I don't have time to read the whole thread to see if you got your answer to this, but one of the more famous Math theorems of the 20th century says exactly this, developed by Kurt Goedel, and known as Goedel's Theorem. To paraphrase (since his phrasing is gibberish to all but a mathematician), "in any logical system with a closed set of rules, there will exist undecidable propositions which are, nonetheless true."
There's a great book by someone with a name like Hofstadler or something close to that, entitled "Goedel, Escher, Bach: An Eternal Golden Braid," which puts it all in laymans terms. Highly recommended, though lengthy.
"...in any logical system with a closed set of rules, there will exist undecidable propositions which are, nonetheless true."
I have always been bemused by the various ways Goedel's Theorem is presented. The one here is intersting because it says, "there will exist undecideable propositions which (Goedel has decided) are, nonetheless true."
It is, "any logical system with a closed set of rules" which lets this theorem out as being applicable to the sciences. The meaning of "truth" in closed systems is merely symbolic, and what is and is not true in such a system is predetermined by the rules of that system. It might be true you get $200 every time you pass "Go" in Monopoly, and there might be situations in Monopoly that are true but not proveable, but that truth is not exactly scientific truth.
Goedel's Theorem is not applicable to any scientific hypothesis. It has meaining only within the context of symbolic logic, which is useful in the design of computer software and with regard to certain questions in mathematics, but "true," in this context, is not what you or I or anyone not specifically thinking in terms of symbolic logic means by true.
At least, that is how I understand it. (I do not mean, by the way, that the discipline that studies things like symbolic logic is not science, but that symbolic logic, "within the limits of a defined closed systems," is not applicable to other sciences, because the rules of a closed system have already determined what will be true and not true, so nothing outside it can be discovered." It is discovering new truths, after all, that is the business of science.)
Thanks again.
Hank
Not at all what Godel proved. For example, plane Euclidean geometry with Hilbert's axioms has no undecidable propositions. Trivially, first-order logic has no undecidable propositions. One needs a system that allows the development of arithmetic to get undecidiablity.
The points are rather subtle. The real line (as a subset of Euclidean geometry) has no undecidable geometrical propositons. Of course, the integers as a subset of the reals do have undecidable propositions. One cannot uniquely select the integers out of the reals.