I think you're too quick to dismiss what is widely considered to be one of the most important and profound results in logic and computability without taking the time to fully appreciate it.
Your procedure of repeated substitution in order to resolve a paradox does little to shed light on the situation since at the heart of the halting problem is that there's no formal system that can determine whether this repeated substitution will ever come to an end.
Sure in your trivial example we could prove that it never comes to an end and determine that the statement is a paradox and label it as such, but there are some statements where it's not clear whether there's a paradox in the first place and there are some statements that are a paradox in one interpretation but in another interpretation are perfectly sensible.
Rest assured, no mathematician is scratching their head wondering whether "This statement is false." is some kind of mysterious statement whose undecidability has profound consequences. The issue is with statements like whether there exist 3 integers, x, y, z such that:
x^3 + y^3 + z^3 = 114
That statement in and of itself may very well be a "paradox" similar to a statement such as "This statement has no proof in theory T." even though on the surface it looks like a perfectly reasonable equation that should either have a solution or not have a solution. I mean either three such numbers exist or they don't exist, right?
And yet... it's possible that for the equation I gave there are solutions only under some interpretations of what we normally call natural numbers and in other interpretations of what we call natural numbers there isn't a solution, and there's no formal system that can filter out one interpretation over another so that there is one and only one unambiguous interpretation of natural numbers that we can always rely upon as the "real" interpretation.
There are perfectly normal looking sentences and problems that on the surface don't appear at all to be paradoxes or self referential, but then become so when you try to pin the question more precisely.
That's where the fascination comes from, from looking at a seemingly normal looking equation or statement about numbers that should just be true or false and realizing that whether it's true or false depends on some very deep and as-of-yet unknown properties of what we even mean when we talk about natural numbers.
PP seems to be asking for an example that is not obviously self-contradictory yet demonstrates a serious, unsurmountable problem, thereby illustrating the incompleteness theorem.
Is the theorem you describe such an example? If so, you have refuted PP with a non-handwaving result.
If not, then you've provided more evidence supporting PP's complaint.
Your procedure of repeated substitution in order to resolve a paradox does little to shed light on the situation since at the heart of the halting problem is that there's no formal system that can determine whether this repeated substitution will ever come to an end.
Sure in your trivial example we could prove that it never comes to an end and determine that the statement is a paradox and label it as such, but there are some statements where it's not clear whether there's a paradox in the first place and there are some statements that are a paradox in one interpretation but in another interpretation are perfectly sensible.
Rest assured, no mathematician is scratching their head wondering whether "This statement is false." is some kind of mysterious statement whose undecidability has profound consequences. The issue is with statements like whether there exist 3 integers, x, y, z such that:
x^3 + y^3 + z^3 = 114
That statement in and of itself may very well be a "paradox" similar to a statement such as "This statement has no proof in theory T." even though on the surface it looks like a perfectly reasonable equation that should either have a solution or not have a solution. I mean either three such numbers exist or they don't exist, right?
And yet... it's possible that for the equation I gave there are solutions only under some interpretations of what we normally call natural numbers and in other interpretations of what we call natural numbers there isn't a solution, and there's no formal system that can filter out one interpretation over another so that there is one and only one unambiguous interpretation of natural numbers that we can always rely upon as the "real" interpretation.
There are perfectly normal looking sentences and problems that on the surface don't appear at all to be paradoxes or self referential, but then become so when you try to pin the question more precisely.
That's where the fascination comes from, from looking at a seemingly normal looking equation or statement about numbers that should just be true or false and realizing that whether it's true or false depends on some very deep and as-of-yet unknown properties of what we even mean when we talk about natural numbers.