> 1: Almost all numbers are transcendental.

Even crazier than that: almost all numbers cannot be defined with any finite expression.

dwohnitmok 7 hours ago | parent | next [-]

This is not necessarily true. It is possible for all real numbers (and indeed all mathematical objects) to be definable under ZFC. It is also possible for that not to be the case. ZFC is mum on the issue.

I've commented on this several times. Here's the most recent one: https://news.ycombinator.com/item?id=44366342

Basically you can't do a standard countability argument because you can't enumerate definable objects because you can't uniformly define "definability." The naive definition falls prey to Liar's Paradox type problems.

	▲	canjobear 2 hours ago \| parent [-]
		I think you're overthinking it. Define a "number definition system" to be any (maybe partial) mapping from finite-length strings on a finite alphabet to numbers. The string that maps to a number is the number's definition in the system. Then for any number definition system, almost all real numbers have no definition.

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zeroonetwothree 6 hours ago | parent | prev | next [-]

Maybe it would be better to say almost all numbers are not computable.

	▲	canjobear 3 hours ago \| parent [-]
		Chaitin's constant is definable but not computable.

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dinosaurdynasty 9 hours ago | parent | prev | next [-]

Leads to really fun statements like "there exists a proof that all reals are equal to themselves" and "there does not exist a proof for every real number that it is equal to itself" (because `x=x`, for most real numbers, can't even be written down, there are more numbers than proofs).

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7 hours ago | parent | prev | next [-]

[deleted]

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bjourne 7 hours ago | parent | prev [-]

Really? Which number can't be defined with a finite expression?