Is there at least one irrational number with the property that it cannot be defined by a finite string of information?

Question

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.

Answer 1

Yes, there are such numbers. Chaitin discusses them extensively, he has a popular book on the subject, Meta Math: The Quest for Omega. They do not form continuous connected segments of the real line, because for example, the rationals are dense (that is, between any two real numbers there is at least one rational -- and in fact there are an infinite number of rationals).

However, these non-computable numbers are also dense in the real line.

You might want to take a look at the "computability theory", "definable number", and "analytical hierarchy" articles in Wikipedia.

Note that this doesn't pose as big a problem for the philosophy of physics as you might think: This doesn't say that you cannot measure a physical constant to whatever precision you might wish.

Answer 2

Be very careful with such statements. For example, take Berry's paradox: Let n be "The smallest number which can not be described with less than 14 words". Should it exist? Then it has been described with 13 words. The number of descriptions is countable, but the number of reals is not. Therefore, most reals are not finitely describable. Does it imply anything for physics? I don't think so...

Answer 3

I don't think the set of all such numbers is well-defined. Using particular words to mean one thing makes defining other things more difficult. Is the set of irrationals that can be defined in English the same as the set that can be defined in French or Chinese?

Answer 4

To be as generous as possible, allow yourself a countably-infinite alphabet and descriptions of finite (but unbounded) length. This seems to describe the human state of affairs, since we seem to able to create many new ways of describing things, but we certainly don't have an infinite amount of time to read a description.

The set of all possible descriptions arising from this scenario is still only countably infinite. (Proof: Fix an ordering on your alphabet and list all descriptions lexicographically.) Since the reals are uncountable, there are uncountably-many real numbers that admit no description in this sense.

Answer 5

Of course you'd need to define precisely how the finite string is interpreted, but the answer is yes. There are only countably many strings and uncountably many reals.

Answer 6

Your question is too vague, especially regarding your requirement of a finite description. Here's an ostensible paradox that can arise if you're not careful:

Suppose there is a real number that cannot be described by any finite sequence of characters. Further, assume that the reals can be well-ordered. Then the set of all reals with no finite description is nonempty, so must have a least element (with respect to the given well-order). But what we've just given is a "finite description" of a number that's not supposed to have a finite description!

The "paradox" just described is redolent of the interesting number paradox and has the same resolution. The point is that descriptions must be fixed ahead of time; the string "the least (with respect to some well-order of the reals) real number that is not finitely describable", if it describes some number, should describe it before you form the set of all non-finitely-describable elements and hence cannot be used to describe any other. This is an issue of syntax vs. semantics; the string already syntactically describes the real, regardless of what its semantics seem to describe.

Thus your question should really be posed like this: fix a finite alphabet $\Sigma$ sufficient to form any English word, sentence, paragraph, etc.; and a function $f: \Sigma^{*} \to \mathbf{R}$, where $\Sigma^{*}$ denotes the set of all finite words over $\Sigma$. Note that a word in this context might actually be what you or I would really call a sentence, paragraph, etc. Define a description of a real number $r$ to be a word $\sigma \in \Sigma^{*}$ such that $f(\sigma) = r$.

With this precise setup, it is trivial to see that some reals have no description, since the collection of all words over a finite (countably infinite, even) alphabet is countable and the reals are uncountable.

To see concretely why there's now no paradox, consider the (definitely nonempty) set of all reals without a description. This set will still have a least element (with respect to any well-ordering of the reals), but now we cannot say that the string $\sigma=$"the least (with respect to some well-order of the reals) real number that is not finitely describable" describes it, since $\sigma$ is already a description of of some other real number.

Edit. I'd like to see someone come up with a paradox without using a well-ordering of the reals (or any other consequence of the axiom of choice). If no one can, I'll ask it as a separate question.

Answer 7

Formally, we say an object $c$ in a structure $M$ is definable (where the language of $M$ is $L$) iff there is formula $\varphi(x)$ in the language s.t. $M \vDash \exists! x \ \varphi(x)$.

If the language is finite, then the set of formulas in the language is countable, and therefore the set of definable objects is also countable. If $M$ is an uncountable structure, then there are many objects in it that are not definable.

Fix a language for the defining real number. If the language is finite, then there are many real number that is not definable.

Also note that if the language is reasonable (i.e. includes $0$, $1$, $+$, and $.$) then all rational numbers are definable, so all undefinable real numbers in such a language will be irrational.