# Is the set of non finitely describable real numbers closed under addition and squaring?

Is the set of non finitely describable real numbers closed under addition and squaring? If so, can someone give a proof? Thanks.

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non finitely describable real numbers = irrational? $\sqrt{2}^2 = 2$ ? – V-X Mar 23 '13 at 22:39
This has a good picture that explains it i.imgur.com/ySI0Lu1.png – User234 Mar 23 '13 at 22:39
What precisely do you mean by "finitely describable"? – Chris Eagle Mar 23 '13 at 22:42
I took it to mean "can be described for a fixed language by a finite sentence". – Ittay Weiss Mar 23 '13 at 22:49
@V-X, ChrisEagle: en.wikipedia.org/wiki/Definable_real_number – anon Mar 23 '13 at 22:49

if $x$ is not finitely describable, then $-x$ is not finitely describable either. But $0=x+(-x)$ is finitely describable so no closure under addition.

Now if $x^2$ is finitely describable, then $x$ is finitely describable (it's described as "the square root (either positive or negative) of the number described by the description of $x$"). So, you do have closure under squaring.

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Isn't your last paragraph an argument for closure under squaring? – Ross Millikan Mar 23 '13 at 22:44
@RossMillikan oooops auto-pilot crashing. Thanks. I'm correcting it. – Ittay Weiss Mar 23 '13 at 22:45

If $x$ is non-describable, so is $-x$. So certainly we do not have closure under addition.

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I believe that the answer is "no," though I might have the wrong definition of "finitely describable."

Take any non-finitely-describable number $N$ and consider the number $K = 1 - N$. This number cannot be finitely describable, since if it were, we could describe $N$ as $1 - K = 1 - (1 - N) = N$. However, $N + K = N + (1 - N) = 1$, which is finitely describable. Thus the non-finitely-describable numbers are not closed under addition.

Hope this helps!

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Hum... if by non-finitely-describable you mean "can't construct a finite description (as e.g. a Turing machine)", they aren't closed with respect to addition: $a + b = 2$ if $a$ is one of yours, $b$ is too (if it wasn't, $a$ could be described). But 2 clearly isn't.

Squares I have no clue.

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