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Ok, so maybe that wasn't the best way of phrasing the question, but I think it's specific enough. Let me explain myself a bit more below in case I am wrong.

So I'm assuming (although I've never checked) that the irrational numbers are defined as simply all reals that are not rational.

I'm asking about the existence, then, of an irrational number that has no finite description. I.e, not only does it not have a finite-number or recurring decimal description, but no other description could be made (counting things such as "The ratio of the circumference and diameter of any circle in a Euclidean plane" as finite).

Clearly, there would have to be infinitely many of these and they would have to form continuous connected regions of the real line, otherwise they would afford descriptions such as "The otherwise-non-finitely-describable number lying between x & y" where x & y bound the "otherwise-non-finitely-describable z, and z can be shown to be the only such number between x & y.

If so, then this would have implications for Laplace's Demon and other similar philosophical arguments since it would be mathematically impossible to have knowledge of all of the universe so long as at least one parameter lay within one of these non-finitely-describable regions.

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    @Craig: I noticed that you re-tagged the question as [computability]. But computability and definability are distinct concepts. For instance, Chaitin's constant (for any given choice of encoding) is definable, but not computable. I would re-tag it to [logic], but perhaps there are better options, and in any case I don't want to start a tag war. But [computability] is wrong.2011-09-07
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    You're making the groundless and false assumption that either undefinable numbers form continuous connected regions or there are x and y such that there exists only a single undefinable number in [x,y]. You forgot the possibility that they are dense but disconnected (which is the actual truth in the matter). Note all connnected regions of reals are intervals & all intervals contain a rational - which is definable. Also, you should be cautious about your levels of description; formally defining a number as undefinable from a specific set of symbols requires a higher level than those symbols.2011-09-07
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    This reminds me of the [All numbers are interesting](http://en.wikipedia.org/wiki/Interesting_number_paradox)-theorem, also mentioned by Quinn below, which in this case also "contradicts" the existence of such numbers.2011-09-08
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    There is a thorough answer to this question at http://mathoverflow.net/questions/44102/is-the-analysis-as-taught-in-universities-in-fact-the-analysis-of-definable-numbe/44129#441292011-09-08
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    Regarding problems with Laplace's demon: What makes you think all the irrationals — let alone any uncountable set — exist (or even are represented?) within the physical universe?2011-09-08
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    I would observe that if there is such a number there's got to be more than one. And, if a finite number, they must somehow be unorderable (if that makes any sense at all).2011-09-08
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    (Got to thinking about it and realized my second statement above was wrong, in the most general sense. With a circular number line (what's the technical term?) you could have a finite number of orderable numbers and still not be able to identify one of them.)2011-09-09

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