A line with as many points as a square

A segment is one-dimensional and a square is two-dimensional, and the natural expectation is that the square has more points — a whole segment’s worth for every position along one side.

Cantor spent three years trying to prove that and then found the opposite. He wrote to Dedekind in 1877 with the correspondence and a sentence to the effect that he could see it and did not believe it — the most quoted line in the subject, and a fair report of what the result does to anybody meeting it.

The construction #

Take a point of the unit square, with coordinates written as decimals:

$$x = 0.a_1a_2a_3\ldots, \qquad y = 0.b_1b_2b_3\ldots$$

Weave them:

$$z = 0.a_1b_1a_2b_2a_3b_3\ldots$$

That is a point of the segment. Unweaving — take the odd places for $x$ and the even ones for $y$ — recovers the point that went in, so different points of the square give different points of the segment. The square injects into the segment.

The other direction is free: send $z$ on the segment to the point $(z, 0)$ of the square. So the segment injects into the square, and by the theorem one rung down the two collections have the same size.

Nothing about the argument used two dimensions. Interleaving three digit strings shows the cube has the same size, interleaving $n$ shows every finite-dimensional space does, and a slightly cleverer weave handles countably many coordinates at once. Dimension does not affect the count, at all.

Where it is not quite a bijection #

The weave is an injection and it is not onto, and the reason is a genuine defect rather than an artefact worth waving through.

A number with two decimal expansions — $0.2000\ldots$ and $0.1999\ldots$ are the same number — has two woven images, and they are nowhere near each other. Weaving the first against a fixed $y$ and the second against the same $y$ gives numbers differing in the third place; the figure computes that difference and asserts it is more than a thousand times the difference between the two expansions.

So the map is not well defined until a convention is chosen — always take the terminating expansion, say — and once the convention is chosen the map is injective but misses points, because nothing weaves to a number whose odd places end in an infinite run of nines.

Two repairs exist and both are standard.

Use the two-injections theorem and never construct the bijection. This is the route above, and it is why that theorem is the rung below rather than a footnote: it converts a pair of imperfect maps into a perfect one without anybody having to write the perfect one down.

Weave blocks rather than digits. Break each expansion into blocks ending at the first non-zero digit, and interleave the blocks. This handles the ambiguity and gives a genuine bijection, at the cost of a construction nobody would guess.

What “the same size” was defined to ignore #

The result is startling and the startle is informative: it says the definition of size ignores something the intuition was using.

It ignores distance. The weave sends nearby points of the square to points of the segment that may be very far apart, and it sends nearby points of the segment to points of the square that may be far apart. Nothing in the definition of a bijection asks about distance.

It ignores order. The square has no natural order and the segment has one; the weave respects neither.

It ignores dimension, which is the headline, and it does so because dimension is not a counting notion at all. It also ignores measure — a set of length zero can have as many points as the line it sits in — and it ignores every structural property that the arithmetic on the rung below already showed it losing.

So the honest statement of the theorem is not the square is no bigger than the segment. It is: the one property that a bijection can see — how many members there are — does not distinguish them. Every property that does distinguish them is a property a bijection is blind to, and the theorem is a measurement of how coarse the notion of size is rather than a fact about squares.

That coarseness is what makes cardinality applicable everywhere, and this essay is the standing demonstration of the cost.

Binary, where the ambiguity is unavoidable #

Doing the weave in base two rather than base ten sharpens both the construction and its defect, and it is the version worth carrying.

A point of the square is two infinite strings of noughts and ones; the weave alternates them into one; unweaving takes the odd and even places. Nothing changes except that the alphabet is smaller and the addresses are longer.

What does change is the size of the ambiguity. In base ten the numbers with two expansions are those ending in an infinite run of nines, which is a countable and easily-avoided nuisance. In base two they are those ending in an infinite run of ones, which is the same countable nuisance — but the repair is harder, because with only two symbols there is no room to encode a block boundary the way base-ten block-weaving does.

That is worth noticing because it says the defect is real rather than notational. No base removes it, and the reason is structural: a decimal expansion is a map from addresses to points that is two-to-one on a countable set, so any construction reading points as addresses inherits that. Every fix is a fix for the double-counting rather than for the weave.

The base-two version also makes the connection to subsets exact. A point of the interval is a subset of the whole numbers — the set of places holding a one — so weaving two points is interleaving two subsets, and the statement that the square has the size of the segment becomes the statement that a pair of subsets of the whole numbers is a subset of the whole numbers. That is the cleanest form of the theorem and it has no geometry in it at all.

What else turns out to be this size #

The continuum absorbs so thoroughly that the list of collections with exactly its size is longer than the list of collections with any other, and it is worth having in hand.

The whole line, and the plane, and every finite-dimensional space. The weave, plus the fact that a segment and a line correspond — $\tan$ does it, or any increasing function with the right ends.

The irrationals alone. Removing a countable collection from the continuum leaves the continuum, by an absorption argument like the ones on the rung below.

The transcendental numbers alone, for the same reason, since the algebraic ones are countable.

Every sequence of whole numbers. A sequence is an address and an address is a point.

The Cantor set — a collection with no length at all, obtained by deleting middle thirds forever, and still the size of the whole line. That one is the sharpest of the list, because it separates size from measure as decisively as this essay separates size from dimension.

