© Copyright Kirk Rader 2023. All rights reserved.

How the Parallel Postulate Changed Everything


Mathematicians from Euclid's time to the start of the modern era were strangely obsessed with the fifth, a.k.a. parallel postulate from his treatise on two-dimensional Geometry, Elements. Over the course of millenia there were countless attempts to prove whether the parallel postulate was really an axiom or, as was suspected for a long time but eventually disproven, actually a theorem hiding simpler axioms under its petticoats. The solution to this question precipitated a revolution in mathematical theory and practice that ultimately birthed the post-modern "information age" as well as contributing to scientific and engineering advances across many disciplines, including General Relativity and Quantum Mechanics.

No, really.

Euclid's Parallel Postulate

Once upon a time, mathematicians in the Western tradition regarded themselves as engaged in a science, not different in kind from chemistry or physics. In this way of regarding mathematical truth, the fact that \( 2 + 2 = 4 \) is proven by empirical observation: if I hold two pebbles in my hand then pick up two more, I will be holding a total of four pebbles. This empiricism was used not only to justify claims about which mathematical statements were true and which were not, it guided the way that mathematical problems were posed and solved. As mathematical theories advanced and techniques expanded to include an ever increasing degree of abstraction, they remained for a very long time rooted in empiricism. Negative numbers, for example, were initially justified by using the kind of reasoning modern bookkeepers would recognize as a double-entry ledger. (e.g. \(-3 + 2 = -1\) is like telling someone who is holding $2 that they owe you $3; after they pay you all they have, they still owe you $1 — and that is how ancient generations of mathematicians actually justified introducing negative numbers into their calculations).

Euclid's Elements epitomizes this view of the nature and practice of mathematics. While providing the model for an axiomitic approach to mathematical proof, his actual method was firmly based in constructions, which are empirically verifiable operations using a physical straight-edge and compass. Euclid's axioms and postulates all hearken directly to such empiricism. They are nearly all very simple declarative statements of easily observable facts or simple steps in a construction. "To extend a point to form a line," "to extend a line to form a plane," and so on.

Then there is the fifth postulate, often referred to as the parallel postulate. It looks nothing like the others. In Euclid's original formulation, it is a nearly paragraph-sized sentence rather than a short phrase and can be difficult to puzzle out without aid of a diagram. Even later attempts to produce simpler formulations did not detract from the degree to which it seems like an outlier compared to the other axioms and postulates.

Euclid's parallel postulate:

If a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

Euclid's Fifth Postulate

From Euclid's time right into the start of the modern era, mathematicians were not happy about this. Over the course of millenia, countless attempts were made to show that the parallel postulate must actually be a theorem, derivable from simpler axioms that look more like the others in Euclid's book. While this produced a number of alternative formulations of the axiom, some of which may be easier to visualize than others, none were a reduction to radically simpler axioms. For example:

Alternative formulation of the parallel postulate:

Given a line and a point not on that line, there is exactly one line that can be drawn through the given point that does not meet the given line when both lines are extended indefinitely in both directions.

Alternate Parallel Postulate

After enough time and failed attempts, Western mathematicians eventually shifted their focus away from trying to prove that the parallel postulate was really a theorem. Instead they strove to produce a formal definition of why it is actually an axiom. By the early 19th Century, mathematical techniques had evolved to the point that allowed the following kind of indirect proof to be carried out:

  1. Assume some different axiom that contradicts Euclid's parallel postulate in some specific way.

  2. Show that the equivalent of Euclid's constructions and proofs that can be carried out using this different version of the parallel postulate are just as internally consistent (i.e. produce no self-contradictions) as Euclid's "classical" proofs.

If the second step of the preceding argument can be accomplished for one or more variations from the parallel postulate assumed in the first step, then this proves that Euclid's fifth postulate was an axiom all along. If it were not an axiom but a theorem, altering it would emerge as a contradiction among the more basic axioms from which it would have been derived.

