Teaching objectives
Are there more whole numbers than even numbers? Can a full infinite hotel take in infinitely many more guests? Can a set have length zero and still contain as many points as the whole line? This Bachillerato/university lab, outside the curriculum, goes through set theory and Georg Cantor’s infinite cardinals: from the idea of a bijection (the same one shepherds used to count sheep with stones) to the strange Cantor set, by way of three classic paradoxes of infinity (Zeno, divergent series, the Ross-Littlewood vase) and Hilbert’s hotel.
What you'll learn
The central idea: infinity does not behave like a large number. A proper part of an infinite set can be the same size as the whole set, and not all infinities are equal — some are strictly larger than others, and above any of them there is always a larger one.
- Sets and bijection (rooms 1-3). What a set is, its cardinal, and the idea of a bijection through the story of the shepherd and his sheep (stones going into and out of a bag).
- Three paradoxes of infinity (rooms 4-6). Achilles and the tortoise (a sum of infinitely many terms can be finite), divergent sums (1-1+1-1+... and Euler’s manipulation; be careful when regrouping), and the Ross-Littlewood vase paradox (infinitely many cards go in and out, and the result depends on the order).
- Countable infinity, ℵ0 (rooms 7-12). Even numbers vs integers, naturals vs integers, Hilbert’s infinite hotel (one more guest, infinitely many more guests, infinitely many delegations via primes), and the rationals listed by diagonals.
- The continuum, uncountable (rooms 13-16). Cantor’s diagonal argument on the reals, why a small interval has the same cardinal as the whole line, why the line and the plane have the same cardinal, and Cantor’s theorem (the power set is always larger).
- The Cantor set (rooms 17-21). Binary and ternary fractions, the construction of the Cantor set by removing middle thirds, why it has measure 0, and why it still has the cardinal of the continuum.
- Wrap-up (rooms 22-24). The Continuum Hypothesis and its independence (Gödel/Cohen), a final review that goes back to the sheep from the beginning, and a closing puzzle: the 3-chip game, where an invariant (2^-(m+n)) proves that certain configurations cannot be reached however long you play.
Key mathematical ideas
- Bijection: two sets have the same cardinal if there is a one-to-one correspondence between them, even if one “looks” larger.
- Countable infinity ℵ0: the cardinal of N, Z and Q; an infinite proper part can have the same cardinal as the whole.
- Cantor’s diagonal argument: building an element that differs from each one in a list at a specific position shows that no list can exhaust the set — it is used both to prove that R is uncountable and for Cantor’s theorem.
- Cantor’s theorem: |P(A)| > |A| always, which produces an infinite tower of ever larger cardinals.
- Cardinal of the continuum c: |R| = |R²| = |(0,1)|; the number of dimensions does not change the number of points.
- Measure vs cardinal: the Cantor set has length 0 but cardinal c — size and quantity are different concepts.
- Convergent vs divergent series: a geometric progression with ratio |r|<1 can be summed safely; a series that does not converge (such as 1-1+1-1+...) has no well-defined sum, however tempting it is to assign it one through algebra.
- Continuum Hypothesis: the question of whether there is a cardinal strictly between ℵ0 and c is independent of the usual axioms of set theory (Gödel 1940, Cohen 1963).
- Invariant (impossibility argument): assigning each state of a game a value that a rule preserves exactly makes it possible to prove that certain states cannot be reached, without having to try every possible move — the same idea, with a different formula, appears in many classic combinatorial problems.
Room-by-room contents
Room 1 · What is a set?
Set notation with braces, the empty set, sets that contain other sets, and the rule that elements are not repeated.
Student tasks
- Write the set {a,b}.
- Write the set of the first 4 prime numbers.
Room 2 · The cardinal
The cardinal as the number of elements; the same cardinal can describe sets of different kinds (numbers, colours).
Student tasks
- Work out the cardinal of several sets, including one with a box inside it and the empty set.
Room 3 · The shepherd and the sheep
The story of counting sheep with stones introduces the bijection: two sets have the same cardinal if there is a one-to-one correspondence between them.
Student tasks
- Find out what material the shepherds used.
- Watch the animation of the sheep and the stones.
- Answer what would happen if there were sheep left over.
Room 4 · Achilles and the tortoise
Zeno’s paradox: Achilles would never catch the tortoise because he has to cover infinitely many ever smaller stretches. It is resolved by adding up the geometric progression.
Student tasks
- Identify the type of sequence formed by the stretches.
- Tell apart what is true and what is false in Zeno’s reasoning.
- Work out where Achilles would catch the tortoise at three times its speed.
Room 5 · Careful with infinite sums
The series 1-1+1-1+... gives different results depending on how it is grouped (Grandi’s paradox); Euler’s algebraic manipulation (s=1-s) gives a third value, 1/2.
Student tasks
- Compare the two groupings.
- Solve s=1-s and think about why none of the three values is “the right one”.
Room 6 · The vase paradox
Ross-Littlewood: at each step 10 cards go in and 1 comes out; at 12 o’clock the vase is empty, because any given card has already come out at its own step. Changing the order in which cards come out changes the result.
Student tasks
- Go through the steps with the slider.
- Answer how many cards are left at 12 o’clock.
- Repeat the reasoning when only the even-numbered cards are taken out.
Room 7 · Which are there more of?
