Can a False Statement Be Proven from True Statements and Accepted Axioms?

In the realm of mathematical logic, the relationship between truth and provability can be quite intriguing. In this article, we delve into the famous Tarski Undefinability Theorem and explore a classic example involving self-referential sentences that seem to challenge the foundations of formal systems. We'll discuss how false statements can appear to be provable when examined through the lens of accepted axioms and true statements.

Introduction to Tarski Undefinability Theorem

The Tarski Undefinability Theorem, introduced by the prominent logician Alfred Tarski, is a cornerstone in metalogic. It asserts that for any sufficiently powerful formal system, there exist statements that are true but cannot be proven within the system itself. This theorem is akin to G?del's Incompleteness Theorems but focuses specifically on the undefinability of truth within a formal system.

The Liar Paradox and Self-Referential Statements

To understand the theorem, we first need to explore the concept of self-referential sentences, a key component in the theorem. Consider the self-referential sentence, “This sentence is not true.” Let's denote this sentence by ( x ). If ( x ) were true, then it would be stating that it is not true, which is a contradiction. Conversely, if ( x ) were false, then it would mean that the sentence inside it is true, which again leads to a contradiction.

Formalizing the Liar Sentence

To formalize the Liar sentence in terms of provability and truth, let's assume the following:

Assumption 1: ( x ) is provable if and only if ( p ) Assumption 2: ( x ) is true if and only if ( p )

From these assumptions, we can derive the following:

Step 3: ( x ) is provable if and only if ( x ) is true.

Using the Law of Excluded Middle (axiom: ( eg text{True}(x) lor eg text{True}( eg x))), we can proceed as follows:

Step 4: If ( x ) is provable, then ( x ) is true. Step 5: If ( x ) is not provable, then ( eg x ) is true (i.e., ( x ) is false).

Combining these steps, we arrive at:

( x ) is true. ( x ) is provable. ( x ) is true.

Thus, ( x ) is both true and provable within the system.

Interpretation and Implications

The steps above show that the Liar sentence, despite being false in a real-world interpretation, can appear as both true and provable within the formal system. This paradox highlights the inherent limitations of formal systems and the potential for inconsistencies when dealing with self-referential statements.

Conclusion

In summary, the Tarski Undefinability Theorem exemplifies the complex relationship between truth and provability in mathematical systems. It demonstrates that within any sufficiently powerful formal system, there can exist statements that are true but cannot be proven, and conversely, that false statements can sometimes appear as provable. This theorem underscores the rich and nuanced nature of mathematical logic and its profound implications for both theoretical and applied mathematics.