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> After all, almost all reals are transcendental It's even worse, almost all reals aren't even computable (let alone transcendental)
by anchpop 7y ago
> After all, almost all reals are transcendental
It's even worse, almost all reals aren't even computable (let alone transcendental)
- ska 7y agoHonestly I don't think that's much worse from a "reals are weird" point of view, although it is another wrinkle that there is a strict subset which is even harder to pin down.
- drdeca 7y agoEven worse, almost all real numbers aren’t definable
- saagarjha 7y agoDoesn’t definability just depend on the system you’re using to generate numbers being able to represent it?
- drdeca 7y agoFor any system we could use to define a real number, (and not just because we are mortal), the system could only define or uniquely specify countably many real numbers. Any countable set of real numbers has measure 0, and therefore, almost all real numbers cannot be defined/specified using that system. This holds for whatever system you use which defines a (possibly partial) function from [ the set of finite (but unboundedly long) sequences of characters over a finite alphabet], to real numbers Or equivalently, from the natural numbers to the real numbers.
- OscarCunningham 7y agoYou can't define definability, because otherwise you could talk about the "least natural number not definable in fewer than n symbols" which is a contradiction for sufficiently large n.
- inflatableDodo 7y agoHence dartboards have the mathematical property that a dart thrown into one is infinitely more likely to land on an integer than on a computable real (or a natural number, but for different reasons).
- saagarjha 7y agoHow so? Aren’t all integers computable real numbers?
- inflatableDodo 7y agoNot in darts. You can prove this to your satisfaction, by attempting to reach the continuum between the integers. The dart will bounce off the metal framey bit and stick into the forehead of local barfly, 'Very Angry Ron'. Then when you graph the result, violent discontinuities will appear in the function, across a wide range of scales. These already have their own special set in category theory and are sometimes referred to, within the field, as; 'Assorted and Unreasonable Associates of Very Angry Ron'. That is, the other side of a reasonably big field, ideally with some cattle in the way.
- heavenlyblue 7y agoGoogle returns this exact post to the query of “Assorted and Unreasonable Associates of Very Angry Ron”.
- tripzilch 7y agoMust be another one of those pesky self-citing researchers.
- inflatableDodo 7y agoPlease forgive me father, for indulging in the sins of Onan.
- aeneasmackenzie 7y agoThat's ok, those ones aren't real.