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I am reading about Gödel's incompleteness theorem and I wanted to clarify something. What is the relation between a tautology and a theorem? If a formula $F$ is a theorem in a consistent axiom system, does that imply that this formula is a tautology, and not all tautologies are theorems?

Or does there exist an equivalence relation between a theorem and a tautology, I am really confused?

mrp
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Lilo
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1 Answers1

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In general, a tautology is a statement true simply by virtue of the structure of the sentence. For example, "All even numbers are even" is a tautology. "All even numbers are integers" is true, but it's not a tautology because the structure of the sentence doesn't tell you anything - you have to know what "even" and "integer" mean. A theorem is something that's true in every universe satisfying whatever axioms you've selected. So "All even numbers are even" and "All even numbers are integers" are both theorems of standard arithmetic, but "All even numbers are less than twelve" is not.

Another way of thinking about it is that a tautology is a formula $F$ that's true in every axiom system, not just one.

My personal rule of thumb is that tautologies are generally not interesting - they add nothing whatsoever to the conversation, and there really aren't any surprising or counterintuitive tautologies. Theorems, however, can be very interesting and unpredictable.

Reese Johnston
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  • Even if the axiom system is very expressive ( like Hilbert one) we know that we cannot have a complete axiom system and Gödel showed that there is no such complete axiom system. Isn't true that the universe of tautologies contains the universe of theorems ??? – Lilo Oct 12 '16 at 11:10
  • Nice answer! So we can say that tautologies are something like the corresponding theorems of logic? – user599310 Jul 20 '20 at 23:27