Cantors Diagonale: mehr als unendlich
About this app
Satz von Cantor zum Anfassen: Jede Figur bekommt eine Teilmenge, die Diagonale leuchtet auf, kippt um und fliegt als neue Zeile D heraus, die in keiner Zeile der Liste steht. „Hänge sie doch an!“ erzeugt sofort eine neue Diagonale, endlich mit Figuren und unendlich für ℕ.
With explain mode: an AI tutor walks you through the app, points out what matters in the scene and answers your questions.
Cantor’s theorem
If every element x of a set M is assigned a subset f(x), you get a list of subsets. Cantor builds the diagonal set D = {x ∈ M | x ∉ f(x)} from it: an element belongs to D exactly when it does not lie in its own subset. D differs from every subset f(x) at position x and therefore does not appear in the list. No map from M onto the power set 𝒫(M) is surjective. For finite sets this means 2ⁿ > n, for the natural numbers: 𝒫(ℕ) is uncountable.
What the app shows
The list is a table of yes/no tiles: row x is the subset that x gets. In finite mode the elements are shapes such as circle, triangle and star, in infinite mode the numbers 1, 2, 3, whose rows and columns get narrower and narrower. In five named steps the diagonal lights up, its tiles flip and fly out as a new row D. At every row a ring marks the spot where D is different. A store shows all 2ⁿ subsets next to the n rows of the list.
Try it yourself
Tap tiles or swipe across them and watch D change at once. Before flipping, the app pauses with the question whether D is already in the list, so a class can vote. With “Just append it!” or by dragging row D into the table, D is inserted. Then a new diagonal appears with a new missing D, a game you cannot win. Under Practice you build D yourself, decide whether a shape belongs to D and try in vain to smuggle D into the list.
Frequently asked questions
- What does Cantor’s theorem say?
- For every set M the power set 𝒫(M), the set of all subsets, has strictly larger cardinality than M. There is no surjective map from M onto 𝒫(M). The proof uses the diagonal set D = {x | x ∉ f(x)}, which differs from every image f(x).
- Why can’t you just append the missing subset?
- Whoever gets D is an element of the list, so the new list has a new diagonal. Flipping it gives a new subset that is missing again. For infinite lists you push D in at the front, with the same result.
- Are there different infinities?
- Yes. The natural and the rational numbers have the same cardinality, both are countable. The set of all subsets of ℕ, however, is uncountable: every list of its elements misses at least one subset. This infinity is larger.
- What can I try in the app?
- Build lists of subsets, play the proof step by step, append D and watch the new diagonal, switch between finite shapes and the natural numbers, show all 2ⁿ subsets and solve practice tasks with feedback in the scene.
Subject: Mathe ohne Zahlen
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