Every continuous function on the line. A continuous function is determined by its values at the fractions, so there are at most continuum-many; and there are at least that many constants.

Every subset of the whole numbers, which is where the continuum comes from in the first place: a subset is a sequence of yes and no, a sequence of yes and no is a binary expansion, and a binary expansion is a point.

What is not on the list is the collection of all subsets of the line, which is strictly larger — and the ladder above this rung is about what, if anything, sits between.

The follow-up question, which took thirty-four years #

If a bijection exists, the obvious next question is whether a good bijection exists — one that is continuous, so that nearby points go to nearby points.

The answer is no, and it is a much harder theorem. Invariance of domain, proved by Brouwer in 1911, says that a continuous injection from an $n$-dimensional space into $n$-dimensional space is open — from which it follows that no continuous bijection exists between spaces of different dimension. Dimension is a topological invariant even though it is not a cardinal one.

The gap between 1877 and 1911 is worth noticing. Cantor’s construction immediately raised the question of whether dimension meant anything, and for thirty-four years nobody could show it did. The tools that settled it — homology, degree, the machinery that also proves that nothing on a sphere can be combed flat — were built partly in order to.

So there are two answers and both are correct. Counting says the square and the segment are the same size. Topology says they are not the same space. The disagreement is not a paradox; it is two questions with two answers, and the interesting part is that the first question had to be asked before anybody realised the second needed an answer.

The curve that does fill the square #

There is a construction that is continuous and does reach every point of the square, and it is worth putting beside the weave because it is exactly what is left when bijectivity is dropped instead of continuity.

Peano’s curve of 1890, and Hilbert’s of 1891, are continuous maps from the segment onto the square. They are not injective — some points of the square are hit more than once, and they have to be, by invariance of domain — but they are continuous and they miss nothing.

So of the two properties, either can be had alone and not both:

• Bijective, not continuous: the weave.
• Continuous, not injective: the space-filling curve.
• Both: impossible, by Brouwer.

The space-filling curve is also the standing example of a curve that is nothing like a curve — continuous everywhere, differentiable nowhere, of infinite length in any sub-segment — and it belongs with the curve with a corner at every point as evidence that continuity alone permits almost anything.

Dimension, which has several meanings and no cardinal one #

Since the theorem says dimension is invisible to counting, it is worth asking what dimension is visible to, and the answer is that it has more than one definition and they agree less often than expected.

Topological dimension is defined by how small a set has to be removed to disconnect a neighbourhood — a point for a line, a curve for a surface. It is a whole number, it is a topological invariant, and it is what Brouwer’s theorem is about.

Linear dimension is the number of independent directions, which is what a basis counts, and it applies to vector spaces rather than to shapes.

Box-counting dimension measures how the number of small boxes needed to cover a shape grows as the boxes shrink, and it is not a whole number in general — the Cantor set’s is $\log 2/\log 3$, and a dimension that is not a whole number is the essay on it.

Three notions, three different jobs, and none of them a count. That is the general lesson of this rung stated more carefully than the headline: the properties that distinguish a square from a segment are structural, and structure is exactly what a bijection is licensed to destroy.

Why the intuition was wrong, precisely #

It is worth diagnosing the original expectation rather than merely overruling it, because the diagnosis explains a class of errors rather than one.

The intuition runs: the square is a segment’s worth of segments, so it has a segment’s worth times a segment’s worth of points, which is more. Every step of that is fine except the last. There are indeed continuum-many segments each with continuum-many points; the product of those two sizes is what has to be computed; and the answer is that the continuum times itself is the continuum — which is exactly the weave, restated.

The error is importing the finite fact that $n \times n > n$. It is true for every whole number above one and false for every infinite size, and the pattern was established one rung down: absorption is what infinite sizes do to multiplication, and the continuum absorbs just as $\aleph_0$ does.

Seen that way the result stops being startling and becomes the expected consequence of a rule already known. That is usually what happens to a startling result, and the interval between the two states is where the interesting mathematics is.

What the picture cannot show #

The figure shows one point going one way, and the claim is about all of them. A correspondence is a function and a function has no picture; what can be drawn is an instance, and an instance shows the mechanism rather than the claim.

It cannot show the failure at ambiguous expansions except by computing it, which is what the caption reports. The two numbers involved differ by less than one unit in the last place drawn and their images differ by a visible amount — a discontinuity, in a picture that has no room to show a limit.

And it cannot show that no continuous version exists. That is Brouwer’s theorem, its proof is homological, and there is no picture of it at all: the absence of a continuous bijection is not visible in any drawing of a bijection that happens not to be continuous.

Where the ladder goes next #

Above: how thoroughly a countable collection can be spread through the line, which is the other half of the lesson that counting sees very little. And then the size above the continuum, and whether anything sits between it and the countable.

One debt. The block-weaving repair that gives a genuine bijection is described in a sentence and not drawn, and it deserves to be — it is a small, checkable construction, and the version everybody quotes is the one that does not quite work.

What the weave was measuring #

Dimension is invisible to counting, because counting was defined to see only whether a pairing exists.

The construction is three lines and the lesson is that the definition of size is coarser than anybody’s intuition about size. Everything that makes a square different from a segment — distance, order, dimension, the possibility of moving continuously — is a structure that a bijection is permitted to destroy, and the weave destroys all of them at once while preserving the only thing it was asked to.