Intereting Posts

Form a space $X$ by identifying the boundary of $M$ with $C$ by a homeomorphism. Compute all the homology groups of $X$.
Derivative of $f(x,y)$ with respect to another function of two variables $k(x,y)$
Derivation of weak form for variational problem
If $(X,d)$ is a metric space then I want to show that limit point compactness and sequential compactness are equivalent.
Intersection of all maximal ideals containing a given ideal
Bijection between binary trees and plane trees?
An inequality on holomorphic functions
Integration and differentiation of Fourier series
how does expectation maximization work?
inverse function of $y=ax\ln(bx)$
The trigonometric solution to the solvable DeMoivre quintic?
Why are there so few Euclidean geometry problems that remain unsolved?
$f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$.
If $p$ is a prime integer, prove that $p$ is a divisor of $\binom p i$ for $0 < i < p$
Asymptotes of $\Gamma(\frac{1}{2} +ix)$ when $\vert x \vert \to \infty$

This question arises from a comment I recently read in another question.

My question is whether we can represent every real number using only finite memory. I will clarify what I mean by *represent using only finite memory* by use of examples:

- $5$ can be represented in finite memory simply by itself as a one-character string.
- Similarly for $1.234583$, which can also be represented by a string of finite length.
- $\pi$ can also be adequately represented in finite memory:
*it is the ratio of any circle’s circumference to its diameter.* - $e$ we can represent as $\displaystyle\lim_{n \rightarrow \infty} \left(1+\frac1n\right)^n$
- $0.818181\ldots$ can be represented as $0.\overline{81}$ or $\frac{9}{11}$.
- $0.010011000111\ldots$ can be represented as the sum of some sequence $a_n$ as $n\rightarrow \infty$.

For all the examples above, an adequate representation of the given real is possible using only finite memory, because we can describe/define exactly the given real using a string of finite length.

- Undecidable language problems.
- Does $\pi$ have infinitely many prime prefixes?
- If $L$ is regular, prove that $\sqrt{L}=\left\{ w : ww\in L\right\}$ is regular
- Do we really need constant symbols in first-order theories?
- Can the Negation of a Conditional Implying the First Atomic Proposition Get Proven in Around 50 Steps?
- In Context of Chomskhy classification of formal languages

So do any reals that cannot be described/represented in finite memory exist? For which their only closed-form expression requires a string of infinite length? (Infinitely many digits?)

Relevant Reading Material Includes:

Is it possible to represent every huge number in abbreviated form?

Every Number is Describable?

- Algorithm to answer existential questions - Reduction
- Proving $ \frac{\sigma(n)}{n} < \frac{n}{\varphi(n)} < \frac{\pi^{2}}{6} \frac{\sigma(n)}{n}$
- If $\gcd(a,b)=d$, then $\gcd(ac,bc)=cd$?
- Solving Pell's equation(or any other diophantine equation) through modular arithmetic.
- Perfect square modulo $n = pq$
- Proof of $ \phi(n) = \sum_{n|d} \mu(d) \cdot\frac nd $
- About Collatz 3n+3?
- How do I prove this sum is not an integer
- Show that there are infinitely many positive integers $N$ that cannot be written in the form $a^n+b^n+c^n$
- Integer solutions to the equation $a_1^2+\cdots +a_n^2=a_1\cdots a_n$

There are in fact *indefinable* real numbers. If you have a language with a countable number of symbols, then any formula $P(x)$ defining a real number will have, at best, a countable number of symbols. By a Cantor-style diagonal argument, you can only define a countable number of reals.

To elaborate: working in standard first-order logic with a given set theory, you can call $r$, a real number, to be *definable* if there is a formula $P(x)$ such that $r$ is the only real number such that $P(r)$ is true. However, note that the collection of formulas with one free variable is countable, whereas the collection of reals is uncountable; hence there are uncountably many undefinable real numbers.

For more information, see this wiki page; Timothy Gowers also has a good expository article that may be of interest.

(So the answer is no, you can’t represent every real number in finite memory, so to speak, if you assume that every description can be represented as a formula in first order logic.)

So do any reals that cannot be described/represented in finite memory exist? For which their only closed-form expression requires infinitely many digits?

Yes, the fact that transcendentals are uncountable implies that there is no finite or even countably infinite way to represent them all. ( Assuming memory is discrete and therefor countable).

Further more, there are numbers that can not be computed, so how does one stores an uncomputable number? ( like the Chaitin’s $\Omega$ constant)

- Is the sum of an algebraic and transcendental complex number transcendental?
- (ZF)subsequence convergent to a limit point of a sequence
- Proving that every vector space has a norm.
- How to do $\frac{ \partial { \mathrm{tr}(XX^TXX^T)}}{\partial X}$
- Is there any non-monoid ring which has no maximal ideal?
- prove $ \frac{1}{13}<\frac{1}{2}\cdot \frac{3}{4}\cdot \frac{5}{6}\cdot \cdots \cdot \frac{99}{100}<\frac{1}{12} $
- What are the interesting applications of hyperbolic geometry?
- Rotation invariant tensors
- There is no Pythagorean triple in which the hypotenuse and one leg are the legs of another Pythagorean triple.
- Expressing $1 + \cos(x) + \cos(2x) +… + \cos(nx)$ as a sum of two terms
- Pointwise a.e. convergence and weak convergence in Lp
- Left and right ideals of $R=\left\{\bigl(\begin{smallmatrix}a&b\\0&c \end{smallmatrix}\bigr) : a\in\mathbb Z, \ b,c\in\mathbb Q\right\}$
- Estimation of sums with number theory functions
- Families of subsets whose union is the whole set
- How many roots have a complex number with irrational exponent?