User:Tsirel/sandbox1

Introduction
The question in the title may seem simple, but is able to cause controversy and trip up professional mathematicians. Here is a quote from a talk "Must there be numbers we cannot describe or define?" by J.D. Hamkins [1]
 * The math tea argument
 * Heard at a good math tea anywhere:
 * “There must be real numbers we cannot describe or define, because there are uncountably many real numbers, but only countably many definitions.”
 * Does this argument withstand scrutiny?

See also two excerpts from A. Tarski [2] quoted in "Maybe there's no such thing as a random sequence" by P.G. Doyle on pp. 6,7. And on Wikipedia one can also find the flawed "math tea" argument on talk pages and obsolete versions of articles. And elsewhere on the Internet. I, the author, was myself a witness and accomplice. I shared and voiced the flawed argument in informal discussions (but not articles or lectures). Despite some awareness (but not professionalism) in mathematical logic, I was a small part of the problem, and now I try to become a small part of the solution, spreading the truth.

Careless handling of the concept "number specified by a finite text" leads to paradoxes; in particular, Richard's paradox.

The paradox begins with the observation that certain expressions of natural language define real numbers unambiguously, while other expressions of natural language do not. For example, "The real number the integer part of which is 17 and the nth decimal place of which is 0 if n is even and 1 if n is odd" defines the real number 17.1010101... = 1693/99, while the phrase "the capital of England" does not define a real number.

Thus there is an infinite list of English phrases (such that each phrase is of finite length, but lengths vary in the list) that define real numbers unambiguously. We first arrange this list of phrases by increasing length, then order all phrases of equal length lexicographically (in dictionary order), so that the ordering is canonical. This yields an infinite list of the corresponding real numbers: r1, r2, ... . Now define a new real number r as follows. The integer part of r is 0, the nth decimal place of r is 1 if the nth decimal place of rn is not 1, and the nth decimal place of r is 2 if the nth decimal place of rn is 1.

The preceding two paragraphs are an expression in English that unambiguously defines a real number r. Thus r must be one of the numbers rn. However, r was constructed so that it cannot equal any of the rn. This is the paradoxical contradiction.

(Quoted from Wikipedia.) --

See also Definability paradoxes by Timothy Gowers.

In order to ask (and hopefully solve) a well-posed question we have to formalize the concept "number specified by a finite text" via a well-defined mathematical notion "definable number". What exactly is meant by "text"? And what exactly is meant by "number specified by text"? Does "specified" mean "defined"? Can we define such notions as "definition" and "definable"? Striving to understand definitions in general, let us start with some examples.

136 notable constants are collected, defined and discussed in the book "Mathematical constants" by Steven Finch [3]. The first member of this collection is "Pythagoras’ Constant, $\sqrt2$"; the second is "The Golden Mean, $\varphi$"; the third "The Natural Logarithmic Base, e"; the fourth "Archimedes’ Constant, $\pi$"; and the last (eleventh) in Chapter 1 "Well-Known Constants" is "Chaitin’s Constant".

Each constant has several equivalent definitions. Below we take for each constant the first (main) definition from the mentioned book.


 * The first constant $$\sqrt2$$ is defined as the positive real number whose product by itself is equal to 2. That is, the real number $$x$$ satisfying $$x>0$$ and $$x^2=2.$$


 * The second constant $$\varphi$$ is defined as the real number satisfying $$\varphi>0$$ and $$\textstyle 1+\frac1\varphi=\varphi.$$


 * The third constant $$e$$ is defined as the limit of $$\textstyle (1+x)^{1/x}$$ as $$x\to0.$$ That is, the real number satisfying the following condition:
 * for every $$\varepsilon>0$$ there exists $$\delta>0$$ such that for every $$x$$ satisfying $$-\delta0 \,\, \exists \delta>0 \,\, \forall x \; \big(\, ( -\delta<x<\delta \,\,\land\,\, x\ne0 ) \Longrightarrow ( -\varepsilon<(1+x)^{1/x}-e<\varepsilon ) \,\big).$$

$$\land$$ "and"      $$\lor$$  "or"      $$\Longrightarrow$$  "implies"      $$\neg$$  "not"      $$\forall$$  "for every"      $$\exists$$  "there exists (at least one)"      $$\exists!$$  "there exists one and only one"          (a longer list). --

We note that these three definitions are of the form "the real number $$x$$ satisfying $$P(x)$$" where $$P$$ is a statement that may be true or false depending on the value of its variable $$x$$; in other words, a property of $$x$$, or a predicate (on real numbers).

Not all predicates may be used this way. For example, we cannot say "the real number $$x$$ satisfying $$x^2=2$$" (why "the"? two numbers satisfy, one positive, one negative), nor "the real number $$x$$ satisfying $$x^2=-2$$" (no such numbers). In order to say "the real number $$x$$ satisfying $$P(x)$$" we have to prove existence and uniqueness:
 * existence: $$\exists x \; P(x)$$ (in words: there exists $$x$$ such that $$P(x)$$);
 * uniqueness: $$\forall x,y \; \big(\, ( P(x) \land P(y) ) \Longrightarrow (x=y) \,\big)$$ (in words: whenever x and y satisfy P they are equal).

In this case one says "there is one and only one such x" and writes "$\exists! x \; P(x)$".

The road to definable numbers passes through definable predicates. We postpone this matter to the next section and return to examples.

