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The compactness theorem states that if every finite subset of a set of logical statements is consistent, then the overall set of statements is consistent.

So, why is the following set of statements (each of which could be formalized under the rules of predicate logic) about a given universe not a counterexample?

  • There exists at least one distinct object in our universe.
  • There exist at least two distinct objects in our universe.
  • There exist at least three distinct objects in our universe.
  • There exist at least four distinct objects in our universe.

...

  • There exist finitely many distinct objects in our universe.

Any finite subset of these statements is consistent, yet the overall set is inconsistent. Is this not a counterexample because the last sentence cannot be formalized in first-order predicate logic?

George Bentley
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1 Answers1

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(each of which could be formalized under the rules of predicate logic)

...

There exist finitely many distinct objects in our universe.

Are you sure that last statement is actually appropriately expressible? Indeed, the compactness theorem shows that it isn't.

(In other words, the very last line of your post is exactly right.)

Noah Schweber
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