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Well, it is a little unclear what you mean by nonconstructive proof still. But you can think of what models of $ZFC+

eg{Con(ZFC)}$ looks like when you consider $ZFC+

eg{Con(ZFC)}$ as a formal theory, which seems to be part of the question you are asking.

First of all a model of the sort of theory you are looking at would have non-standard natural numbers else the inconsistency would be witnessed making $ZFC$ itself inconsistent. Also to make matters even more confusing, there maybe models of what it believes to be $ZFC$ in there! (This is weird nesting sort of argument because we can develop set theory inside of set theory inside of set theory inside of set theory........)

Again, note that what I'm talking about is a little tricky. I'm assuming that you are working within set theory and formalize, a separate copy if you will, of ZFC in there. Now you can do the proof of the incompleteness theorem, which gives you that (under the assumption that $ZFC$ is consistent) $ZFC+

eg{Con(ZFC)}$ is consistent. So you can look at those models of that theory.

Also it should be noted: considered as a formal theory, the incompleteness theorem applies to $ZFC+

eg{Con(ZFC)}$ too. So as a result we have that $ZFC+

eg{Con(ZFC)}+{Con(ZFC+

eg{Con(ZFC)})}$ and $ZFC+

eg{Con(ZFC)}+

eg{Con(ZFC+

eg{Con(ZFC)})}$. In fact this idea can be used to show that $ZFC+

eg{Con(ZFC)}$ if consistent has continuum many different completions to complete theories. So even at a countable level there are a maximum number of non-isomorphic models. As can be seen already, it will be a fairly complicated thing to study, as what exactly the model thinks about certain things will be dependent on what exactly it believes about statements like ${Con(ZFC+

eg{Con(ZFC)})}$ etc.

If you are interested; the study of non-standard Peano Arithmetic is well developed. As far as I know their interest is different from what you are asking about and I don't know how much of the ideas would carry over, but it seems as good a starting point as any.