Read a short article about Georg Cantor as an argument, check its facts, learn the mathematics it summarises (bijection, the diagonal, Cantor's theorem), and test a draft that uses it to talk about language learning. Written 2026-11-10.
Chapter 6 teaches the Toulmin model. This page applies it to a short popular article, “Georg Cantor and the infinity of infinities”, a blog post by Jeremy (surname not recorded), shared through a LinkedIn short link that this page cannot open; the text used is the copy given to this course. The article is not reproduced here. Read it first, then use this page to (a) reconstruct its argument, (b) check its facts, (c) learn the mathematics it summarises, and (d) practise the language of definitions.
The page also contains a second text, a draft post by the course author, On Crossing Infinities, which uses Cantor to talk about how children learn language. Part 6 tests that draft against the mathematics, and Part 5 separates what the sources show about Cantor’s own languages from what is only plausible.
One reader’s reconstruction. Check it against the article and correct it.
| Box | What the article offers |
|---|---|
| Claim | Cantor proved that infinity comes in different sizes, and that the sizes form an endless hierarchy. |
| Grounds | A definition of “same size” for infinite sets (a one-to-one pairing); one worked pairing of the natural numbers with the integers; the diagonal argument and Cantor’s theorem, both named and linked but not proved. |
| Warrant | If two finite sets are the same size exactly when their members can be paired off, the same test may be used for infinite sets. This step extends a definition; it is not forced, and it is the step Cantor’s critics could question. |
| Backing | A list of reference pages and videos, and links to standard definitions. The article does not cite Cantor’s own papers. |
| Qualifier | The article says it explains “briefly” and covers cardinality only. Its last sentence then drops all qualification (“absolutely mind-blowing”). |
| Rebuttal | The opposition of Kronecker is told as a story of delays and insults. The article does not state what the objection to infinite sets was, so a reader cannot weigh it. |
Status codes: internal means the page can check it from the article’s own text; read means found in a source read for this page; general means from general knowledge, not checked here; so verify it before you rely on it.
| Statement in the article | Status | Note |
|---|---|---|
| The family left Russia for Germany in 1856. | read | Sinkevich (2018) reports the father’s request for leave abroad on 4 May 1856 and the departure that year. |
| A caption says Cantor lived in the house on the plaque “from 1845 till 1854”. | internal | This differs from 1856 in the text. Both can be true (a house is not a city), but a careful writer would say so. |
| Cantor was educated in a German-speaking world. | read | He attended the German Lutheran St Petri-Schule from about age 8. Its timetable also listed Russian and French. |
| Kronecker’s opposition delayed the 1874 paper and he called Cantor a “charlatan”. | general | Widely told. The article gives no source; find a historian (for example Dauben) before repeating it. |
| Wittgenstein was a harsh contemporary critic. | general | Wittgenstein was born in 1889, when Cantor was 44. His main writing on mathematics came decades after Cantor died in 1918. “Later critic” is more accurate than “contemporary”. |
| Before Cantor people believed N and R have the same size. | general | An overstatement: puzzles about infinite collections being pairable with their parts were known long before. Say what the article means: no one had a theory of different sizes. |
| The diagonal argument proves R is bigger than N. | read | Correct. Cantor’s first proof (1874) was different; the diagonal argument is the later one of 1891. |
| f(n) = (−n−1)/2 for odd n and n/2 for even n pairs N with Z. | internal | Correct if N starts at 0 (checked in Part 4). |
Task (120 words). Choose one row marked general. Say what source you would look for, and write the sentence you would use in your own paper if the source confirmed it, with the hedge you would keep if it did not.
Two sets have the same cardinality if there is a bijection between them: a rule that sends each member of one set to exactly one member of the other, hitting every member once. For infinite sets this is the only available test. Here is the article’s example, with N = {0, 1, 2, …}:
| n | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|---|
| f(n) | 0 | −1 | 1 | −2 | 2 | −3 | 3 | −4 | 4 |
Every integer appears once: the negatives come from the odd n and the non-negatives from the even n. So N and Z are the same size, although Z seems to have “twice as many” members. Task 1: find f(9) and f(10), then say which n gives −100.
Suppose five numbers between 0 and 1 are listed, and we look at their first five digits:
| r1 | 0.31415 |
|---|---|
| r2 | 0.27182 |
| r3 | 0.57521 |
| r4 | 0.41421 |
| r5 | 0.14142 |
The diagonal digits (first of r1, second of r2, and so on) are 3, 7, 5, 2, 2. Change each one by a rule: write 5, unless the digit is 5, and then write 6. This gives d = 0.55655. It differs from r1 in the first digit, from r2 in the second, and so on, so it differs from every number on the list. With an infinite list the same construction gives a number not on it. The article says only “change it”; the 5-or-6 rule matters because it avoids the digits 0 and 9, and so avoids the problem that 0.4999… and 0.5000… are the same number.
