About the Gödel's incompleteness theorems and about Markov chains

Hello,


As you have noticed , i have just written an interesting article about why random sampling in Statistics works for estimation and its role in both Mathematical Statistics and Monte Carlo Simulation , and here it is:

https://myphilo10.blogspot.com/2025/04/about-why-random-sampling-in-statistics.html

And i think that understanding how random sampling in Statistics works for estimation is also interesting for politics and economy too.

And now i will talk more about the Gödel's incompleteness theorems and about Markov chains that are also interesting for politics and economics too:


So, I have searched more on internet the much more precise and correct Gödel's First incompleteness theorem, and here it is:

"Any consistent formal system F within which a certain amount of elementary arithmetic can be carried out is incomplete; i.e., there are statements of the language of F which can neither be proved nor disproved in F"

And in mathematics, a statement is a declarative sentence that is either true or false but not both. A statement is sometimes called a proposition. The key is that there must be no ambiguity. To be a statement, a sentence "must" be true or false, and it cannot be both.

So that means that we know that the statement is true or false but it can not be proven true or false, so we then logically infer that we can not prove the consistency of the system , so the statement can be that it is like an axiom in mathematics that is true but that we can not prove by such logical inference or deduction, so then the system remains really useful even if it's incomplete by Gödel's incompleteness theorems, so i think that Gödel's incompleteness theorems are not so problematic.


More of my thoughts about eigenvectors and eigenvalues and Markov chains in mathematics ..

So now i will create an interesting mathematical tutorial about eigenvectors and Markov chains in mathematics, and notice carefully my way of creating it, and here it is:

Note: "Lambda" means the 11th letter of the Greek alphabet that is used as both a symbol and a concept in various fields of science, mathematics and computing.

So , say v is an eigenvector of a matrix A with eigenvalue Lambda. Then Av=Lambda*v.


So there is an important thing to know in mathematics and it is that mathematical eigenvectors show in phenomenons that exhibit a "stable" behavior with time, and in Markov chains we search also for an eigenvector where the system will stabilize, so we have to solve:

A*vector(v) = I*vector(v) [1]

The identity matrix I has 1 as its only eigenvalue. All vectors are associated eigenvectors since

I*v = v = (1)v

for all v.


A is the transition matrix and v a vector.

So since the Eigenvalue is 1, that means we have to solve the following system of equation that will give you the eigenvector where the system will "stabilize" its behavior:

(A - I)*vector(x)= vector(0)

I is the identity matrix.


So now here is my tutorial about Markov chains in mathematics:

In mathematics, many Markov chains automatically find their own way to an equilibrium distribution as the chain wanders through time. This happens for many Markov chains, but not all. I will talk about the conditions required for the chain to find its way to an equilibrium distribution.

So if in mathematics we give a Markov chain on a finite state space and asks if it converges to an equilibrium distribution as t goes to infinity. An equilibrium distribution will always exist for a finite state space. But you need to check whether the chain is irreducible and aperiodic. If so, it will converge to equilibrium. If the chain is irreducible but periodic, it cannot converge to an equilibrium distribution that is independent of start state. If the chain is reducible, it may or may not converge.

So i will give an example:

Suppose that for a course you are currently taking there are two volumes on the market and represent them by A and B. Suppose further that the probability that a teacher using volume A keeps the same volume next year is 0.4 and the probability that it will change for volume Bis 0.6. Furthermore the probability that a professor using B this year changes to next year for A is 0.2 and the probability that it again uses volume B is 0.8. We notice that the transition matrix is:

| 0.4 0.6 |
|
| 0.2 0.8 |


So the interesting question for any businessman is whether his market share will stabilize over time. In other words, does it exist a probability vector (t1, t2) such that:

(t1, t2) * (transition matrix above) = (t1, t2) [2]

and notice that the mathematical system [2] looks like the mathematical system [1] above.

So notice that the transition matrix above is irreducible and aperiodic,
so it will converge to an equilibrium distribution that is (t1, t2) that
i will mathematically find, so the system of equations of [2] above is:

0.4 * t1 + 0.2 * t2 = t1
0.6 * t1 + 0.8 * t2 = t2

this gives:

-0.6 * t1 + 0.2 * t2 = 0
0.6 * t1 - 0.2 * t2 = 0

But we know that (t1, t2) is a probability vector, so we have:

t1 + t2 = 1

So we have to solve the following system of equations:

t1 + t2 = 1
0.6 * t1 - 0.2 * t2 = 0

So i have just solved it with R, and this gives the vector:

(0.25 , 0.75)

Which means that in the long term, volume A will grab 25% of the market
while volume B will grab 75% of the market unless the advertising campaign does change the transition probabilities.


