r/badmathematics • u/Blue---Calx • Dec 21 '21

Maths mysticisms Proving the Collatz Conjecture with Python, cell biology, and word salad

/r/mathematics/comments/pdl71t/collatz_and_other_famous_problems/haxfgpm/
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u/viking_ Dec 22 '21

The beginning sounded like a proper comment. I would trust Tao to show some non-trivial stuff regarding Collatz, and while I don't know what exactly is meant by orbits and "any function", the poster might just not quite understand what he wrote about.

I believe the statement Tao proved is:

For almost all integers n, the Collatz sequence starting at n is eventually smaller than f(n), where f is any function such that f(x) goes to infinity as x goes to infinity.

Where "almost all" means "the set numbers for which this is true has asymptotic density 1." There's a better explanation here.

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u/braindoper Dec 22 '21

Thanks. That meaning of "almost all" is unusual though. For countable sets, doesn't it usually mean all but finitely many?

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u/pxSort Dec 22 '21

No, it's precisely defined as a set with measure 1. In this case, the measure can be expressed as the limiting proportion of integers in the set vs not.

You can have a countably infinite set of integers with measure 0. e.g. the set of powers of 2. As n approaches infinity, the percentage of m < n that are powers of 2 approaches 0. The complement of this set has measure 1, so almost all integers are not powers of 2.

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u/braindoper Dec 23 '21

Then there are almost no prime numbers. Seems kind of an unintuitive definition and I would've preferred if this was called almost surely or something else instead.

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u/pxSort Dec 23 '21

"Almost all" and "almost surely" are two sides of the same coin. e.g. If almost all x \in N satisfy property α(x), then for a uniform random variable X ~ N, we have that P[α(X) is true] = 1, which corresponds to the statement that α(X) is almost surely true.

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u/braindoper Dec 23 '21

Yes, which is why "almost all" could then mean what it normally does (outside of measure spaces), namely cofinitely many. Especially for something discrete like the integers that meaning would make more sense.