Gödel Number Usefulness to Determine an Infinite Set's Completeness

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Gödel numbers are used to encode wffs of formal systems that are strong enough in order to deal with Arithmetic.

In my question, Gödel numbers are used to encode wffs as follows:

Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by taking the union of x with its singleton {x}, is also a member of A.

So, syntactically x → xU{x} is the bijective membership function of A.

(Some remark

xU{x} can be replaced by {x}, as follows:

Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any x is a member of A, the set formed by {x}, is also a member of A.

In this post xU{x} is used, but it can easily be replaced by {x} without changing the context of the question)Now we are using also Semantics (adding some meaning) by establish some models about this function, as follows:Model 1:

Let any x be an axiom (wff that is not proven) in A.

Let any xU{x} be a theorem (wff that is proven) in A.

Let A be an infinite set of wffs, where Infinity is taken in terms of Actual Infinity (A is taken as a complete whole).

Each wff (wff that is proven (some xU{x})) is encoded by a Gödel number, where 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 all wffs are already in A and therefore all Gödel numbers are already in A (because Infinity is taken in terms of Actual Infinity) there is a Gödel number of an axiom (some x) that proves G (some xU{x}) in A, which is a contradiction in A. Therefore, A is inconsistent.Model 2:

Let any x or any xU{x} be axioms (wffs that are not proven) in A.

G axiom states: "There is no number m such that m is the Gödel number of a proof in A, of G"

Since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Actual 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 wffs 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).

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So, in both models infinitely in terms of Actual Infinity (an infinite set that is taken as a complete whole) does not establish an interesting formal system.

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My question is:

Do Gödel numbers can be used in order to determine the usefulness of an infinite set as a complete whole (according to the given models)?

Please pay attention to the following remarks, before you reply:

Since A is a set of infinitely many wffs that are taken as a complete whole (this is exactly what Actual Infinity is about) there cannot be a Gödel number that is not already in A, whether some wff is an axiom or a theorem in A (see Model 1). So, one can't use G as a wff that is unproven in A, as done in case of GIT, since if one does this, one deduces in terms of Potential Infinity, which is not a part of my question.

 

[Valkarie]

Also, in Model 2, since G is already an axiom in A (where A is an infinite set of axioms, such that Infinity is taken in terms of Actual 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 wffs that do not need any proof in A). So, one can't use G as a wff that needs a proof in A, as done in case of GIT, since if one does this, one deduces in terms of Potential Infinity, which is not a part of my question.I would like to find out if Gödel numbers can be used in order to determine the usefulness of an infinite set as a complete whole (according to the given models).Thanks in advance.