement 11 3, you cannot simply use the statement itself as evidence. Doing so would result in a logically inconsistent proof that would allow you to prove almost anything, leading to the absurd conclusion that 1 3.

Let's illustrate this with an example:

Example: A Circular Proof

Consider the theorem: 1 1 3.

Proof:

11 3 (by the above theorem)

Therefore, 1 1 3. Q.E.D. (End of Proof).

This proof is invalid because it relies on the original assumption, creating a circular argument. Such a proof is not only logically untenable but also unacceptable in mathematical and logical reasoning.

Building on Proven Theorems

While you cannot use a theorem to prove itself, you can build proofs on top of existing theorems. This is a common and valid practice in mathematics. For instance, to prove a theorem for all real numbers, you might start by proving it for natural numbers, then integers, then rational numbers, and finally extend it to real numbers.

This method of progression from specific to general is often used in various branches of mathematics, such as number theory, algebra, and analysis. By proving a theorem for a subset of a larger set and then extending it to the whole set, you ensure the proof is logically sound and consistent.

For example, let's consider the theorem: For all real numbers x, x^2 - 2x 1 (x - 1)^2.

Proof:

Start by proving the theorem for natural numbers:

Let x be a natural number. Then, x^2 - 2x 1 (x - 1)^2. This can be verified by direct substitution or simple algebraic manipulation.

Next, prove the theorem for integers:

Let x be an integer. We have: x^2 - 2x 1 x^2 - 2x 1 (still true for integers).

Then, prove the theorem for rational numbers:

Let x be a rational number. We have: rational number manipulation confirms that x^2 - 2x 1 (x - 1)^2.

Finally, extend the proof to all real numbers:

Using the density of rational numbers in the real numbers, we can extend the proof to all real numbers. This approach demonstrates that the theorem holds for all real numbers based on its proven validity for the dense subset of rational numbers.

Exceptions to the Rule

While the rule of not using a theorem in its own proof is fundamental, there are specific contexts in mathematics where self-referential structures can be used. These contexts, such as certain areas of category theory or formal logic, might allow for a more nuanced approach to proofs.

Category Theory

In category theory, you might find instances where a structure is defined in such a way that a theorem can inform the very structure being defined. However, even in these cases, the theorem is not directly being used in its own proof but rather in the process of defining the context within which it operates.

For example, in category theory, the concept of initial object is defined such that it satisfies a universal property. The property itself can sometimes be the starting point for reasoning within the category, but the proof of a theorem about initial objects does not rely on the initial object being an initial object.

Logical Frameworks

In certain formal logical systems, self-referential statements can be handled through careful definitions and methods such as G?del's incompleteness theorems. However, these are still subject to rigorous scrutiny to ensure that they do not lead to contradictions or circular arguments.

To illustrate, let's consider a statement in a formal system that defines self-referential structures:

Example: G?del's Statement

Consider G?del's statement G which states "G is not provable." If G were provable, then it would contradict its own statement, and if it were not provable, it would be true, leading to a contradiction. This kind of self-referential structure is carefully handled to avoid circularity and ensure logical consistency.

Conclusion

In summary, while it is not acceptable to use a theorem in its own proof due to the risk of creating a circular argument, the logical and mathematical structure of mathematics allows for proofs to be built on top of each other. This approach ensures that each theorem is established as a building block, contributing to a larger, consistent framework of mathematical knowledge.

The nuances of mathematical proof underscore the importance of rigour and logical consistency. Understanding these principles not only enhances your ability to construct valid proofs but also deepens your appreciation for the elegance and power of mathematical reasoning.