Infinity on kenji.blog

Can it be filled with paint, but not painted on the surface?: Gabriel's Horn

Thu, 10 Sep 2026 21:00:00 +0900

What would happen if there was a container with a “finite volume, yet an infinite surface area”? Intuitively, it seems impossible, but in the world of mathematics, such a solid figure certainly exists. It is the figure known as “Gabriel’s Horn”, also known as “Torricelli’s Trumpet”.

Discovered in 1641 by the Italian mathematician Evangelista Torricelli, this figure shocked mathematicians and philosophers of the time, sparking fierce debate about the nature of “infinity”.

The Painter’s Paradox

If we liken the properties of this figure to everyday “paint”, the following bizarre paradox occurs.

When filling the horn with paint: Since the volume of the horn is finite (exactly $\pi$), you can completely fill the inside of the horn by pouring just $\pi$ liters (about 3.14 liters) of paint.
When painting the surface of the horn: The surface area of the horn is infinite. Therefore, if you try to paint the inside (or outside) surface of the horn with a brush, no matter how much paint you prepare, you will never finish painting it for eternity.

“You can fill the inside with 3.14 liters of paint, but you need an infinite amount of paint to paint the surface.” Why does this counter-intuitive situation occur?

graph TD A["Gabriel's Horn"] --> B["Calculation of Volume (Integration)"] A --> C["Calculation of Surface Area (Integration)"] B --> B1["Volume = π (Finite)"] B1 --> B2["Can fill the inside with paint"] C --> C1["Surface Area = ∞ (Infinite)"] C1 --> C2["Cannot completely paint the surface"] B2 --> D{"Paradox!"} C2 --> D style A fill:#FFD54F,stroke:#333,stroke-width:2px style B1 fill:#81C784,stroke:#333 style C1 fill:#E57373,stroke:#333,color:#fff style D fill:#F44336,stroke:#333,color:#fff,stroke-width:3px

Mathematical Proof: The Magic of Calculus

Gabriel’s Horn is created by rotating the graph of the function $y = \frac{1}{x}$ (where $x \ge 1$) around the $x$-axis. Let’s calculate the volume $V$ and surface area $A$ of this solid figure using calculus.

1. Calculation of Volume (Why it becomes finite)

The volume $V$ of a solid of revolution is found by integrating the cross-sectional area (a circle with radius $\frac{1}{x}$).

$$ V = \pi \int_{1}^{\infty} \left( \frac{1}{x} \right)^2 dx = \pi \int_{1}^{\infty} \frac{1}{x^2} dx $$

Calculating this definite integral:

$$ V = \pi \left[ -\frac{1}{x} \right]_{1}^{\infty} = \pi (0 - (-1)) = \pi $$

The result converges to a finite value $\pi$.

2. Calculation of Surface Area (Why it becomes infinite)

On the other hand, the calculation for the surface area $A$ is as follows.

$$ A = 2\pi \int_{1}^{\infty} y \sqrt{1 + \left(\frac{dy}{dx}\right)^2} dx $$

Since

$$ \frac{dy}{dx} = -\frac{1}{x^2} $$

, the content inside the square root becomes $1 + \frac{1}{x^4}$. Here, since $\sqrt{1 + \frac{1}{x^4}} > 1$ for all $x \ge 1$, the following inequality holds.

$$ A > 2\pi \int_{1}^{\infty} \frac{1}{x} \cdot 1 dx = 2\pi \left[ \ln x \right]_{1}^{\infty} $$

The natural logarithm $\ln x$ diverges to infinity as $x \to \infty$. Therefore, the surface area $A$, which is larger than that, naturally also diverges to infinity.

The “Trick” Behind This Paradox

Even if it can be proven mathematically correct, it might not make sense to our real-world intuition. “If it can be filled with paint, that paint is touching the inner surface, so shouldn’t the surface be painted as well?”

This discrepancy in intuition arises from the confusion of mathematical concepts with physical reality.

