Posted: Sep 15, 2015 10:09 am
by crank
scott1328 wrote:I have listened to the video and read the paper you quoted. And in a video and paper aimed at popularizing deeply technical ideas, I would expect such language and metaphors to make his ideas accessible. I do not believe he is claiming what you think he is.

And if he is claiming that reals have infinite precision, then he is playing fast and loose with the mathematics.

Chaitin does make a particularly egregious error in the paper you cited

So in fact there are more uncomputable reals than computable reals. From Cantor's theory of infinite sets, we see that the set of uncomputable reals is just as big as the set of all reals, while the set of computable reals is only as big as the set of whole numbers. The set of uncomputable reals is much bigger than the set of computable reals.

#{uncomputable reals} = #{all reals} = ℵ1
#{computable reals} = #{computer programs} = #{whole numbers} = ℵ0
ℵ1 > ℵ0

In the text I've quoted Chaitin asserted that the cardinality of the reals, typically referred to as C, is equivalent to Aleph-1. This assertion, called the Continuum Hypothesis is formally undecidable in standard formulations of mathematical set theory, that he would assert it as true in his popular paper, leads me to doubt the seriousness of this entire paper.

I think he's claiming what he says he's claiming, I can't read his mind, can you? You directly assert what is contradictory to exactly what he says.

Now, the ℵ1 vs C question, what do you think more likely, that a mathematician of Chaitin's caliber made a mental slip, or that he is unaware of such a profound issue in the area of infinities?

Plus, I went to wiki, they say it quite differently from what you say:

\aleph_1 is the cardinality of the set of all countable ordinal numbers, called ω1 or (sometimes) Ω. This ω1 is itself an ordinal number larger than all countable ones, so it is an uncountable set. Therefore \aleph_1 is distinct from \aleph_0. The definition of \aleph_1 implies (in ZF, Zermelo–Fraenkel set theory without the axiom of choice) that no cardinal number is between \aleph_0 and \aleph_1. If the axiom of choice (AC) is used, it can be further proved that the class of cardinal numbers is totally ordered, and thus \aleph_1 is the second-smallest infinite cardinal number.

I appears you can use the terminology the way Chaitin did, there are different ways of using the aleph-numbers depending on what assumptions you want to use.

You obviously know a hell of a lot more maths than I do, but you keep making boneheaded errors, and instead of trying to discuss the issues, you mostly have tried to make me look stupid and wrong. I had a perfectly reasonable conversation with Thommo, we came to agreement on almost everything. I said upfront, I didn't know the proper terminology, but what I was saying was plain enough for the most part, and we were able to clear up the confusion. I was wrong about maths treating integers differently than reals, but only a little, the mind surely does treat them differently. I was wrong about using 'will' when the probability was 100%, that was a technical error. Everything else there was no real error in what I said.

The infinite precision thing is a good example. You need infinite information to specify an uncomputable number, that is what he's talking about. You can't express one, no one disputes this and everyone states this. To express any real, you need an infinite string of coefficients, these are seldom explicitly stated, like they are in Cali's definition, but they are there, by implication, by convention, by whatever, but they are there, in order to ensure the repeating fraction or the zeros. This is not claiming you need infinite information, like I've explained repeatedly. That is only true of the uncomputables. You keep objecting to the 'information' ideas, and the 'infinite precision' idea, but give no reasoning, and you are at odds with most physicist and mathematicians. In order to distinguish between two numbers, you can't ever know if they are different without this infinity of digits. You can't even make sense of these things without first having to just think of the number and its infinity of digits as being there, all at once, like on a number line. As any two point get closer and closer together, you have to look closer and closer to see if there is a difference, in other words, go further and further out to the right of the decimal. In effect, take in the whole thing at once to see if the numbers are the same, but this requires all the digits, an infinity of them. You can't get the granularity fine enough, in other words, look at enough digits, to ever guarantee distinguishing two uncomputables. I can't think of any more ways to explain what is obvious to me.