The key here is to separate the idea of "internal consistency" from "empirical truth." The Non-Euclidean Geometries invented by Bolyai, Lobachevsky, Riemann et al. each produce different theorems that contradict those described by Euclid, but as long as the axioms for each such Non-Euclidean Geometry do not contradict themselves for a given version of the parallel postulate, they demonstrate that each such variation on the theme of "Geometry" is just as good, in some sense, as every other — including Euclid's own. Carrying out proofs in Riemann's Elliptic Geometry is like doing Euclid-style constructions on the surface of a spheroid where all lines forming "great circles" around the circumference will eventually intersect. Lobachevsky's Hyperbolic Geometry assumes, on the other hand, that you can draw more than one line through any point none of which intersect some other line. This is harder to visualize than Elliptic Geometry. In this case it is like doing Euclid-style constructions on a surface with a hyperbolic curvature. Neither variation corresponds to what happens when, like Euclid, you use a straight-edge and compass on a flat surface. But neither Lobachevskian nor Riemannian Geometries produce self-contradictory results. And so Euclid's fifth postulate was formally demonstrated to have been an axiom all along.

Mathematicians of their day rejoiced at Bolyai's, Lobachevsky's and Riemann's accomplishments for multiple reasons. They could finally regard the whole question pertaining to the status of the parallel postulate as satisfactorily settled. They could start playing with the new mathematical techiques developed for this enterprise, seeking additional areas of mathematical exploration. But there was a reckoning to be made as well: in order to accept this highly desirable result, mathematicians were forced to alter fundamentally their understanding of their own profession. Mathematics came to be understood as being concerned with validity (internal consistency) rather than (empirical) truth. Euclid's theorems do not merely form an interesting set of mutually consistent mathematical formulas. Euclid's reliance on "constructions" demonstrate that his theorems describe properties of objects and operations carried out in the real world. Non-Euclidean Geometries have no such correspondence to observable reality (at least not at the scale at which human senses operate). Not only does every valid Non-Euclidean Geometry contradict Euclid's "common sense" Geometry, each contradicts every other. But if validity is the goal rather than correspondence to empirical truth, that does not matter at all. A "mathematical truth" was accepted as proven even though that required abandoning the need or ability to assert anything at all about empirical truth solely on the basis of mathematics.

This is more than a mere matter of terminology (substituting "validity" as if it were a synonym of "truth"). Mathematicians knew they could not have their cake and eat it. In order for the indirect proof to work at all, you must first accept that the only thing that is significant mathematically about any set of axioms is the set of valid theorems they imply. You cannot impose any additional constraint such as that when you break out your straight-edge and compass and start drawing out the steps of a construction you will get the shapes and angles the axioms dictate. If the latter kind of empirical verification is retained from Euclidean Geometry, then the Non-Euclidean Geometries fail before the check for validity even becomes relevant. The term meta-mathematics was coined to refer to what now had to be regarded as special cases where a given mathematical theory could be shown to correspond in some way to empirical reality. Euclid's constructions provide the meta-mathematical theory that connects it to common sense experience that is not available to Non-Euclidean Geometries.

Formal Linguistics, Set Theory and Infinite Quantities

The change in outlook from truth to validity opened the way to vast new realms of mathematical inquiry. If mathematicians could no longer rely on counting their fingers or drawing constructions to prove the correctness of their formulas, they needed a new set of robust tools to replace empiricism. Suddenly, the nature of "validity," itself, became a subject of study along with an examination of the nature of axiomatic systems and the formal languages used to define and apply them when carrying out mathematical proofs. The detachment from empiricism also gave permission for mathematicians to consider subjects that had formerly been forbidden because they referred to mathematical objects and states of affairs that could not exist in the real world.

For example, right up to the middle of the 19th Century it was considered "out of bounds" for mathematicians to posit the existence of "completed infinities" in their calculations. After all, you can never hold an infinite number of pebbles in your hand. When Leibniz and Newton separately invented Calculus in the 17th Century, they each had to perform some mental gymnastics that allowed them to treat "infinitessimal" quantities as mathematically sound while still shying away from the mathematical inverses of infinitessimals, i.e. infinite quatities. As a foreshadowing of how the world of science and mathematics would react to Non-Euclidean Geometry a couple of centuries later, Leibniz' and Newton's contemporaries found Calculus too compelling and useful to ignore, while consciously repressing the cognitive dissonance required to embrace infinitessimals. That repression finally ended when the generation of mathematicians following the invention of Non-Euclidean Geometries fully embraced the consequences: go ahead and construct proofs based on infinite quantities or any other formerly forbidden category of mathematical object. So long as your results are internally consistent then your mathematics are valid. There may or may not be a meta-mathematical theory mapping your results to properties of the real world. If so, that may be a serendipitous side-effect, of potential use to scientists and engineers (among whom mathematicians could no longer count themselves), but such a corresondence to reality is no concern of "pure" math in this new way of looking at things.

