Mark Gritter (markgritter) wrote,
Mark Gritter

Godelian Axe-Grinding

Via goodexperience, of all places,

Godel proved that any consistent mathematical system that is powerful enough to prove anything interesting will include true statements that can't be proven via that system. So from the standpoint of any logic, there will be elements that must be taken "on faith".

I strongly disagree with this so-called "best summation".

Goedel sentences are not "true". The "natural interpretation" from the construction of G may suggest that we accept it as true. But either G or not-G can be added to the axioms of the proof system without contradiction. So G is "true" only if you come in with preconceived notions about what numbers ought to be--- and avoiding that was the whole point of inventing formal proof systems.

Nor are Goedel sentences about taking things "on faith" --- the axioms of the system must be agreed to, which could already be considered a leap of faith. But their truth is really more a matter of aesthetics. Faith only comes into it if you appeal to some Platonist ideal of the natural numbers and their mystical properties. But there is no one true geometry, no one true set theory, and no one true number system. There are many!
Tags: logic, math, rant
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