
Non Repeating Decimals - ColinWright
https://www.solipsys.co.uk/new/NonRepeatingDecimals.html?se26hn
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PaulHoule
It has always bugged me that people call the numbers that Cantor 'constructs'
the real numbers.

Underlying it is the axiom of choice, which says that it is meaningful to talk
about the results of processes that would take an infinite amount of time to
complete. (e.g. choose an infinite number of digits that create a unique
number.)

A number like 0 or 75 or 1/3 has a name, in fact all the integers and
rationals have names. You can pick one out individually and talk about it. You
might encounter it in some way.

Some irrationals (sqrt 11) and transcendentals (4 pi) have names. Most don't,
and you will never see them as individual numbers, only as members of an
interval.

~~~
umanwizard
You do not need the axiom of choice to construct the real numbers:
[https://www.quora.com/Is-the-axiom-of-choice-necessary-to-
in...](https://www.quora.com/Is-the-axiom-of-choice-necessary-to-introduce-
real-numbers-as-Dedekind-cuts)

However, I agree with the general thrust of your argument: it seems weird, in
hindsight, to call them the "real" numbers since most of them do not
meaningfully correspond to anything in physical reality. Especially since most
of them are non-computable; i.e. there does not exist any computer program
that outputs their digits in order. It seems difficult to say anything
meaningful about those at all!

In case you haven't seen it, the proof of this is simple: there are only
countably many computer programs, and uncountably many real numbers. Thus
almost all real numbers are non-computable.

In fact I am not convinced that there is _anything_ in physical reality that
meaningfully corresponds to an "uncountable set" in mathematics.

The real numbers are useful because their behavior _as a set_ matches our
intuition about how numbers should behave, even though most individual members
don't. I believe this is what you are alluding to in your last paragraph.

But anyway, they were named "real numbers" in opposition to "imaginary
numbers", not rationals or computables or anything like that. Which is even
more unfortunate because the construction of complex numbers from real numbers
is extremely straightforward and not mysterious at all (just define them as
pairs of reals with a particular multiplication operation).

