Incompleteness: Do supernatural numbers function logically as unknowns?

Let T be standard arithmetic, T' the same theory of arithmetic constructed
within T, and let G be Goedel's undecidable statement, "My reflection G' in
T' is unprovable." Its negation, ~G, is "G' is provable in T'."

T is called omega-consistent if, whenever P(1), P(2), ... are provable for
all numerals 1, 2, ... it is not the case that an exception Ex~P(x) is
provable.

Then, if T is omega-consistent, G is neither provable nor disprovable, and
we may add either G or ~G as an axiom to T. If we add ~G, then, if I have
been correct, T' turns out inconsistent within T + ~G. Most people seem to
prefer G over ~G, believing that the choice of G is due to a noncomputable
property of human knowledge.

But suppose we add ~G. In that case, we arrive at what are called
"supernatural numbers," one of which is the proof of G' in T'. Let's call
this supernatural proof 'x'. Then x has the property of being inductively
inaccessible outside T + ~G (it does not occur in the sequence 1, 2, ...),
but inductively *accessible* in T + ~G. How is this so?

I believe I've found the answer. The idea is that x, although not occurring
in the list 1, 2, ..., inherits all the inductive properties of 1, 2, ... in
T + ~G, because it functions logically as an unknown. Of course, it is a
number of its own (we call it 'x'), but it *acts* like a variable that can
take on any of the values 1, 2, ... For this reason, any property of 1, 2,
.... also applies to x, whatever it may be (and we call it the "new number
x").

It is as if we have "removed" the property of its inductive inaccessibility
in the theory T + ~G by giving it the status of functioning logically as an
unknown.

Have I understood correctly?


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