Simultaneously, the new emphasis on validity begged the question: what exactly is "validity," in the first place? Previously, it had been considered both necessary and sufficient to demonstrate the correctness of a mathematical theory to show how it described properties of real things. Even abstrations like negative numbers were justified by treating them the way that bookkeepers treat balance sheets. Oddities like what happens when you try to divide any number by zero were regarded as special cases needing no further explanation or inquiry than, "that can't happen in the real world, so just skip over it." But once approaches were accepted like those which used Non-Euclidean Geometries to prove something about classical Geometry, such empiricism was no longer necessary nor sufficient. The nature of mathematics and mathematical formulas in and of themselves suddenly became a topic of great interest to mathematicians. Thus were Symbolic Logic, Set Theory and Formal Linguistics born as areas of intense mathematical interest.

Symbolic Logic is a mathematical system for determining whether or not a set of statements are mutually consistent, i.e. form a valid argument. Set Theory is an extension of Symbolic Logic that considers the mathematical properties of collections of things. Formal Linguistics is the study of the properties of the sets of symbols and rules for combining them used to express mathematical formulas (or, later, computer programs) in meaningful and useful ways.

Cantor used this new freedom and these novel approaches to consider the following question: is there more than one infinite quantity? If Cantor had been born even a generation earlier, he would have been laughed out of his doctoral program for even posing such a question. Infinite quantities were "right out" (to quote Monty Python) as mathematical objects in the first place, let alone the question of what properties they might have. But by Cantor's day, the world of academic mathematics was ready to give almost anything a go.

The school-yard understanding of "infinity" is that once you reach it, that's all there is. If a grade-schooler says to her best friend, "I love you," the friend may reply, "I love you more!" The natural response is, "I love you twice as much!" Which elicits, "I love you times 10!" Then "...times 100!" Eventually, one of the amicable combatants will end the game with, "...times infinity," to which, on the school yard at least, there is no retort since (as every child understands) once you reach "infinity" you cannot count any higher. Cantor understood this school-yard intuition as meaning that what most people think of as \(\infty\) corresponds to the cardinality of the set of natural numbers. As a Set Theoretician would say, when you count up indefinitely from 1 (or 0, depending on your tastes and the context of any given discussion), you are defining the set of natural numbers by intention and the cardinality of that set is \(\infty\).

To understand Set Theory's terminology like "cardinality," "intension vs. extension" and so on, consider the letters of the Latin alphabet as a set of symbols. There are twenty-six of them as used in English, so the cardinality of this set is 26. "Cardinality" can be loosely understood as "the number of members of a given set." However, "cardinality" is used to allow the notion to be extended to sets with infinitely many members. Historically, people were hesitant to refer to infinite quantities as "numbers," so a different term was invented for "number of members" when that number's magnitude was potentially infinite.

There is a lot more underlying these distinctions of terminology related to the original motivation for creating Set Theory in relation to ideas regarding the nature of numbers and of mathematics, itself. Mathemeticians use terms like "cardinal numbers," "cardinality" when talking about magnitudes, i.e. quantities. They use terms like "ordinal numbers," "ordinality" when talking about the relationships (ordering) among such magnitudes. Sets composed of ordered magnitudes can be shown to have specific properties of great use in Set Theory. At a deeper level, separating the concept of "cardinality" from the concept of "number" allowed early Set Theoreticians to use the concept of "cardinality of a set" to define the concept of "number" in purely Set Theoretical terms. But I digress....

Sets are defined by enclosing their contents in curly braces \(\left(\{ \dots \}\right)\) in Set Theory. For a set defined by extension, what is inside the curly braces is an explicit list. For example the set of "letters of the alphabet" as used in English is

\[ \{A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z\} \]

as demonstrated by the "...now I've learned my A, B, C's..." chant every American grade-schooler is taught. What makes the preceding collection of letters enclosed in curly braces "the set of letters of the alphabet" is simply the assertion that the explicitly given list is the given set. This is what it means to define a set by extension. Only finite sets can be defined by extension, because only a finite number of items can be listed explicitly.

