Just how can there be specific polynomial formulas for which the presence of integer remedies is unprovable?

This answer recommends that there are specific polynomial formulas for which the presence (or nonexistence) of integer remedies is unprovable. Just how can this be?

2019-05-05 05:28:53
Source Share
Answers: 2

The response to this inquiry relies on just how the trouble is specified, yet the solution is no, at the very least without specifying the trouble in a deceptive means.

Given that my first remedy was entirely off the mark, I have actually removed it and also uploaded this new one.

Take into consideration polynomial p. If it has an integer remedy, after that the remedy will become located by arbitrary presuming. So if it is difficult to confirm the existential standing, there have to be no remedy.

Currently we understand from this link that there is a polynomial, q, that is unresolvable in the integers iff ZFC corresponds. It is popular that ZFC can not confirm its very own uniformity. So if it ZFC corresponds, after that q is unresolvable, yet we can not confirm this as after that we can confirm ZFC. So it feels like it is exact to claim that if math corresponds, we have a polynomial without integer origins, yet we can not confirm it. Nonetheless, if we are thinking mathematics corresponds, we can utilize this to confirm that the formula is unresolvable (without a doubt that is what we have actually done). So, it actually isn't exact an exact declaration in all.

To better make clear, when taking into consideration mathematical fact, there are 2 standard means of watching it. The first is where we are thinking that are axioms hold true, which always suggests thinking uniformity. If we show any kind of trouble amounts uniformity, after that we consider it to be real.

The various other is where we are taking into consideration an official set of declarations, of which the axioms have actually been specified to be real and also seeing which declarations can be acquired to be real. From this point of view, we do not in fact recognize whether the axioms correspond or otherwise. Actually, Godel's 2nd incompleteness theory reveals that no "non - unimportant" atomic system can confirm its very own uniformity. So revealing a trouble amounts uniformity is in fact the like revealing that the trouble is unprovable by the atomic system.

The complication originates from thinking ZFC corresponds to remove one opportunity in a selection, yet not permitting this presumption to be made use of as an axiom in the evidence.

2019-05-09 00:59:23

Edit : This has actually been clarified much better by Joel David Hamkins ; cf. this MO thread.

This is due to the fact that offered any kind of first - order formula $\phi(x_1, \dots, x_n)$ with $n$ parameters in the integers, there is a polynomial $P(x_1, \dots, x_n, z_1, \dots, z_m)$ which can be confirmed to have an origin in $z_1, \dots, z_m$ in the integers if and also just if $\phi(x_1, \dots, x_n)$ - - - this is a variation of Matiyasevich's remedy to Hilbert's tenth trouble.

Currently, it's feasible to construct within any kind of official system specific first - order solutions which can not be confirmed or refuted (as an example, the declaration "this official system corresponds"). This is the 2nd incompleteness theory. These first - order solutions represent polynomials by the first paragraph.

Specifically, one can show that you can construct a polynomial $P(z_1, \dots, z_m)$ representing the declaration "math * is irregular" converted to a formula. Hence, if math corresponds, there's no mathematical evidence that this polynomial does not have an origin.

*Math = ZFC below.

For extra on these lines, see Ebbinghaus - Frum - Thomas's really obtainable publication on mathematical logic.

2019-05-08 06:36:53