Even numbers vs integers: an explicit bijection (×2 / ÷2) is built between the two sets, showing that they have the same cardinal, ℵ₀.
Student tasks
- Predict which set is larger.
- Watch the animated bijection.
- Compare multiples of 10 and multiples of 2.
Room 8 · More naturals or integers?
A zigzag bijection between N and Z that splits the naturals into consecutive pairs.
Student tasks
- Look at the correspondence.
- Work out the formula for 2n-1, knowing that 2n→n.
Room 9 · The infinite hotel: one more guest
With Hilbert’s hotel full, moving every guest from room N to room N+1 frees room 1.
Student tasks
- Find the reassignment formula for one new guest.
Room 10 · The infinite hotel: infinitely many more guests
The N+1 trick is not enough for infinitely many new guests; N→2N (or something similar) is needed to leave infinitely many gaps.
Student tasks
- Find a formula that makes room for infinitely many new guests.
Room 11 · The infinite hotel: infinitely many delegations
Housing infinitely many delegations, each with infinitely many guests, requires prime powers, p_m^k, to make sure there are no collisions.
Student tasks
- Rule out formulas that are not bijections (with a counterexample).
- Work out the fourth odd prime.
- Find the final formula with primes.
Room 12 · The rationals, Q
Listing the positive fractions by diagonals (constant numerator+denominator) shows that |Q|=|N|, even though the first correspondence is not injective.
Student tasks
- Predict whether there are more rationals than naturals.
- Watch the listing by diagonals.
- Explain why it is not a bijection yet.
Room 13 · The irrationals: Cantor’s diagonal
The diagonal argument: assuming a complete list of the reals in (0,1), a number is built that differs from each one on the list in one digit, which shows that R is uncountable.
Student tasks
- Watch the construction of the diagonal number.
- Explain why that number cannot be on the list.
- Build a diagonal number from new data.
Room 14 · A piece of the line, as big as the whole line
A geometric bijection (an arc plus a projection) shows that the interval (0,1) has the same cardinal as the whole real line.
Student tasks
- Watch what happens to the points close to the endpoints.
- Explain why the interval has to be open.
Room 15 · The line and the plane
Interleaving the decimal digits of x and y gives a bijection between the square (0,1)×(0,1) and the interval (0,1), which shows that |R²|=|R|.
Student tasks
- Interleave digits to build z from x and y.
- Split a given z into its x and y.
Room 16 · Cantor’s theorem
First by counting subsets (2ⁿ) and then with a rigorous proof by contradiction (the set of “traitors”), it is shown that |P(A)|>|A| always, even for infinite sets.
Student tasks
- Count subsets with the interactive slider.
- Identify the traitors in a concrete example.
- Follow the contradiction showing that D cannot be the image of any x.
Room 17 · Binary and decimal fractions
Every real number in (0,1) has a binary expansion, which gives a concrete version of the previous room (subsets of N ↔ reals) and a simpler version of the diagonal argument.
Student tasks
- Convert between binary and decimal.
- Build an inverted diagonal in binary.
- Identify the subset of N that a sequence represents.
Room 18 · Base-three fractions
Base-3 notation and the key observation: the numbers whose ternary expansion uses only 0 and 2, with no 1 at all.
Student tasks
- Convert between base 3 and decimal.
- Identify which numbers use only 0s and 2s.
Room 19 · The Cantor set
Construction by repeatedly removing middle thirds; it is identified with the numbers in [0,1] whose base-3 expansion uses only 0 and 2 (left/right at each iteration).
Student tasks
- Watch the construction step by step.
- Count the pieces after step 4.
- Locate a point in the Cantor set by choosing left/right at each iteration.
Room 20 · The Cantor set has measure 0
The sum of the lengths removed is a geometric series that converges to 1: no length is left.
Student tasks
- Work out what the series of removed lengths converges to.
Room 21 · The Cantor set has the cardinal of the continuum
Relabelling the ternary digits (0,2)→(0,1) gives a bijection between the Cantor set and (0,1): it has measure 0 but cardinal c.
Student tasks
- Convert a Cantor point to binary.
- Convert a binary number back to a Cantor point.
Room 22 · The continuum hypothesis
Is there any cardinal between ℵ₀ and c? Cantor’s conjecture, and its independence as shown by Gödel (1940) and Cohen (1963).
Student tasks
- Give an intuition about the question.
- Understand what it means for the answer to be independent of the axioms.
Room 23 · Review: the zoo of infinities
A final review with true/false questions on all the ideas of the lab and a visual close that brings back the sheep animation from room 3.
Student tasks
- Answer the set of true/false questions.
- Choose the sentence that best sums up the lab.
Room 24 · The 3-chip game
Final puzzle: 3 chips at (1,1),(1,2),(2,1) on an infinite grid; touching one splits it into (m+1,n) and (m,n+1) if both are free. After 25 moves the same interactive board is used to show why it is impossible to empty the 3 starting squares: each square (m,n) is given the value 2^-(m+n), the 3 starting squares add up to 1/2 and the whole board to 1, and each split preserves the sum exactly, so no finite number of chips can ever fill up the 1/2 outside them.
Student tasks
- Try to empty the 3 starting squares (25 moves).
- Work out the sum of the values of the rows of the board.
- Explain why it is impossible with a finite number of chips.
Rooms to project
The most striking ones to show and discuss in class.