If our axiom set T is independent of a property P about numbers then by definition there is nothing inconsistent about the theory T1 = “T and P” and also nothing inconsistent about the theory T2= “T and not P”.
To say that they are not inconsistent is to say that they are satisfiable, that they have possible models. As T1 and T2 are inconsistent with each other, their models are different.
The single zero-based chain of numbers without nonstandard numbers is a single model. Therefore, if there exists a property about numbers that is independent of any theory of arithmetic, that theory of arithmetic does not logically exclude the possibility of nonstandard elements.
By Godel’s incompleteness theorems, a theory must have statements that are independent from it unless it is either inconsistent or has a non-recursively-enumerable theorem set.
Each instance of the axiom schema of induction can be constructed from a property. The set of properties is recursively enumerable, therefore the set of instances of the axiom schema of induction is recursively enumerable.
Every theorem of Peano Arithmetic must use a finite number of axioms in its proof. We can enumerate the theorems of Peano Arithmetic by adding increasingly larger subsets of the infinite set of instances of the axiom schema of induction to our axiom set.
Since the theory of Peano Arithmetic has a recursively enumerable set of theorems it is either inconsistent or is independent of some property and thus allows for the existence of nonstandard elements.
If our axiom set T is independent of a property P about numbers then by definition there is nothing inconsistent about the theory T1 = “T and P” and also nothing inconsistent about the theory T2= “T and not P”.
To say that they are not inconsistent is to say that they are satisfiable, that they have possible models. As T1 and T2 are inconsistent with each other, their models are different.
The single zero-based chain of numbers without nonstandard numbers is a single model. Therefore, if there exists a property about numbers that is independent of any theory of arithmetic, that theory of arithmetic does not logically exclude the possibility of nonstandard elements.
By Godel’s incompleteness theorems, a theory must have statements that are independent from it unless it is either inconsistent or has a non-recursively-enumerable theorem set.
Each instance of the axiom schema of induction can be constructed from a property. The set of properties is recursively enumerable, therefore the set of instances of the axiom schema of induction is recursively enumerable.
Every theorem of Peano Arithmetic must use a finite number of axioms in its proof. We can enumerate the theorems of Peano Arithmetic by adding increasingly larger subsets of the infinite set of instances of the axiom schema of induction to our axiom set.
Since the theory of Peano Arithmetic has a recursively enumerable set of theorems it is either inconsistent or is independent of some property and thus allows for the existence of nonstandard elements.