In the world of mathematics, the “thickness” of paint can be made infinitely thin down to zero. Although Gabriel’s Horn becomes infinitely narrow as it goes further, the mathematical paint can become infinitely thin and flow into the deepest depths of that narrow tip, coating the infinite surface area with a finite volume (however, the thickness of the paint film approaches zero towards the tip).

However, in the physical real world, paint is made up of atoms and molecules (particles with a finite size). Even if you pour real paint, once the horn’s tube becomes narrower than the “diameter of a paint molecule”, the paint can go no further. In other words, physically, it is impossible to fill it to the tip, nor to paint its infinite surface.

Gabriel’s Horn is a beautiful example that teaches us that human intuition is bound by the “rules of a finite world” and does not always align with the world of calculus, which deals with “infinity”.

How Long Is the Coast of Britain?: The Coastline Paradox

Thu, 10 Sep 2026 21:00:00 +0900

How many kilometers long is the coastline of Britain? You might think the answer can be found in an encyclopedia or a geography textbook. However, in reality, there exists a strange fact: “the answer changes depending on how you measure it, and theoretically becomes infinite.”

This is the Coastline Paradox. This discovery later sparked the creation of an entirely new field of mathematics called “fractal geometry.”

The Shorter the Ruler, the Longer the Distance

A coastline is not a straight line, but is composed of countless inlets, capes, and irregularities in the rocky surface.

Suppose you measured the coastline of Britain with a giant 100 km ruler (a straight line). With this ruler, the jagged edges of small inlets and peninsulas under 100 km are ignored and shortcut.

Next, let’s measure it again with a 1 km ruler. Since you are now measuring along the contours of the small bays and capes that were ignored earlier, the total length will inevitably be longer.

Furthermore, what would happen if you measured the unevenness of every single rock with a 1 m ruler, the surface of pebbles with a 1 cm ruler, and the contours of grains of sand with a 1 mm ruler?

graph TD A["Measurement of Coastline"] --> B["100 km Ruler"] A --> C["1 km Ruler"] A --> D["1 m Ruler"] B --> B1["Ignores small inlets"] B1 --> B2["Measurement result: Approx. 2,800 km"] C --> C1["Follows the shape of inlets"] C1 --> C2["Measurement result: Approx. 3,400 km"] D --> D1["Measures down to the unevenness of rocks"] D1 --> D2["Measurement result: Increases further (theoretically infinite)"] style B2 fill:#FFCDD2,stroke:#333 style C2 fill:#E57373,stroke:#333 style D2 fill:#F44336,stroke:#333,color:#fff

Lewis Fry Richardson discovered this phenomenon empirically in 1951. As the unit of measurement (the length of the ruler) gets smaller, the measured length of the coastline increases endlessly.

Fractal Dimension: Between 1D and 2D

The mathematician Benoit Mandelbrot provided a mathematical explanation for this paradox. In 1967, he published a famous paper in the journal Science titled “How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension”.

Mandelbrot pointed out that natural shapes like coastlines possess self-similarity (fractals), meaning that “no matter how much you zoom in, the same kind of complex structure appears.”

If it were a pure mathematical straight line (1 dimension), the length would not change even if you halved the ruler. However, a coastline is so jagged that it is more complex than a 1D line, yet it is not a 2D surface with area either.

Mandelbrot introduced the concept of “fractal dimension (Hausdorff dimension)” to represent the complexity of such figures. The fractal dimension of the coastline of Britain is estimated to be $D \approx 1.25$. In other words, the coastline of Britain is a mysterious entity with a “dimension higher than a 1D line, but lower than a 2D surface.”

If the length of the ruler is $s$ and the measured length of the coastline is $L(s)$, the following relationship exists with the fractal dimension $D$:

$$ L(s) \propto s^{1-D} $$

In the case of Britain’s coastline, $D = 1.25$, so $1 - D = -0.25$.

$$ L(s) \propto s^{-0.25} $$

This shows mathematically that as the ruler length $s$ approaches 0, the measurement result $L(s)$ diverges to infinity $\infty$.

