Set Theory on kenji.blog

Richard's Paradox: The Contradiction Caused by Infinite Decimals and the "Diagonal Argument"

Thu, 10 Sep 2026 12:00:00 +0900

1. A List of Numbers Definable in Words

“Richard’s Paradox”, published by the French mathematician Jules Richard in 1905, is akin to a relative of the “Berry Paradox” introduced earlier. However, this one is more mathematical and contains a deep contradiction that feels like peering into the abyss of infinity.

First, imagine collecting all the “real numbers between 0 and 1 (decimals) that can be completely defined by English sentences”.

For example, numbers like these:

“Zero point five” $\rightarrow$ $0.5$
“One third” $\rightarrow$ $0.333333...$
“The number formed by the decimal digits of pi” $\rightarrow$ $0.14159265...$

The combinations of sentences that can be expressed in English are simply rearrangements of characters found in a dictionary, so we can assign an “order” to them. (For example, arranging them in order of length, and if they have the same length, arranging them in alphabetical order.)

In this way, we have created an infinitely continuing numbered list (1st, 2nd, 3rd…) of “all real numbers definable in English”.

$$ \begin{align*} r_1 &= 0.\mathbf{3}333... \\ r_2 &= 0.5\mathbf{0}00... \\ r_3 &= 0.14\mathbf{1}5... \\ r_4 &= 0.777\mathbf{7}... \\ &\vdots \end{align*} $$

Within this list, “every possible real number definable in English” should be perfectly included without a single exception.

2. The Demonic Technique: “Diagonal Argument”

Here, Richard performs a terrifying operation. He artificially creates a “completely new number $X$” that avoids all the numbers currently in the list.

The method is simple:

Look at the 1st decimal digit of the 1st number in the list (in the above example, $3$). Add $1$ to it, and make that the 1st digit of $X$ ($3+1=4$).
Look at the 2nd decimal digit of the 2nd number in the list (in the above example, $0$). Add $1$ to it, and make that the 2nd digit of $X$ ($0+1=1$).
Look at the 3rd decimal digit of the 3rd number in the list (in the above example, $1$). Add $1$ to it, and make that the 3rd digit of $X$ ($1+1=2$).

*If the original digit is $9$, let’s assume it loops back to $0$.

The new number $X$ created by this method (in the example above, $X = 0.4128...$) will absolutely never match any number in the list. This is because the $n$-th decimal digit of $X$ is intentionally shifted from the $n$-th decimal digit of the $n$-th number. (This technique is called the “Diagonal Argument”, devised by the genius mathematician Cantor to prove the infinite size of real numbers.)

3. The Completion of Richard’s Paradox

Now, here comes the paradox.

We have just created a new number $X$. And the “rule” for creating this $X$ is perfectly explained (defined) by the English sentences I just wrote above.

In other words, $X$ is a “real number definable in English”.

However, remember the initial premise. “Real numbers definable in English” were supposed to be all comprehensively included in the initial list ($r_1, r_2, r_3...$). Yet, $X$ was constructed so that it does not match any number in the list.

$X$ must exist within the list (because it was defined in English).
$X$ must not exist within the list (because it was constructed using the diagonal argument to differ from all numbers in the list).

A perfect contradiction! This is Richard’s Paradox.

4. Why Did the Logic Collapse? (The Trap of Meta-language)

The reason this paradox arose, similarly to the Berry Paradox, lies in the confusion of “levels of language.”

To perform mathematics rigorously, one must clearly separate the “list of numbers in question (object language)” from the “rules that talk about the properties of that list from the outside (meta-language)”.

Richard’s list is a collection of “definitions of computable numbers”. However, the rule to create the new number $X$, “look at the $n$-th digit of the $n$-th number in the list”, is a “meta-language” operation that cannot be executed without looking down at the list itself from the outside.

Richard’s Paradox exploded into self-contradiction because it secretly tried to slip the “meta-linguistic number $X$ created by manipulating the list from the outside” into the “inside list”.

5. Passing the Baton to Gödel

This Richard’s Paradox sent a massive shockwave through the mathematical community of the time. “Human language (and logical systems) can easily cause self-contradiction if we are not careful. How can we make mathematics perfect and free of contradiction?”

In 1931, it was the 25-year-old genius mathematician Kurt Gödel who brought a final resolution to this problem. Gödel perfectly translated and reproduced the structure of this paradox, which Richard caused using the “ambiguity of language,” by using “rigorous mathematical formulas (Gödel numbering)”.

The result derived from this was the famous “Gödel’s Incompleteness Theorems”. It was a monumental discovery proving the limits of human knowledge: “No matter how rigorously mathematical rules are established, ’truths that can neither be proved nor disproved’ will inevitably arise within those rules (mathematics is incomplete).”

