3 ms·

This seems to sort of line up with something I've been thinking recently: the difference between nameable and unnameable reals. Some "real" numbers may have no

by shkkmo 4y ago

This seems to sort of line up with something I've been thinking recently: the difference between nameable and unnameable reals. Some "real" numbers may have no finite definition (i.e. there exists no finite algorithm by which any finite digit can be calculated in finite time), I tend to think of these as cthulu numbers. The total number of possible names for real numbers seems easily countable (just list the possible papers describing the proof that a specific one is nameable, by length and alphabetically), and the real numbers that actually have names are just a subset (though determining which ones those are is an exercise left to the reader.) Thus the uncountable portion of the reals seems to me to be comprised of cthulu numbers. That or the nameable numbers are what we mean by the reals and these cthulu numbers are something else entirely.

ComplexSystems 4y ago
The "Cthulhu numbers" are basically, informally, just "random" numbers. The easiest way to build one is to choose a number at random in the interval [0,1], i.e. by flipping a coin countably many times and choosing each bit in the binary expansion based on the result. With probability one, the number you get will not be the output of any Turing machine, nor described by any particular formula, etc - it will just be a random string of bits whose bits are chosen "for no logical reason." This means it will be one of your Cthulhu numbers. Do we really want to get rid of these and only have non-random structured numbers? Because it's certainly a useful idea to have idealized infinite strings of structureless random bits in plenty of situations. But if you like, you can always try Gödel's constructible universe L.
OscarCunningham 4y ago
One complexity is that it's impossible to define 'definable'. Otherwise you could apply Cantor's construction to define an undefinable number.
jbaber 4y ago
You're noticing the algebraic numbers + all the transcendentals that you happen to need throughout your life (\pi, e, etc.) are countable. Sounds kind of deep, but plenty of proofs apply to uncountable sets. We finite little humans just live, necessarily, in a countable world.
jerf 4y ago
The more conventional name for what you are calling "nameable" is "computable": https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number The computable numbers are indeed countable, because they are defined by a set of Turing Machines, which are themselves countable. They're an interesting set of numbers, but they're a complicated thing to do work with because of the "holes" in them. For instance, see the section in that article labelled "Use in place of the reals". To see how bad missing things like supremums can be, consider something like this video: https://www.youtube.com/watch?v=vV7ZuouUSfs https://www.youtube.com/watch?v=vV7ZuouUSfs That video is about the difficulty/impossibility of defining calculus over the rational numbers, which is a simpler lens to see the problems that can arise when you're unable to define some simple properties that you may be so used to you don't realize it. Imagining how that can go wrong with a different set of "holes" in the number line is not too much of a strain after that. For all their paradoxes and oddities, and the unlikelihood of them corresponding to anything actually "real" in the real world, the real numbers are popular in math for a reason.
- puffoflogic 4y ago
  > The more conventional name for what you are calling "nameable" is "computable": GP appears to be talking about definable reals, not computable. For example [any particular] Chaitin's constant is definable but not computable.

by shkkmo 4y ago

ComplexSystems 4y ago

The "Cthulhu numbers" are basically, informally, just "random" numbers. The easiest way to build one is to choose a number at random in the interval [0,1], i.e. by flipping a coin countably many times and choosing each bit in the binary expansion based on the result. With probability one, the number you get will not be the output of any Turing machine, nor described by any particular formula, etc - it will just be a random string of bits whose bits are chosen "for no logical reason." This means it will be one of your Cthulhu numbers. Do we really want to get rid of these and only have non-random structured numbers? Because it's certainly a useful idea to have idealized infinite strings of structureless random bits in plenty of situations. But if you like, you can always try Gödel's constructible universe L.

OscarCunningham 4y ago

One complexity is that it's impossible to define 'definable'. Otherwise you could apply Cantor's construction to define an undefinable number.

jbaber 4y ago

You're noticing the algebraic numbers + all the transcendentals that you happen to need throughout your life (\pi, e, etc.) are countable. Sounds kind of deep, but plenty of proofs apply to uncountable sets. We finite little humans just live, necessarily, in a countable world.

jerf 4y ago

The more conventional name for what you are calling "nameable" is "computable": https://en.wikipedia.org/wiki/Computable_number https://en.wikipedia.org/wiki/Computable_number The computable numbers are indeed countable, because they are defined by a set of Turing Machines, which are themselves countable. They're an interesting set of numbers, but they're a complicated thing to do work with because of the "holes" in them. For instance, see the section in that article labelled "Use in place of the reals". To see how bad missing things like supremums can be, consider something like this video: https://www.youtube.com/watch?v=vV7ZuouUSfs https://www.youtube.com/watch?v=vV7ZuouUSfs That video is about the difficulty/impossibility of defining calculus over the rational numbers, which is a simpler lens to see the problems that can arise when you're unable to define some simple properties that you may be so used to you don't realize it. Imagining how that can go wrong with a different set of "holes" in the number line is not too much of a strain after that. For all their paradoxes and oddities, and the unlikelihood of them corresponding to anything actually "real" in the real world, the real numbers are popular in math for a reason.

puffoflogic 4y ago
> The more conventional name for what you are calling "nameable" is "computable": GP appears to be talking about definable reals, not computable. For example [any particular] Chaitin's constant is definable but not computable.