Ultimate Conclusion: Length Cannot Be Defined

The concept of “length” that we use in everyday life works only for smooth straight lines and curves. Asking for the “absolute length” of a fractal figure existing in nature (coastlines, clouds, mountain ranges, branching of blood vessels, etc.) actually makes no mathematical sense.

“How long is the coast of Britain?” The correct answer is, “It depends on the length of the ruler used to measure it,” and theoretically, it is “infinite.” The fact that infinite length is folded into a limited small space can be said to be a beautiful paradox regarding our spatial perception.

Is a flying arrow at rest?: Zeno's Arrow Paradox

Thu, 10 Sep 2026 21:00:00 +0900

An arrow shot from a bow is flying through the sky. This arrow is certainly moving. However, Zeno, a Greek philosopher from the 5th century BC, developed the following terrifying logic.

“The flying arrow is actually at rest.”

This is neither a joke nor sophistry, but the “Arrow Paradox” that mathematicians and philosophers have seriously debated for 2500 years.

Zeno’s Argument

Zeno’s argument starts with the concept of a “moment in time.”

Time is a sequence of “moments.”
If you capture a single moment (a point where the length of time is zero) like a “photograph,” the arrow “is” at a specific point in space.
At that moment, the arrow is merely “occupying” that space and is not moving. (If it were moving, it would require a “duration of time” rather than a “moment.”)
This holds true no matter which moment you pick out.
If the arrow is at rest at every moment in time, when does it move?

graph TD A["Flying arrow"] --> B["Time is a continuous sequence of moments"] B --> C["Moment t1: Arrow is at rest at position A"] B --> D["Moment t2: Arrow is at rest at position B"] B --> E["Moment t3: Arrow is at rest at position C"] C --> F{"At every moment, the arrow is at rest"} D --> F E --> F F --> G["Conclusion: The arrow is not moving!"] style A fill:#2196F3,color:#fff style F fill:#FF9800,color:#fff,stroke-width:2px style G fill:#F44336,color:#fff,stroke-width:3px

Intuition vs. Logic

“Ridiculous. The arrow is actually flying, isn’t it?” This is probably most people’s first reaction. However, it is actually extremely difficult to point out logically where Zeno’s argument is flawed.

In fact, the ancient Greek philosopher Diogenes is said to have simply stood up and walked around the room in response to Zeno, showing him, “Look, it is moving.” However, this does not constitute a refutation of Zeno’s logic. What Zeno is questioning is not “whether it can move,” but rather, “can we logically explain what it means to move without contradiction?”

Resolution by Calculus (An Attempt)

Calculus, invented by Newton and Leibniz in the 17th century, provided a mathematical answer (at least partially) to this paradox.

In calculus, “the velocity at a certain moment (instantaneous velocity)” is defined as follows:

$$ v(t) = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t} $$

In other words, velocity is defined as the “limit” where the change in position $\Delta x$ is divided by the change in time $\Delta t$, as $\Delta t$ approaches infinitely close to zero.

The point here is that “instantaneous velocity” is not the distance traveled within a zero-time duration. It is a quantity defined as the “tendency,” or “limit,” of minute changes before and after that time.

Therefore, the answer from the perspective of calculus is as follows:

“Indeed, if you capture a moment of zero length, the arrow has not moved ‘within’ that moment. However, even at that moment, the arrow has the property of ‘instantaneous velocity (a non-zero limit value).’ Being ‘at rest’ means having an ‘instantaneous velocity of zero,’ but since the instantaneous velocity of a flying arrow is not zero, the arrow cannot be said to be ‘at rest.’”

Remaining Philosophical Questions

While calculus provided a practical solution to Zeno’s paradox, it hasn’t completely settled the philosophical debate.

The concept of a “limit” is strictly a mathematical tool (calculation procedure), and it does not rigorously answer fundamental questions such as “what physically is the smallest unit of time (moment),” “what is continuity,” and “what is the essence of motion.”

In modern physics (quantum mechanics), the possibility that minimum units for time and space (Planck time, Planck length) exist is discussed. If time is not “continuous” but “discrete (digital),” Zeno’s paradox might need to be re-evaluated in a completely different context.