And here is what the AI (Artificial intelligence) from Google that is called Gemini 2.5 Pro , that is advanced, answered me about my above thoughts about mathematics:

"Okay, Amine, let's break down your thoughts section by section. You're engaging with some deep and fascinating topics here!

**1. Gödel's Incompleteness Theorems:**

* **Your Understanding:** You've correctly stated Gödel's First Incompleteness Theorem. Your interpretation is largely sound:
* Yes, the theorem implies there are statements within sufficiently complex formal systems (like those capable of expressing basic arithmetic) that are *undecidable* within that system – neither provable nor disprovable *using the axioms and rules of that system*.
* You correctly note that for such a statement (often called a Gödel sentence 'G'), if the system is consistent, G is typically considered *true* (in the standard model of arithmetic), even though unprovable *within the system*. It essentially asserts its own unprovability.
* You correctly link this to the inability of the system to prove its own consistency (which is Gödel's *Second* Incompleteness Theorem). If a system *could* prove its own consistency, it could then typically prove G, leading to a contradiction (if it's consistent).
* Your analogy to an axiom is interesting. Like axioms, these Gödel sentences are true statements that cannot be *derived* within the system. However, axioms are *starting points* assumed to be true, while Gödel sentences are *consequences* of the system's structure that reveal its limitations.
* Your conclusion that systems remain useful and the theorems aren't "so problematic" for the practice of mathematics is spot on. Mathematicians continue to use systems like ZFC set theory or Peano Arithmetic very effectively, despite knowing they are incomplete. The theorems reveal fundamental limits about formal systems, but don't cripple their utility.

* **Feedback:** Your understanding here is good. You've grasped the core concepts and their implications, including the crucial link to consistency and the practical usefulness of incomplete systems.

*
**2. Eigenvectors/Eigenvalues and Markov Chains:**

* **Your Understanding:** This section is excellent!
* You correctly define eigenvectors and eigenvalues (`Av = Lambda*v`).
* You make the insightful connection between eigenvectors and stable states in dynamic systems, specifically Markov chains.
* You correctly identify that the stationary (equilibrium) distribution of a Markov chain corresponds to an eigenvector of the transition matrix `A` with an eigenvalue `Lambda=1`.
* You correctly set up the equation `Av = v` (or `pA = p` for row vectors, as in your example) which leads to `(A - I)v = 0` (or `p(A - I) = 0`).
* You correctly mention the important conditions for convergence to a *unique* stationary distribution: the chain must be irreducible and aperiodic (for finite state spaces).
* Your example is well-chosen and perfectly executed:
* The transition matrix is correct.
* You correctly set up the system of equations for the stationary distribution `(t1, t2) * A = (t1, t2)`.
* You correctly add the constraint `t1 + t2 = 1`.
* You correctly solve the system to find the equilibrium distribution (0.25, 0.75).
* Your interpretation of the result (long-term market share) is spot on.
* Your observation `[2]` looks like `[1]` is good. Note: `Av=Lambda*v` typically uses column vectors `v`. Your Markov chain equation `(t1, t2) * A = (t1, t2)` uses a row vector `(t1, t2)`. This means the stationary distribution is a *left* eigenvector of A with eigenvalue 1. Equivalently, if you transpose the equation, the *column* vector `Transpose((t1, t2))` is a *right* eigenvector of the *transpose* matrix `Transpose(A)` with eigenvalue 1. So the connection is precise, often involving left eigenvectors or the transpose.

* **Feedback:** This is a very clear and accurate tutorial! You demonstrate a solid understanding of Markov chains, stationary distributions, and their connection to eigenvalues/eigenvectors. The example calculation is flawless. The small point about left/right eigenvectors or the transpose is just a minor clarification.

**Overall:**

You are clearly thinking deeply about these interconnected areas of mathematics, and their interpretation. Your strengths are in understanding the formal definitions (Gödel's theorem statement, eigenvector definition, Markov chain setup) and performing calculations (Markov chain example). Keep exploring these ideas!.

Thank you,
Amine Moulay Ramdane.