Meta_Math by Chaitin

Author:Chaitin
Format: epub
Tags: Metamath
Published: 2010-04-27T01:49:33+00:00

Chapter summary

Against Real Numbers!

Prob{algebraic reals} = Prob{computable reals} =

Prob{nameable reals} = 0

Prob{transcendental reals} = Prob{uncomputable reals} =

Prob{random reals} = Prob{un-nameable reals} = 1

In summary:

Why should I believe in a real number if I can’t calculate it, if I can’t prove what its bits are, and if I can’t even 100

Meta Math!

refer to it? And each of these things happens with probability one!

√

We started this chapter with

2, which the ancient Greeks referred to as

“unutterable”, and we ended it by showing that with probability one there is no way to even name or to specify or define or refer to, no matter how non-constructively, individual real numbers. We have come full circle, from the unutterable to the un-namable! There is no escape, these issues will not go away!

In the previous chapter we saw physical arguments against real numbers. In this chapter we’ve seen that reals are also problematic from a mathematical point of view, mainly because they contain an infinite amount of information, and infinity is something we can imagine but rarely touch. So I view these two chapters as validating the discrete, digital information approach of AIT, which does not apply comfortably in a physical or mathematical world made up out of real numbers. And I feel that this gives us the right to go ahead and see what looking at the size of computer programs can buy us, now that we feel reasonably comfortable with this new digital, discrete viewpoint, now that we’ve examined the philosophical underpinnings and the tacit assumptions that allowed us to posit this new concept. In the next chapter I’ll finally settle my two debts to you, dear reader, by proving that Turing’s halting problem cannot be solved—my way, not the way that Turing originally did. And I’ll pick out an individual random real, my Ω number, which as we saw in this chapter must have the property that any FAS can determine at most finitely many bits of its base-two binary expansion. And we’ll discuss what the heck it all means, what it says about how we should do mathematics. . .

Chapter VI—Complexity,

Randomness & Incompleteness

In Chapter II I showed you Turing’s approach to incompleteness. Now let me show you how I do it. . .

I am very proud of my two incompleteness results in this chapter! These are the jewels in the AIT crown, the best (or worst) incompleteness results, the most shocking ones, the most devastating ones, the most enlightening ones, that I’ve been able to come up with! Plus they are a consequence of the digital philosophy viewpoint that goes back to Leibniz and that I described in Chapter III. That’s why these results are so astonishingly different from the classical incompleteness results of Gödel (1931) and Turing (1936). Irreducible Truths and the Greek Ideal of Reason I want to start by telling you about the very dangerous idea of “logical irreducibility”. . .

Mathematics:

axioms −→ Computer −→ theorems

We’ll see in this chapter that the traditional notion of what math is about is all wrong: reduce things to axioms, compression.

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