But what about a set like that of all the natural numbers? Prior to Cantor's day, the answer would have been, "there is no such thing as the set of natural numbers because that would represent a completed infinity." But the new interest in things like Set Theory and the new freedom to explore valid theories whether or not they could be empirically verified allows one to define "the set of natural numbers" by intension. That is to define rules by which anything in the universe can be tested to determine whether or not it belongs in the set. So the set of natural numbers can be defined by intension using a rule.

Definition of the set of natural numbers by intention:

Anything is a natural number if and only if it is the number 0 or is the result of adding 1 to a natural number.

Using the formal language of Set Theory, the preceding definition becomes:

\[ \mathbb{N} = \{ x | (x = 0) \lor \exists y ((y \in \mathbb{N}) \land (x = y + 1)) \} \]

The preceding formula of Set Theory would be read in English as "the set of all \(x\)'s such that \(x\) equals zero or there is a \(y\) where \(y\) is an element of \(N\) and \(x\) equals \(y\) plus one."

  • \(|\) - "such that" when defining a set by intenstion
  • \(\lor\) - disjunction, i.e. "inclusive or"
  • \(\land\) - conjunction, i.e. "and"
  • \(\exists\) - existential generalization, i.e. "there is" or "there exists"
  • \(\in\) - set membership, i.e. "is a member of" or "in"

By the preceding rule, 1 is in the set of natural numbers because it is the result of \(0 + 1\). Given that, so is 2, because it is the result of \(1 + 1\). The number 1.5 is not a natural number because there is no way to produce it by adding 1 to any natural number. The letter "A" is not a natural number because the arithmetic addition operation does not even apply to it. And so on for everything in the universe, including abstractions like numbers.

Note that the preceding rule is self-referential in that it refers to the set \(\mathbb{N}\) when defining \(\mathbb{N}\) by intension. Such self-reference is characteristic of the definitions of infinite sets, but must be used with care. Self-referential definitions can easily slip into circular arguments when abused. They can also lead to problems like the "liar paradox." The sentence, "I am lying," is a well-formed utterance in English. It consists of meaningful English words, correctly arranged according the rules of English syntax and grammar. But it cannot be said to be either true or false. If it were true, then I would be lying. But since what I am saying is that I am lying, to be true it must be false and vice versa. It cannot be both true and false, so it must be neither. Later, we will see how such paradoxical self-reference came to be used to prove important things about the nature of sets of numbers and, by Gödel, the nature of formal languages by treating them as sets of numbers.

This allows "the set of natural numbers" to be well defined, but what about its cardinality? Cantor begins his exploration of infinite quantity simply by stipulating that the set of natural numbers when defined by intention has a cardinality, which he called (\(\aleph_{0}\)) (pronounced "aleph null"):

\[ \aleph_{0} = |\mathbb{N}| \]

where the "magnitude" bars surrounding the name of a set refers to its cardinality, i.e. \(|\mathbb{N}|\) simply means the "the cardinality of the set \(\mathbb{N}\)."

It is not a coincidence that Set Theory uses the same notation for "cardinality" as that used for "magnitude" in Number Theory. One of the original motivations for defining Set Theory in the first place was as part of an attempt to "derive" Arithmetic, in some sense, from pure Logic. Sets with their cardinalities became the "proxies" for numbers with their magnitudes in that context. (And that is why a different term, "cardinality," for "number of members" was introduced in the first place, as well; i.e. to avoid circularity when using the concept of cardinality when providing a definition of the concept of number.) That whole enterprise failed, but in a far more interesting way than if it had succeeded. What was actually shown was not that Logic, Arithmetic, Algebra, Geometry, Calculus etc. existed in a hierarchy where each level in the hierarchy was somehow derivable from the next level down. (But that is where the term "higher math" comes from, which you will still hear non-mathematicians use to this day.) What was shown is that any formal language with certain properties can be used to "model" the axioms of any other such language. I.e. Set Theory is just as "derivable" from Arithmetic as Arithmetic is from Set Theory — but it is not really "derivation" at all but, rather, "representation."

Since there is no limit to the number of times you can add 1 to a natural number, \(|\mathbb{N}|\) represents an infinite quantity. But is \(|\mathbb{N}|\) "all there is" when it comes to infinite quantities, as in the school-yard intuition regarding \(\infty\)?

