The Lowenheim-Skolem theorem says (loosely interpreted) that even if you CLAIM to be talking about uncountably infinite things, there's a perfectly self-consistent interpretation of your talk that refers only to finite things (e.g. your definitions and proofs themselves).

Only in first-order logic. In second-order logic, you can actually talk about the natural numbers as distinguished from any other collection, and the uncountable reals.

Amusingly, if you insist that we are only allowed to talk in first-order logic, it is impossible for you to talk about the property "finite", since there is no first-order formula which expresses this property. (Follows from the Compactness Theorem for first-order logic - any set of first-order formulae which are true of unboundedly large finite collections also have models of arbitrarily large infinite cardinality.) Without second-order logic there is no way to talk about this property of "finiteness", or for that matter "countability", which you seem to think is so important.