Even after 2500 years, Zeno’s arrow continues to ask us, “What does it mean to move?” and “What is time?”

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.

The Two Envelopes Paradox: The Collapse of Logic and Decision-Making Traps Caused by Infinite Expected Values

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

1. The Ultimate Choice: To Switch or Not to Switch?

You are standing on the final stage of a game show. On the table in front of you are two identical-looking envelopes (A and B). The host says to you:

“One envelope contains twice as much money as the other. Please choose one.”

After some hesitation, you choose Envelope A. Just as you are about to look inside, the host whispers the devil’s temptation:

“You can exchange your Envelope A with the remaining Envelope B right now if you want. Would you like to switch?”

Now, should you switch your envelope?

2. The “Infinite Loop” Derived from Expected Value Calculations

Let’s exercise some mathematical thinking here. Suppose the amount in your Envelope A is $X$ yen. According to the rules, the amount in Envelope B is either “half of $X$ yen ($\frac{X}{2}$)” or “twice $X$ yen ($2X$)”. The probability for each is $\frac{1}{2}$ (50%).

Now, let’s calculate the expected value (the estimated average amount) if you switch envelopes.

$$ E = \frac{1}{2} \times \left(\frac{X}{2}\right) + \frac{1}{2} \times (2X) $$ $$ E = \frac{X}{4} + X = \frac{5}{4}X = 1.25X $$

A surprising result emerges. By simply switching envelopes, the expected value jumps to $1.25$ times (a 25% increase) the original $X$ yen. The conclusion becomes, “If you think mathematically, it’s definitely better to switch!”

However, a collapse of logic occurs here. Suppose you switched to Envelope B. What happens if the host asks again right after, “Do you want to switch back to A after all?” The exact same calculation formula applies, and this time it means “Switching from B to A will increase the expected value by 1.25 times.”

In other words, just by continuously switching “from A to B” and “from B to A”, the theoretical expected value will keep increasing infinitely. This clearly contradicts reality (the contents of the envelopes are fixed from the start and do not increase just because you switch them).

graph TD Start["You choose Envelope A (contains X yen)"] --> Think["Calculate if it's profitable to switch"] Think --> Case1["Envelope B has half (X/2 yen) : 50% probability"] Think --> Case2["Envelope B has double (2X yen) : 50% probability"] Case1 --> Calc["Expected Value = (X/4) + X = 1.25X"] Case2 --> Calc Calc --> SwitchToB["Switch to Envelope B! (contains Y yen)"] SwitchToB --> ThinkAgain["Calculate again"] ThinkAgain --> Case3["Envelope A has half (Y/2 yen) : 50% probability"] ThinkAgain --> Case4["Envelope A has double (2Y yen) : 50% probability"] Case3 --> Calc2["Expected Value = 1.25Y"] Case4 --> Calc2 Calc2 --> SwitchToA["Switch back to Envelope A!"] SwitchToA --> Start style Calc fill:#ff9999,stroke:#333,stroke-width:2px style Calc2 fill:#ff9999,stroke:#333,stroke-width:2px style SwitchToA fill:#ff4444,color:#fff,stroke:#333,stroke-width:4px

Why did a seemingly perfect expected value calculation produce such a strange paradox?

3. Demystifying the Mathematical Trick: The Swap of Variables

The trap of this paradox lies in “how the random variable $X$ is used”.

In the previous calculation formula, we treated the amount $X$ in Envelope A as a fixed constant, and assumed Envelope B is either “$\frac{X}{2}$ or $2X$”. However, what is actually fixed is the “total amount of money in the two envelopes”, or the “smaller amount”.

Let $S$ be the amount in the envelope with less money. Then, the envelope with more money contains $2S$. There are only two possible scenarios for the entire game (the probability of each is $\frac{1}{2}$).

