On Apr 22, 7:53 am, fom <fomJ...@nyms.net> wrote: > On 4/22/2013 2:03 AM, WM wrote: > > > > > [Gregory Chaitin: "How real are real numbers?" (2004)] > >http://arxiv.org/abs/math.HO/0411418 > > Page 12 > > "Why should we believe in real numbers, if most of > them are uncomputable?
What does it mean to "believe in real numbers"? How can you define "most" of an infinite set? You cannot take aggregates of infinite sets in general. Consider the puzzle of the probability a random chord in a circle is shorter than the side of an inscribed equilateral triangle. There are different answers depending on how you choose from that infinite set. It is not well-defined to say that "most" of those chords are shorter. Likewise taking 2 boxes and putting a random number of pennies in one and twice that in another leads to a self-contradiction when you consider whether a randomly chosen box has an expected amount greater than the other. If you have N pennies then the other has on average 2N+(N/2)/2=5/4N so switching always increases the expected number!
Not all functions are computable. Not all sets are r.e. and not all are recursive. So what? That just means we have wonderful theories about functions that no program computes and sets that no program enumerates or no program decides.
> Why should we believe in real > numbers, if most of them, it turns out, are maximally > unknowable like Omega?"
Omega is not a real number. It is a function from a universal turing machine to an infinite series.
> footnote with answer > > "In spite of the fact that most individual real > numbers will forever escape us, the notion > of an arbitrary real has beautiful mathematical > properties and is a concept that helps us > to organize and understand the real world. > Individual concepts in a theory do not need > to have concrete meaning on their own;
What does it mean for a concept to "have a concrete meaning of its own"? If something is meaninless, how can you call it a concept?
No, don't tell me to real more of his silliness. He says that there are unprovable truths because mathematics is random. Gad, I hope math isn't random. And no, the Godel sentence is not unprovable. Godel himself proves it. It just isn't provable in his system of logic (which is similar to PA.) As Godel says in the same paper, what is not provable in his system is provable by "metamathematical" reasoning. It just shows that his system can't prove as much as the reasoning that he uses in the paper itself.
Besides, the version of Godel's Theorem that Chaitin refers to is the weaker version given by Godel in passing (in the introduction of the article) and is not even the main theorem. And of course there are extensions such as Rosser's 1936 paper making any discussion of improving upon Godel's weaker, simpler proof without significance. It's like saying you made an improvement over Newtonian Physics or you have a better buggy whip.
Chaitin says he read and understood Godel's paper as a child, but as an adult he has never touched the stronger more complex theorem in that paper, nor the 2nd Incompleteness Theorem, nor Rosser's extension, and even gives a faulty exposition of Turing's simple proof.
> it is > enough if the theory as a whole can be > compared with the results of experiments." > > [Gregory Chaitin: "How real are real numbers?" (2004)]http://arxiv.org/abs/math.HO/0411418