Richard’s Paradox began as a mere contradictory play on words, and eventually evolved into the ultimate weapon to shatter the “absoluteness” of mathematics itself.

Hilbert's Grand Hotel: How to Accommodate Infinite New Guests in a Fully Booked Hotel

Thu, 10 Sep 2026 06:00:00 +0900

1. Welcome to the Ultimate Hotel

The great German mathematician David Hilbert devised an interesting thought experiment to illustrate how far the concept of “infinity” is from human intuition.

Imagine that somewhere in the universe, there is a hotel called “Hilbert’s Grand Hotel.” This hotel has an infinite number of rooms, numbered 1, 2, 3, and so on.

One day, there was a massive event in the universe, and every single room in this infinite hotel was occupied, making it “fully booked.” Then, an exhausted traveler arrived and asked the front desk, “Could you please find me a room?”

A normal hotel would have no choice but to refuse, saying, “We are sorry, but we are fully booked.” However, this is the Grand Hotel. The manager smiled and said, “Certainly. We will have a room ready for you right away.” How can they accommodate a new guest when the hotel is already full?

2. Case 1: How to Accommodate One New Guest

The manager made an announcement over the intercom to all the guests currently staying at the hotel:

“Attention all guests. Please move to the room whose number is ‘plus 1’ of your current room number.”

What happens then?

The guest in room 1 moves to room 2.
The guest in room 2 moves to room 3.
The guest in room 3 moves to room 4.
The guest in room $n$ moves to room $n+1$.

graph LR subgraph "Before Moving (Fully Booked)" R1["Room 1
(Guest A)"] R2["Room 2
(Guest B)"] R3["Room 3
(Guest C)"] R4["..."] end subgraph "After Moving" NewR1["Room 1
(Empty!)"] NewR2["Room 2
(Guest A)"] NewR3["Room 3
(Guest B)"] NewR4["Room 4
(Guest C)"] end R1 -->|Move| NewR2 R2 -->|Move| NewR3 R3 -->|Move| NewR4 NewGuest["New Guest"] -->|Check-in| NewR1 style NewR1 fill:#aaffaa,stroke:#333,stroke-width:2px style NewGuest fill:#ffaaaa,stroke:#333,stroke-width:2px

Since there are an infinite number of rooms, the scenario where “the guest in the last room is kicked out” never occurs. Everyone successfully moves to the next room. And brilliantly, Room 1 becomes vacant. The new traveler was able to stay in Room 1 safely.

In the world of infinity, $\infty + 1 = \infty$ holds true. Even if you take out “one” from the “whole (infinity),” the size of the whole does not change.

3. Case 2: How to Accommodate an Infinite Number of New Guests

Well, the next day, the hotel was fully booked once again. Then, incredibly, an infinite bus carrying an “infinite number of passengers” arrived. The passengers who got off the bus pressed the front desk, saying, “We need rooms for everyone!”

If they asked for the “plus 1” move like yesterday, it would take forever. However, the manager didn’t panic. He made another intercom announcement.

“Attention all guests. Please move to the room whose number is ‘multiplied by 2’ of your current room number.”

What happens then?

The guest in room 1 moves to room 2.
The guest in room 2 moves to room 4.
The guest in room 3 moves to room 6.
The guest in room $n$ moves to room $2n$.

Through this move, the infinite number of guests who were already staying fit perfectly into “all the even-numbered rooms.” And miraculously, “all the odd-numbered rooms (Room 1, 3, 5…)” became completely vacant!

Since there are an infinite number of odd numbers as well, the manager can accommodate everyone by guiding the passengers of the infinite bus sequentially from the front to Room 1, Room 3, Room 5, and so on.

In the world of infinity, $\infty + \infty = \infty$ holds true. Even if you add infinity to infinity, the size remains the same “infinity.”

4. Case 3: What if an Infinite Number of Infinite Buses Arrive?

Furthermore, the next day. Once again, the hotel is fully booked. And then, astonishingly, “an infinite number of infinite buses, each carrying an infinite number of passengers,” arrived in a continuous line.

Bus 1 has an infinite number of people, Bus 2 has an infinite number of people, Bus 3 has an infinite number of people… this goes on for an infinite number of buses. Even the manager seems like he might panic, but he was a mathematical genius. He came up with the idea of using “prime numbers.”

