[DeathArrow]

>Laurence Kirby and Jeff Paris[1] showed that it is unprovable in Peano arithmetic (but it can be proven in stronger systems, such as second-order arithmetic). This was the third example of a true statement about natural numbers that is unprovable in Peano arithmetic, after the examples provided by Gödel's incompleteness theorem and Gerhard Gentzen's 1943 direct proof of the unprovability of ε0-induction in Peano arithmetic.

It seems math is never perfect but always perfectible. A perfect system wouldn't have paradoxes. One common example is Russel paradox.

We arrive at different conclusions by choosing a different set of axioms and constructing everything else based on that set. We can have parallels that intersect and parallels that don't.

[p-e-w]

We don't want perfect systems but useful ones. A perfect axiom system wouldn't have true but unprovable statements either, yet, as we learned a while ago, any such "perfect" system would be unable to express even basic arithmetic.

[DeathArrow]

>We don't want perfect systems but useful ones.

I think that is the difference between science and engineering. Science strives for the ultimate truth while engineering cares about useful stuff.

[p-e-w]

During the past century, it has been demonstrated again and again that the "ultimate truth" either doesn't exist or cannot be attained (Incompleteness Theorem, Uncertainty Principle, Observable Universe, not to mention a million philosophical and psychological problems).

If that is what science strives for, it's a lost cause. Fortunately, lots of valuable things can be achieved without chasing such lofty, unattainable goals.

[Karellen]

If "ultimate truth" cannot be attained, but can be asymptotically approached, is striving for it still a lost cause if the knowledge gained in the approach might be useful?

I'll never be able to run a marathon as fast as Eliud Kipchoge. That doesn't mean it's a lost cause for me to try to get my marathon time as close to his as possible - I can still achieve valuable things despite the goal being lofty and unattainable. Further, I might achieve more through chasing an unattainable goal, than I would if I'd set my sights lower.

It's also worth remembering the aphorism that people saying: “It can’t be done,” are always being interrupted by somebody doing it.

[lanstin]

Ones reach should exceed one’s grasp, else what’s a heaven for.

[baq]

Actually… reaching the very limits of knowability is the crown achievement of science and your first two examples have very important engineering implications.

Given these economies, perhaps it makes sense to say wherever in science we aren’t at the boundary of knowable, there’s still something worth discovering.

[klyrs]

I disagree strongly with your reading of those results. Each says that the "ultimate truth" is complicated -- that not every question has an easy answer. That doesn't say there isn't an ultimate truth, just "that question cannot be answered" is the ultimate truth. Exploring the bounds of knowability is incredibly important to the dual side of science; exploring the bounds of knowledge. In my lofty opinion, the very purpose of humanity is to bring these bounds together.