Theorem. For every set X, there is no function from X onto its power set P(X) (the set of all subsets of X). So |X| < |P(X)|.
Proof. Suppose f maps X onto P(X). Let D be the set of those x in X that are not members of f(x). Since f is onto, D = f(y) for some y. Is y in D? If y is in D, then by the definition of D, y is not in f(y) = D. If y is not in D, then y is in f(y) = D. Both cases are contradictions. So no such f exists. ■
For finite sets this is easy to see, because a set with n members has 2n subsets:
| n = |X| | 0 | 1 | 2 | 3 | 4 | 5 |
|---|---|---|---|---|---|---|
| |P(X)| = 2n | 1 | 2 | 4 | 8 | 16 | 32 |
The proof above shows the same thing for infinite sets, where counting no longer works. Applied to N it gives a set P(N) strictly bigger than N, and applying it again gives P(P(N)), and so on without end. Task 2: write the proof’s two cases in your own words, using the connectors suppose, since, so and both cases.
Orthogenesis/NestedInfinities/EightChecks.lean as a < 2 ^ a for any cardinal a. This page does not re-run that file.| Question | Documented | Plausible, not shown | Not found |
|---|---|---|---|
| Which languages was he exposed to as a child? | Lived in St Petersburg until about age 11. A German Lutheran school from about age 8, whose timetable listed Russian, German and French (English and Latin optional). A Danish-born father and a St Petersburg-born mother of Hungarian and Viennese descent. (Sinkevich 2018) | That several languages were around him at home. | Which language the family spoke at home. |
| Did he write in Russian? | A biographer quotes Grattan-Guinness that he seemingly never did, though he must have known it (MacTutor). His father did petition the authorities in Russian in 1854. | Any text of his in Russian. | |
| In what language did he write set theory? | German. French versions appeared in Acta Mathematica vol. 2 (1883), translated by others, among them Poincaré (Grattan-Guinness 1970). Gerbaldi translated part of the 1895 survey into Italian. Jourdain made the English edition of 1915. | Marotte’s French translation of 1899 (not confirmed in the pages read). | |
| Did he learn Latin? | His 1867 dissertation has a Latin title. At school he did not take the optional Latin. | That the text itself is in Latin. | The language of the dissertation text. |
| Was a childhood in a work-focused, adult-minded society behind his later ideas? | His father wanted an engineer; the family lived by trade; he was a gifted young violinist. | That children were handed more language than they could yet control. | Any source on child-rearing in that place and time, or a link to his mathematics. |
Task (150 words). Write a paragraph for a paper using only the Documented column. Then write a second paragraph that adds one sentence from the Plausible column, with a hedge from Part 7, and compare how each reads.
The draft post makes two claims: that a child’s vocabulary and the sentences they can form are infinite sets, and that applying grammar “crosses another” infinity. Test them in order.
| Step | What is true | Size |
|---|---|---|
| Words a child has learned at a given age | A finite list. | finite |
| Words the child’s system can form (compounding, endings) | Unbounded, because a compound can be compounded again. Each word is a finite string of parts. | ℵ0 (countably infinite) |
| Sentences, with no length limit | Unbounded; each is a finite string of words. | ℵ0, the same size as N |
| Languages (sets of sentences) | Every subset of the sentences is a candidate language. By Cantor’s theorem there are more of these than sentences. | 2ℵ0 (uncountably infinite) |
Strings over a finite list of 3 parts show the pattern: there are 3, 9, 27, 81 strings of length 1, 2, 3, 4, and 120 in all up to length 4. Listing them by length and then alphabetically gives each string a place in the list, so the set of all finite strings can be paired with N. That is why adding grammar to vocabulary does not make a bigger infinity. The bigger one appears one level up, at the set of possible languages.
Limits of the argument. This is about formal languages. The number of languages humans can actually learn is far smaller, and nothing here shows how children find one. The sentence “Cantor would have agreed” cannot be checked: no source read for this page shows Cantor writing about language acquisition. “Cantor’s theorem lets us say it exactly” can be checked, and is what Part 4 proves.
Task (200 words). Rewrite the draft’s opening paragraph, keeping the author’s point about learning at the front, so that each claim about infinity has the right size in the table above.
| Function | Phrases from this page |
|---|---|
| Defining | X is said to have … if and only if …; two sets have the same cardinality if there is a bijection between them |
| Proving | suppose …; let D be …; since f is onto; both cases are contradictions; so no such f exists |
| Hedging a claim about people | it seems that; the sources do not show; it is plausible that; I found no source for |
| Register | “mind-blowing” is a reaction, not a claim. Replace it with a sentence about what the theorem lets us do. |
Task (80 words). Rewrite the article’s final sentence in an academic register, without any word that names a feeling.