The manager gave the following instructions:

Movement of guests already staying in the hotel Let the current room number be $n$. Have them move to room “$2^n$”. (Room 1 $\rightarrow$ Room 2, Room 2 $\rightarrow$ Room 4, Room 3 $\rightarrow$ Room 8…) This accommodates all the current guests.
Guiding guests from Bus 1 (Infinite people) Let the guest’s seat number be $n$. Guide them to room “$3^n$”. (Room 3, Room 9, Room 27…)
Guiding guests from Bus 2 (Infinite people) Use the next prime number, 5, and guide them to room “$5^n$”. (Room 5, Room 25, Room 125…)
Guiding guests from Bus $k$ (Infinite people) Use the $(k+1)$-th prime number, $P$, and guide them to room “$P^n$”.

Thanks to a powerful mathematical theorem known as “the uniqueness of prime factorization (any number can be expressed as a combination of prime factor multiplications in only one way),” the room numbers $2^n, 3^n, 5^n, 7^n \dots$ will absolutely never overlap with anyone else.

In this way, the manager brilliantly managed to accommodate a staggering number of guests—“Infinity $\times$ Infinity”—into a single infinite hotel!

5. Infinite Sets Have Different “Sizes” (Cantor’s Theorem)

What Hilbert’s Grand Hotel teaches us is the fact that “countably infinite (infinity that can be counted by assigning numbers like 1, 2, 3…)”, no matter how many times it is added together or multiplied, will ultimately fit within the same size of “countably infinite” framework.

However, the mathematician Georg Cantor discovered an even more terrifying truth. “Natural numbers” and “fractions” can all be accommodated in this infinite hotel. But if guests of “real numbers (all decimals, including irrational numbers)” arrive, even this infinite hotel will absolutely not be able to accommodate all of them.

It has been proven that the number of real numbers is fundamentally a “larger (higher-level) infinity” than the number of rooms in the infinite hotel (countably infinite). Although often lumped together under the word “infinity,” there actually exists a hierarchical structure (cardinality) within infinity, ranging from a “small infinity” to an “infinity so large it is absolutely unreachable.”

6. Conclusion: The “Infinity” That Destroys Human Intuition

Hilbert’s Grand Hotel vividly illustrates how the “common sense of the finite” cultivated in our daily lives simply does not apply in the “world of infinity.”

“The whole is greater than the part” “No one can enter a fully booked hotel” “If you add infinity to infinity, it gets bigger”

All these obvious intuitions are brilliantly betrayed. The world of infinity is a treasure trove of paradoxes (truths that contradict intuition). Mathematicians did not fear these paradoxes; instead, they subdued them with the power of logic, classified them, and built the beautiful system of modern set theory.

The next time you are turned away because “the hotel is fully booked,” try to imagine, “What if this hotel were Hilbert’s Grand Hotel?”

Banach-Tarski Paradox: Cut one sphere and get two spheres of the same size?

Thu, 10 Sep 2026 02:00:00 +0900

1. A Magical Theorem: 1 = 1 + 1 ?

Imagine you have a solid gold sphere (ball) right in front of you. You cut this sphere into several pieces with a knife. Then, you reassemble those pieces like a puzzle. You don’t stretch, bend, or add any new gold to the pieces. You just move them around and put them together.

However, when you look at the completed puzzle, you end up with “two solid gold spheres of exactly the same size as the original one”.

You might think, “That’s absurd! It violates the law of conservation of mass, and it’s an alchemist’s delusion!” In the real physical world, it is absolutely impossible. However, in the world of pure mathematics (geometry and set theory), this is proven as a logically 100% correct theorem.

This is the “Banach-Tarski Paradox”, proven in 1924 by two mathematicians, Stefan Banach and Alfred Tarski.

2. Accurately Understanding the Claim of the Paradox

When the theorem proven by Banach and Tarski is expressed in mathematically precise words, it goes like this:

Banach-Tarski Theorem Any solid sphere $S$ in 3-dimensional space can be decomposed into a finite number of disjoint pieces. Then, by reassembling those pieces (using only rotations and translations), it is possible to create two solid spheres that have exactly the same radius as the original sphere $S$.

Even more surprisingly, applying this theorem leads to the following conclusion:

By decomposing a single pea into a finite number of pieces and reassembling them, you can create a sphere exactly the size of the sun. (Also known as the pea and the sun paradox)

Why is such magic mathematically permitted? The secret is hidden in two keywords: “Infinity” and the “Axiom of Choice”.

3. The Mysterious Properties of “Infinity”

The first step to understanding this paradox is to learn about the strange properties of “infinite sets”.

In the “finite” world we normally deal with, the whole is always strictly greater than the part. For example, if you take out the even numbers (5 numbers) from the numbers 1 to 10 (10 numbers), the count is halved.

