This is the first post in a series introducing the Theory of Axiomatism — a new framework for analyzing large systems, built on a formal mathematical model. Instead of analyzing GDP, budgets, stock market indicators, and the like, the theory introduces a specific metric: Axiomatic Complexity (S). This metric is derived using set theory, deontic logic, and Zadeh's fuzzy sets. It allows us to predict which systems retain the capacity for coherent action and which paralyze themselves through internal contradictions.

In 2021, the world watched as a global superpower with the largest military in human history lost a twenty-year war to an insurgent force that had no tanks, no air force, and no advanced technology. This unprecedented situation was accompanied by humiliating images of the American army fleeing from Kabul airport, abandoning tens of billions of dollars worth of equipment.

More than four years have passed, and it seems America has once again stumbled into a war with an opponent it doesn't consider a serious rival.

At the root of this overconfidence are purely quantitative metrics: GDP, military budgets, and other conventional measures of economic and military potential.

Every US president boasts that their country's economy is the strongest, that its military might is unprecedented in history. But how many long-term wars has this unparalleled army and economy actually won in the last fifty years?

What if it's not about the size of the army or the volume of GDP? What if strength is measured in something more fundamental? I'm not talking about the fighting spirit of soldiers or their level of dedication — those are subjective things. I'm talking about something mathematically measurable, something that could be calculated before a conflict even begins, allowing us to predict with logical certainty who will win.

Moreover, armed with such a metric, we could create something like "Super-Subjects" — entities whose hidden power, invisible to mainstream economists and political scientists, would surpass all their opponents. And this applies not only to states, but to political parties, movements, organizations, and even business projects.

This sounds arrogant, as if another unrecognized genius has found the philosopher's stone. However, unlike other political theories that claim to explain everything, my theory is a rigorous formal mathematical model, which I will introduce in the appropriate section.

I'm talking about the Theory of Axiomatism.

Let's start from the beginning. My theory was born from mathematical Set Theory.

Set theory was created to formally describe how we categorize reality. Its concept is simple: a set is a collection of distinct objects, thought of as a single whole.

The simplest example: imagine a shelf with vegetables. A tomato, a cucumber, and a bell pepper are different objects. But our brain instantly groups them into the set "Vegetables." We draw an invisible boundary in space that separates these objects from "Fruits" or the "Meat section."

Set theory is, essentially, about where we draw boundaries. We take the chaos of billions of individual objects and group them into categories: "my family," "citizens of the country," "democratic values" — which then come to be thought of as unified wholes.

Historically, this theory, created by Georg Cantor, relied on intuitive understanding and a single powerful rule: for any property P(x) that can be expressed in mathematical language, there exists a set A that consists exactly of all objects x having property P(x). Simply put, a set is defined by the property of its elements. Any property we can formulate creates a set.

Now let's take the modern system of international relations in the broad sense — including the system of values, principles, and what is commonly called the "world order" — and project Cantor's rule onto it.

We get: any imperative that can be formulated from the premises of such a system becomes an integral part of that system, a subset of it. Moreover, in our world, the more such imperatives we can formulate within a system, the better — the more "universal" it is considered. But this is precisely the moment when the system begins to disintegrate. It becomes so complex that it can no longer process all the imperatives flowing into its core. The complexity of the system generates its entropy, which grows exponentially. And if such a system encounters an obstacle — another system that, unlike it, is as simple as a bicycle — it seems that Goliath will easily defeat David. But the truth is that the simpler system always wins. Later we'll see how this is expressed mathematically, but for now, let's delve deeper.

Modern political systems operate as Cantorian machines (political Cantorianism): they complicate the system, formulating as many imperatives within it as they possibly can. Let me give an example.

Before the start of the Afghan war (2001-2021), the coalition set itself an enormous number of goals: victory over the Taliban, nation-building, creation of democratic institutions, fighting religious hegemony in public life, combating drug trafficking, minimizing coalition military casualties, reducing the financial burden on the budget.

We see a huge number of noble imperatives, but the truth is that together they lead to system paralysis. We see that many of these imperatives simply contradict each other. For instance, the goal of defeating the Taliban contradicts the imperative of minimizing military casualties. Fighting religious hegemony in society contradicts the goal of building a democratic state (because the majority, consisting of religious Afghans, would effectively be denied representation — if you don't deny them representation, the first honest election would be won by a cleric). And so on.

Meanwhile, the Taliban had only one task: to survive.

Fast forward to today. Right now, the US is engaged in a war with Iran, declaring the following goals: regime change, destruction of missile production, dismantling the nuclear program, as well as protecting Israel, protecting Arab allies, and preventing the collapse of the Iranian state — which would trigger new waves of millions of refugees heading west.

Iran's task: to survive.

But let's return to Cantor for a moment to explain this phenomenon. His idea was noble. He wanted to create a maximally open system that would be understandable, logical, and attractive. His idea failed spectacularly and was later called "naive".

What happened?

Its vulnerability was exposed by Bertrand Russell, who formulated his famous paradox. What is the essence of this paradox? Russell divided all conceivable sets into two types:

Ordinary sets, which do not contain themselves. For example, take a box of vegetables. The box itself is not an element of the set called "box of vegetables" because it's cardboard, not a vegetable.
Extraordinary sets, which contain themselves. Imagine the "set of all abstract ideas." The concept of this set is itself an abstract idea, so it must lie inside this box.

Russell demonstrated his paradox with the example of the barber. So: in a village lives a barber who shaves all those, and only those, who do not shave themselves. Question: does he shave himself? If he does, he violates the rule (not shaving those who shave themselves). If he doesn't, he also violates the rule (he must shave those who don't shave themselves).

This is Russell's Paradox. As soon as we try to draw a boundary around everything that has no boundaries, logic breaks down.

This paradox is not just an amusing puzzle. In mathematics, it led to the greatest crisis, and in life, it's exactly what happens to political systems.

The mathematical world emerged from this crisis in 1908 thanks to Ernst Zermelo. The essence of the solution: Zermelo forbade creating sets "out of nowhere" based on a simple description. He introduced a rule: you cannot create the "set of all objects with property X". You can only take an already existing set and "filter out" the subset of elements you need.

You cannot simply say, "Let there exist a club of all people who do not belong to any clubs". By Zermelo's rules, you must first have the "set of all residents of the town," and only from that set can you pick out those who do not belong to clubs.

Zermelo essentially established a "ceiling" for systems, forbidding them from being infinitely all-encompassing.

For the world of international politics, the conclusion is this: it is impossible to create a comprehensive global system (an empire). In trying to embrace the unembraceable, the system plants a logical bomb under itself, and as it attempts to realize such an ambition, internal entropy will simply tear it apart.

To prevent Russell's paradox from detonating within a system and destroying its internal logic, we need a filter — one that screens the system's goals and imperatives in such a way as to prevent its collapse.

And this filter exists…

This is the first part. In the following parts, I'll try to sketch out the mathematics (there will be formulas, but nothing too scary), and show what this "filter" might look like and how it could work.

For now, I'd welcome any criticism, especially if you can point out where I'm wrong or what has already been discussed before me. Thanks for reading.