Enjoy A Problem Course in Mathematical Logic. Read Definition 6.4, Definition 6.5, and Definition 6.6 (Edit: They are on PDF pages 47-50, book pages 35-38.). It means that, within each model of the axioms, it is the case that every object in the model has the specified property. The natural numbers happen to be a model of first-order Peano arithmetic.
Let me ask you what “every x” means in first-order ZFC set theory. Answer carefully—it has a countable model.
What does “Every x” mean in the absence of set theory?
Enjoy A Problem Course in Mathematical Logic. Read Definition 6.4, Definition 6.5, and Definition 6.6 (Edit: They are on PDF pages 47-50, book pages 35-38.). It means that, within each model of the axioms, it is the case that every object in the model has the specified property. The natural numbers happen to be a model of first-order Peano arithmetic.
Let me ask you what “every x” means in first-order ZFC set theory. Answer carefully—it has a countable model.