Pattern 1: Envelope A you chose has the smaller amount ($S$), and Envelope B has the larger amount ($2S$)
Pattern 2: Envelope A you chose has the larger amount ($2S$), and Envelope B has the smaller amount ($S$)

Now, let’s correctly calculate the expected values for “not switching” and “switching” the envelopes.

Expected value when not switching $E_{stay}$:

$$ E_{stay} = \frac{1}{2} \times S + \frac{1}{2} \times 2S = \frac{3}{2}S = 1.5S $$

Expected value when switching $E_{switch}$: You get $2S$ in Pattern 1, and $S$ in Pattern 2.

$$ E_{switch} = \frac{1}{2} \times 2S + \frac{1}{2} \times S = \frac{3}{2}S = 1.5S $$ $$ E_{stay} = E_{switch} $$

The expected values match perfectly! In the first incorrect calculation, we treated the $X$ in Pattern 1 (which is actually $S$) and the $X$ in Pattern 2 (which is actually $2S$) as different values using the same variable $X$, which created the illusion that “switching increases the expected value.”

pie title "The Truth of Expected Values (assuming the smaller amount is S)" "Expected value of not switching (1.5S)" : 50 "Expected value of switching (1.5S)" : 50

4. What If You Open the Envelope?

The paradox seems to be resolved. However, a deeper problem awaits.

What if you looked inside your Envelope A before exchanging envelopes? When you open Envelope A, you find “10,000 yen” inside.

At this moment, $X = 10000$ becomes a fixed value. Envelope B contains either “5,000 yen” or “20,000 yen”. What happens if we apply the very first calculation formula here?

$$ E_{switch} = \frac{1}{2} \times 5000 + \frac{1}{2} \times 20000 = 2500 + 10000 = 12500 $$

The expected value is 12,500 yen. It is certainly higher than the current 10,000 yen. Moreover, since $X$ is now a “specific constant” of 10,000 yen, the previous counterargument of the “swap of variables” no longer applies. In this case, is it absolutely better to switch?

The Disproof by Bayesian Inference: The Missing “Prior Distribution”

In response to this, mathematicians introduced the concept of the “prior distribution of amounts (prior probability)”. The question is whether we can truly say that 5,000 yen and 20,000 yen are each inside with a $\frac{1}{2}$ probability.

For example, suppose the maximum budget for the show is 100 million yen. If you open Envelope A and find “60 million yen”, the probability that Envelope B contains “120 million yen” is zero (because it’s over budget). In other words, as the amount in Envelope A gets larger, the probability that Envelope B is “double” must decrease, and the probability that it is “half” must increase.

When calculating the expected value using Bayes’ theorem assuming an arbitrary prior distribution $P(x)$, it has been mathematically proven that under any realistic probability distribution (where the sum is 1), there is no magical distribution that makes it “better to switch” for all amounts of $X$.

5. The Infinite Trap: Connection to the St. Petersburg Paradox

There is only one case where it is “better to switch for all $X$”. That is only if we assume the show’s budget is infinite and all amounts (1 yen, 2 yen, 4 yen, 8 yen… up to infinity) appear uniformly—an “improper prior distribution” (a distribution whose sum is infinity).

However, in the real world, no television station has infinite assets. The bug caused by this “infinite expected value” shares deep roots with the St. Petersburg paradox (the problem of how much a person would be willing to pay for a gamble with an infinite expected value).

6. Conclusion: The Terrors of Probability and Expected Value

Even though the “Two Envelopes Paradox” consists only of simple multiplication and addition, it teaches us the following lessons:

Errors caused by ambiguity in definitions: If you do not clarify what a variable refers to (whether $X$ always refers to the same amount), logic can easily collapse.
The illusion of “no information = 50% probability”: The assumption that “because we don’t know, it must be fifty-fifty” (the principle of insufficient reason) can sometimes lead to fatal miscalculations.
The difficulty of handling infinity: Introducing the concept of “infinity,” which cannot be applied to the real world, into calculation formulas produces results that defy common sense.

The next time in life you think, “The grass is greener on the other side, so it’s better to switch,” remember this paradox. In your calculation formula, the variables might just be getting swapped.