Cantor's answer, as it turns out, is, "no, there are infinitely many infinite quantities which differ in magnitude in the same way that the magnitudes of finite numbers differ." To prove this, we need to intoduce some additional jargon: mapping and countability. A mapping is a rule by which elements in one set are associated with elements in another set. For finite sets defined by extension, such mappings can be defined by extension as well. A finite mapping can be represented as a set of ordered pairs where the first member of a pair is from one set and the second member of a pair is from the other set.

As an example, consider two sets, \(n\) and \(l\) where

\[ \begin{align*} \text{Let } n & = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 23, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26 \} \\ \text{and } l & = \{ A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z \} \end{align*} \]

Obviously, \(n\) is the set of natural numbers from 1 through 26 while \(l\) is the set of the 26 letters of the Latin alphabet as used in English. An obvious mapping from \(n\) to \(l\) would be:

\[ \begin{align*} \text{Let } \alpha = \{ & <1, A>, <2, B>, <3, C>, <4, D>, <5, E>, <6, F>, <7, G>, \\ & <8, H>, <9, I>, <10, J>, <11, K>, <12, L>, <13, M>, <14, N>, \\ & <15, O>, <16, P>, <17, Q>, <18, R>, <19, S>, <20, T>, <21, U>, \\ & <22, V>, <23, W>, <24, X>, <25, Y>, <26, Z> \} \end{align*} \]

This is simply associating each letter with its ordering when reciting the "A, B, C" song, which is also the conventional sorting order when alphabetizing entries in a telephone directory, a book's index or the like. This kind of pairing-off between elements of sets is called a "mapping." The set from which the first member of each pair comes is called the "domain" of the mapping. The set from which the second member of each pair comes is called the "co-domain" or "range." Another name for a mapping is a "function."

Mappings are categorized by how they associate members of the domain and range. Any "pairing off" between elements of two sets is a mapping, no matter how incomplete or haphazard:

\[ \text{Let } \beta = \{ <3, A>, <17, B>, <17, Q>, <25, B> \} \]

The set \(\beta\) only associates a few elements of \(n\) with even fewer elements of \(l\), since it associates both the numbers 17 and 25 with the letter B as well as associating 17 with Q. Yes, \(\beta\) is a mapping, but a very arbitrary and incomplete one that does not correspond to anything meaningful or useful.

The set \(\alpha\) is different. Note that every element from the domain is included exactly once. That is what is called an injective or "into" mapping. In addition, every element of the range is included exactly once, making it a surjective or "onto" mapping. Since it is both injective and surjective, it is a bijective or "one-to-one" mapping. Mappings that are injective or surjective have special properties that can make them useful in making claims about the nature of relationships between sets. Bijective sets have the most interesting and powerful relationships. A bijective mapping represents some function or computation that, given any element of either set, allows you to choose or compute exactly the member of the other set to which it corresponds. In particular, two sets must have exactly the same cardinality in order for there to be a bijective mapping because every element of both sets is included in the mapping exactly once.

As defined, \(\alpha\) is bijective because \(n\) only includes the subset of natural numbers from 1 through 26 which gives it the same cardinality as \(l\). But what if we replace \(n\) with \(\mathbb{N}\) in the definition of \(\alpha\)? In that case, \(\alpha\) remains surjective but is no longer injective, since only a subset of the domain, \(\mathbb{N}\), is included in the mapping. Since it is no longer surjective, it can no longer be bijective. Any set for which there is a surjective mapping from \(\mathbb{N}\) is said to be countable. Any set for which there is a bijective mapping with \(\mathbb{N}\) is said to be countably infinite. Reasoning by analogy with the original, bijective, definition of \(\alpha\), any countably infinite set has, by definition, the same cardinality as \(\mathbb{N}\). I.e.

\[ \begin{align*} \forall s (I(s) \implies |s| = |\mathbb{N}|) \\ \text{where } I(s) = s \text{ is countably infinite} \end{align*} \]

or, "for every \(s\), if \(s\) is countably infinite then the cardinality of \(s\) is the same as the cardinality of \(\mathbb{N}\)."