This definition involves geometry. True, a lot of equivalent definitions in terms of numbers are well-known; in particular, according to the mentioned book, this area is equal to $$\textstyle \, 4\int_0^1 \sqrt{1-x^2} \, dx = \lim_{n\to\infty} \frac4{n^2} \sum_{k=0}^n \sqrt{n^2-k^2} \, .$$ However, in general, every branch of mathematics may be involved in a definition of a number; existence of an equivalent definition in terms of (only) numbers is not guaranteed.
 * The fourth constant $$\pi$$ is defined as the area enclosed by a circle of radius 1.

The last example is Chaitin's constant. In contrast to the four constants (mentioned above) of evident theoretical and practical importance, Chaitin's constant is rather of theoretical interest. Its definition is intricate.


 * The last constant $$\Omega$$ is defined as the sum of the series $$\textstyle \Omega = \sum_{N=1}^\infty 2^{-N} A_N$$ where $$A_N$$ is equal to 1 if there exist natural numbers $$x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9$$ such that $$f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0,$$ otherwise $$A_N=0;$$ and $$f$$ is a polynomial in 10 variables, with integer coefficients, such that the sequence $$A_1, A_2, \dots$$ is uncomputable.

Hilbert’s tenth problem asked for a general algorithm that could ascertain whether the Diophantine equation $$f(x_0,\dots,x_k)=0$$ has positive integer solutions $$(x_0,\dots,x_k),$$ given arbitrary polynomial $$f$$ with integer coefficients. It appears that no such algorithm can exist even for a single $$f$$ and arbitrary $$x_0,$$ when $$f$$ is complicated enough. See Wikipedia: computability theory, Matiyasevich's theorem; and Scholarpedia:Matiyasevich theorem.

The five numbers $$\sqrt2, \varphi, e, \pi, \Omega$$ are defined, thus, should be definable according to any reasonable approach to definability. The first four numbers $$\sqrt2, \varphi, e, \pi$$ are computable (both theoretically and practically; in fact, trillions, that is, millions of millions, of decimal digits of $$\pi$$ are already computed), but the last number $$\Omega$$ is uncomputable. How so? Striving to better understand this strange situation we may introduce approximations $$A_{M,N}$$ to the numbers $$A_N$$ as follows: $$A_{M,N}$$ is equal to 1 if there exist natural numbers $$x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9$$ less than $$M$$ such that $$f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0,$$ otherwise $$A_N=0;$$ here $$M$$ is arbitrary. For each $$N$$ we have $$A_{M,N}\uparrow A_N$$ as $$M\to\infty;$$ that is, the sequence $$A_{1,N}, A_{2,N}, \dots$$ is increasing, and converges to $$A_N.$$ Also, this sequence $$A_{1,N}, A_{2,N}, \dots$$ is computable (given $$M,$$ just check all the $$(M-1)^9$$ points $$(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9),$$ $$0<x_1<M, \dots, 0<x_9<M$$). Now we introduce approximations $$\omega_M$$ to the number $$\Omega$$ as follows: $$\textstyle \omega_M = \sum_{N=1}^M 2^{-N} A_{M,N}.$$ We have $$\omega_M\uparrow\Omega$$ (as $M\to\infty$), and the sequence $$\omega_1, \omega_2, \dots$$ is computable. A wonder: a computable increasing sequence of rational numbers converges to a uncomputable number!

For every $$N$$ there exists $$M$$ such that $$A_{M,N}=A_N;$$ such $$M$$ depending on $$N,$$ denote it $$M_N$$ and get $$\textstyle \sum_{N=1}^\infty 2^{-N} A_{M_N,N}=\Omega;$$ moreover, $$\textstyle \Omega-\sum_{N=1}^K 2^{-N} A_{M_N,N}\le 2^{-K}$$ for all $$K.$$ In order to compute $$\Omega$$ up to $$2^{-K}$$ it suffices to compute $$\textstyle \sum_{N=1}^K 2^{-N} A_{M_N,N}.$$ Doesn't it mean that $$\Omega$$ is computable? No, it does not, unless the sequence $$M_1,M_2,\dots$$ is computable. Well, these numbers need not be optimal, just large enough. Isn't $$\textstyle M_N=10^{1000N}$$ large enough? Amazingly, no, this is not large enough. Moreover, $$\textstyle M_N=10^{10^{1000N}}$$ is not enough. And even the "power tower"$$M_N=\underbrace{10^{10^{\cdot^{\cdot^{10}}}}}_{1000N}$$ is still not enough!

Here is the first paragraph from a prize-winning article:
 * "Does the equation $$\textstyle x^3+y^3+z^3=29$$ have a solution in integers? Yes: (3, 1, 1), for instance. How about $$\textstyle x^3+y^3+z^3=30$$? Again yes, although this was not known until 1999: the smallest solution is (−283059965, −2218888517, 2220422932). And how about $$\textstyle x^3+y^3+z^3=33?$$ This is an unsolved problem."

Given that the simple Diophantine equation $$\textstyle N+x^3+y^3-z^3=0$$ has solutions for $$N=30$$ but only beyond $$10^9$$ we may guess that the "worst case" Diophantine equation $$f(N,x_1,x_2,x_3,x_4,x_5,x_6,x_7,x_8,x_9)=0$$ needs very large $$M_N.$$ In fact, the sequence $$M_1,M_2,\dots$$ has to be uncomputable (otherwise $$\Omega$$ would be computable, but it is not). Some computable sequences grow fantastically fast. See Wikipedia: Ackermann function, Fast-growing hierarchy. And nevertheless, no one of them bounds from above the sequence $$M_1,M_2,\dots\,$$ Reality beyond imagination!

Every computable number is definable, but a definable number need not be computable. Computability being another story, we return to definability.