u/LorenzoGB

Is the Compactness Theorem equivalent to Proposition 2? I ask because of the following:

1. For all X1, if X1 is a theory in FOL and for all X2, if X2 is a finite subset of X1 then X2 is consistent, then X1 is consistent.

2. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite or X1 is a theory in FOL and X2 is not a subset of X1 or X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

3. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not finite then X1 is consistent.

4. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is not a subset of X1 then X1 is consistent.

5. Therefore, for all X1 there exists X2 such that if X1 is a theory in FOL and X2 is consistent, then X1 is consistent.

With regard to generalized quantifiers, the following deduction is invalid: There are exactly three cats. All cats are mammal. Therefore, there are exactly three mammals. This is because the generalized quantifier exactly three is non-monotonic. Yet suppose I interpret the quantifier exactly three as the following: There exist X1, X2, and X3 such that X1, X2, and X3 are cats and none of them are identical to each other. Then the reasoning is valid because exactly three mammals can be interpreted as there exist X1, X2, and X3 such that X1, X2, and X3 are mammals and none of them are identical to each other. So can non-monotonic generalized quantifiers be interpreted as strings of existential quantifiers of whatever cardinality?