According to Modern Mathematics (where the majority of mathematicians agree about the notion of actual infinite sets, as established mostly by George Cantor) an inductive set (as given by ZF(C) Axiom Of Infinity) has an accurate cardinality, which implies that it is complete (no one of its members is missing).
In other words, by ZF(C) Axiom Of Infinity there exists at least one infinite AND complete set (if we agree with the notion of actual infinity, as mostly established by Cantor).
Now, assume a complete set of infinite axioms (according to the reasoning of actual infinity, as established mostly by Cantor and agreed by the majority of modern mathematicians).
But by Gödel's First Incompleteness Theorem such set of axioms must be inconsistent as follows:
Set A (which is strong enough is order to deal with Arithmetic) is a set of infinitely many axioms (where each axiom is written by finitely many symbols) which is established by using ZF(C) Axiom Of Infinity on ZF(C) itself, such that Infinity is taken in terms of Platonic Infinity (By Platonic Infinity there exists a set of infinitely many things as a complete whole (without using any process)).
Some example: The infinite set of all natural numbers is taken in terms of Platonic infinity.
Now all we care is about the set of all infinitely many wffs (in terms of Platonic Infinity) that are established in A .
Each wff has some Gödel number, where at least one of these wffs, called G, states "There is no number m such that m is the Gödel number of a proof in A , of G" (since G needs a proof, it is not an axiom but a theorem).
Since all wffs are already in A and all Gödel numbers are already in A (because Infinity is taken in terms of Platonic Infinity) there is a Gödel number of a proof of G in A , which contradicts G in A , exactly because A is complete (as shown) and therefore inconsistent.
So the problem is actually the notion of a complete set of infinity many things in terms of Platonic Infinity, and in order to save the consistency of A, ZF(C) Axiom Of Infinity is taken in terms of Potential Infinity (process is used, exactly as done in case of GIT in its standard sense).
But then ZF(C) Axiom Of Infinity can't be used in order to establish sets in terms of Platonic Infinity (for example: the notion of The infinite set of all natural number is logically inconsistent).
Gödel was a Platonist (he agreed with Actual infinity in terms of Cantor (which is actually Platonic Infinity)) and his main motivation behind his Incompleteness Theorems was to logically demonstrate that formal systems that are strong enough in order to deal with Arithmetic, can't be complete AND consistent and also can't prove their own consistency (which means that many "interesting" formal systems can't deal with Platonic realms).
But Gödel's First Incompleteness Theorem also proves that the very notion of Actual infinity in terms of Platonism (which is also Actual infinity in terms of Cantor) does not hold logically.
There is a non-interesting solution about the discussed subject, as follows:
G states: "There is no number m such that m is the Godel number of a proof in A , of G"
If G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Platonic Infinity) it is actually a wff that is true in A , which does not have any Gödel number that is used in order to encode G's proof (since axioms are true wff that do not need any proof in A ).
But then no proof is needed and mathematicians are out of job (therefore it is an unwanted solution).
Also please be aware of the following:
1) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then ZF(C) Axiom Of Infinity is taken in terms of Platonic Infinity OR Not (useless tautology).
2) If ZF(C) Axiom Of Infinity is not necessarily taken in terms of Platonic Infinity, then it can't be used in order to establish even the set of all natural numbers (which means that N (and | N |) is not necessarily established by ZF(C)).