In this link they show how the compactness theorem fails in english:
(first page with the starbucs example):
My question is: Will it fail as well when we replace 'there are finitely many starbucks' with 'there are more girls than boys' in formal language? what are the other propositions I should add to the list (or set) instead of the other propositions mentioned there (there is at least X starbucks etc.), in order to show that every subset is compossible and yet the compactness theorem fails.