Countability, infinite or otherwise, has other interesting implications. For example, any set whose cardinality is less than or equal to \(|\mathbb{N}\|) and for which there is a "natural ordering" among its members is countable since you can arrange the members of that set in their natural order and then map the first element to 0, the second element to 1 and so on.

A "naturally ordered" set is simply one for which its elements have some kind of intrinsic sorting order. Sets of numbers are ordered by the "less than" operation. If one accepts the "A, B, C" song as canonical, English words can be sorted alphabetically, and so on.

For Cantor, the test of the school-yard intuition that once you reach \(\infty\), that is as big as a quantity can be thus became: is the cardinality of every infinite set equal to \(|\mathbb{N}|\) (which Cantor referred to as \(\aleph_0\))? For sets of numbers, which are ordered, this is equivalent to asking if every such set is countably infinite. His answer was that while some infinite sets are countable, others are not. For example, the sets of integers and of rational numbers are infinitely countable, even though the former includes negative numbers and the latter consists of pairs of natural numbers and so each might have been assumed to have a larger cardinality than \(|\mathbb{N}|\). So far, so good for naive intuition regarding \(\infty\)!

Cantor's approach hinged on identifying sets that were infinite and naturally ordered but not countable. To show that the set of real numbers, \(\mathbb{R}\), is such a set, start by assuming that you can define a countable set of real numbers between any two real numbers in the continuum (points on the number line). It does not matter what rule you use to construct this list, since the crux of Cantor's argument is to show that any such rule cannot represent all the points of the real number line between the chosen end points. As for any countable set, we can look at it as a mapping from natural numbers to elements of our set. Real numbers are represented as potentially infinite sequences of digits, as in:

\[ \begin{align*} <0, \ &0.427 \dots> \\ <1, \ &0.613 \dots> \\ <2, \ &0.590 \dots> \\ \vdots & \end{align*} \]

Cantor's argument for showing that any such list cannot include all the real numbers within the selected range proceeds by describing a rule for constructing a new sequence of digits representing a real number that cannot already be in our list. In particular, he presents a simple rule for adding each successive digit to the new real number based on the sequences of digits representing the real numbers that are already included. Specifically, the rule is that the \(n \text{th}\) digit in the new sequence must differ from the \(n \text{th}\) digit in the \(n \text{th}\) entry in the original list.

\[ \begin{align*} <0, \ &0.\mathbf{4}27 \dots> \\ <1, \ &0.6\mathbf{1}3 \dots> \\ <2, \ &0.59\mathbf{0} \dots> \\ \vdots & \\ \hline <r, \ &0.\mathbf{502 \dots}> \end{align*} \]

In the preceding example, the first digit in our new sequence, \(r\), is 5. The only signfigance to 5 here is that it is different from 4, the first digit in the first entry in the original list. The second digit is 0, which differs from the second digit, 1, in the second entry. The third digit is 2, which differs from the third digit, 0, in the third row. And so on. The new sequence of digits is guaranteed to differ in at least one decimal place from every sequence in the original list, so it cannot already be included. If we add \(r\) to our list, we can repeat the procedure to construct yet another new sequence, ad infinitum. This means that there are infinitely more real numbers between any two given ones than can be bijectively mapped to the natural numbers. \(|\mathbb{R}|\) is infinitely larger than \(|\mathbb{N}|\). So much for naive intuition regarding \(\infty\).

Note some interesting aspects to Cantor's argument. It assumes that the continuum, i.e. the "real number line," is infinitely subdivisable, i.e. there are an infinite number of real numbers between any two real numbers. It also assumes that any sequence of digit symbols corresponds to a real number and that any two real numbers with distinct magnitudes will be represented by distinct sequences of digits. We know that some real numbers, like \(\pi\) would require an infinitely long sequence of digits to represent precisely. So Cantor's argument turns on being able to reason sensibly about infinite sets of infinitely long sequences of digits, none of which could actually be produced in the real world in a finite amount of time. Some people say that "Cantor proved that distinct infinite quantities exist." What he actually claimed is that "if infinite quantities are assumed to exist at all, they must have certain characteristics including that there are more than one of them" which is not at all the same thing. An empricist does not have to disagree with the validity of Cantor's argument to still object to the literal reality of "completed infinities." This distinction is important to keep in mind when talking about meta-mathematical theories. When constructing mathematical models of reality, there is always the danger of mistaking features of the model with features of reality. In fact, one could as easily interpret Cantor's argument as a reductio ad absurdum argument for why the "set of real numbers" cannot actually be said to exist.