However, this common sense does not apply in the world of “infinity”. Which are more numerous: all “natural numbers” (1, 2, 3, 4, …) or all “even numbers” (2, 4, 6, 8, …)? Intuitively, since even numbers are only half of the natural numbers, you might feel there are more natural numbers. However, try making pairs as follows:

1 $\rightarrow$ 2
2 $\rightarrow$ 4
3 $\rightarrow$ 6
$n \rightarrow 2n$

In this way, for every natural number, you can exactly pair it with an even number that is exactly twice its value (a one-to-one correspondence). There are no numbers left over. In other words, mathematically, “the number of natural numbers (infinity)” and “the number of even numbers (infinity)” are exactly the same size!

Even though we supposedly took out half (even numbers) from the whole (natural numbers), the size remains unchanged. In infinite sets, it can happen that “a part is equal to the whole”. The Banach-Tarski theorem can be said to be the ultimate form of applying this “magic of infinity” to sets of “points” in 3-dimensional space.

4. Points in Space are Cut “Immeasurably”

When you cut a real object (like gold or an apple) with a knife, the pieces always have a “volume”. However, a sphere in mathematics is a “collection of an infinite number of points” with no volume in themselves.

Banach and Tarski grouped (divided) these infinite points in a very special and complex way. The way they are divided is so complex and scattered that they become a state where “volume can no longer be measured (non-measurable set)”.

graph TD S["Original sphere S (Volume V)"] -->|Special decomposition| P1["Piece 1 (Volume unmeasurable)"] S --> P2["Piece 2 (Volume unmeasurable)"] S --> P3["Piece 3 (Volume unmeasurable)"] S --> P4["Piece 4 (Volume unmeasurable)"] S --> P5["Piece 5 (Volume unmeasurable)"] P1 -->|Rotation and Translation| S1["New sphere 1 (Volume V)"] P2 -->|Rotation and Translation| S1 P3 -->|Rotation and Translation| S1 P4 -->|Rotation and Translation| S2["New sphere 2 (Volume V)"] P5 -->|Rotation and Translation| S2 style S fill:#ffddaa,stroke:#333,stroke-width:2px style S1 fill:#aaddff,stroke:#333,stroke-width:2px style S2 fill:#aaddff,stroke:#333,stroke-width:2px

Once each piece becomes a hazy collection of points that “do not have (or cannot be measured for) volume”, they can escape the constraint of the physical rule (additivity of measure) that says “the sum of the pieces must equal the original volume”.

And by cleverly rotating and combining those pieces of hazy points, the “magic of infinity” completes two spheres packed exactly with the same points as the original sphere. In fact, it has been proven that this operation of “making two spheres from one sphere” is possible by dividing the original sphere into just 5 pieces.

5. The Root of It All: What is the “Axiom of Choice”?

So, why is a “decomposition so complex that its volume cannot be measured” mathematically possible? It is because we accept the “Axiom of Choice”, a rule that forms the foundation of modern mathematics.

Roughly speaking, the Axiom of Choice is the following rule:

Concept of the Axiom of Choice When there are items in many boxes, the rule says “you can choose exactly one item from each box and form a new set”.

If the number of boxes is finite, anyone can do it normally. However, if there are an “infinite” number of boxes, humans cannot finish the operation of “choosing one by one” an infinite number of times. Even so, the Axiom of Choice admits that “it is acceptable to assume the chosen set exists”.

This axiom was extremely convenient and essential in constructing modern mathematics. Most mathematicians accepted this rule, thinking, “Well, it’s obvious.”

However, accepting this Axiom of Choice means admitting the existence of the “scattered, hazy set of points whose volume cannot be measured (non-measurable set)” mentioned earlier. And as a result, the Banach-Tarski theorem, which states that “one sphere becomes two”, is derived as a logical necessity.

6. Conclusion: The “World Beyond Intuition” Painted by Mathematics

The Banach-Tarski paradox is not a paradox in the sense that “there is a contradiction in logic”. It is a paradox in the sense that while the logic is 100% correct, the derived conclusion violently contradicts human intuition and physical laws.

When this theorem was published, some mathematicians argued, “If such an absurd conclusion is reached, the Axiom of Choice must be wrong!” However, today, many mathematicians accept the Axiom of Choice, and the Banach-Tarski theorem is also accepted as a “bizarre but beautiful property held by 3-dimensional space and infinite sets”.

Since the physical world we live in is made of “finite-sized particles” called atoms, we cannot turn a pea into the size of the sun. However, on the canvas of “mathematics” created by the human brain, the size of a point is zero, and infinite operations are allowed.

The Banach-Tarski paradox can be said to be one of the masterpieces of modern mathematics, teaching us how effortlessly the concept of “infinity” leaps over naive human intuition.