Let me explain.

The Computability Problem

As stated above, the definition of a countably infinite set is that there is a bijective (one-to-one) function that maps natural numbers to elements of that set. The accepted understanding of Cantor's arguments is that the set of real numbers is not countable because any mapping from the natural numbers is at best injective and can never be surjective. That means that there are more real numbers, in some sense, than natural numbers. Infinitely more. In fact, a consequence of Cantor's argument is that the percentage of real numbers for which an injective mapping function is possible is so small compared to the real numbers denoted by the continuum (all the points on the number line), the ones that do correspond to a countable set must be very special in some way. And so a new branch of mathematics was born, Computability Theory.

To fully understand the motivation for Computability Theory, you first must know something about Formal Linguistics. As already noted, Formal Linguistics is the study of the kinds of artifical languages used to express mathematical formulas and computer programs. Among the most famous results in Formal Linguistics are Gödel's "incompleteness theorems." One of the things Gödel proved along the way is that the number of well-formed formulas for any language of the kind used for mathematical purposes is countable. I.e. given a finite set of symbols like numeric digits, the decimal point, operators (e.g. \(+, \ -, \ \times, \ \div \)) and the rules for combining them into useful formulas of arithmetic, you will only ever be able to produce countably many well-formed formulas in that formal language.

This means that one can re-frame the understanding that "the set of real numbers is not countable" as "most real numbers are not computable" — i.e. there are no mathematical formulas for computing their values. This is because there are only countably many formulas but the set of all real numbers is not countable so there are uncountably many real numbers for which there is no formula on the basis of which they could be computed. Computability Theory attempts to answer the question, "what makes a real number computable?"

Computability Theory to the Rescue?

But how does one go about characterizing what makes a number computable? (Leaving aside, for now, the question of how "real" a number can be if it cannot be computed, i.e. identified or characterized in any way.) Starting in the late 19th Century, various philosophers of mathematics and philosophically-inclined mathematicians worked on this question. The question was considered settled when the work of two such people, Church and Turing, converged during the first half of the 20th Century (Church was Turing's Doctoral advisor).

  • Church's approach was purely formal. He invented a mathematical language, the Lambda Calculus, for defining any possible calculation. Church's thesis was that the set of computable numbers is the set of results of evaluating all well-formed formulas of the Lambda Calculus. This way of characterizing computability is very much in keeping with what had by then become the mainstream "all that matters is validity" view of mathematics, which encourages such formalism.

  • Turing's approach starts with thought experiments involving hypothetical "Turing Machines." (Turing called these "a-machines" in his dissertation and they are now recognized as being a particular type of "finite state automaton." It was Church who first referred to them as "Turing Machines.") In this way, Turing's work actually aligns well with the classical, "empiricist" view of mathematics, at least conceptually. A Turing Machine is a theoretical mechanical device built according to certain rules and which produces a particular sequence of symbols (e.g. digits in the representation of a real number) as the result of its operation. If Turing Machines could be constructed (which they could not be in reality because each would require an infinitely long paper tape on which to print its output), Turing's thesis is that the outputs of all possible such machines would be the set of computable numbers.

Computability Theory was considered settled and the world moved on to other interests when it was shown that Church's and Turing's theses are mathematically equivalent: given the rules governing the behavior of any possible Turing Machine, one can define a formula of the Lambda Calculus that will carry out the equivalent calculation and produce the same result. Any formula of the Lambda Calculus can be used as the "blueprint" for "building" a Turing Machine. Since two intuitively compelling definitions of "computability" were shown to produce the same results (by way of the now-familiar reasoning based bijective mappings between infinite sets), the consensus among those interested in this question was that the "problem" was solved, and there was, once again, much rejoicing.

As an unintended side-effect, Turing and Church had between them provided the mathematical model and mental framework for programmable digital computers and for the programming languages used to control them. But before we delve further into the ongoing legacy of their work, note something interesting: neither Church's nor Turing's work actually directly addresses what makes computable numbers different from all those other supposedly-real-but-not-computable ones. Instead, they provide a means for characterizing an exhaustive set of computable numbers while saying nothing at all about non-computable ones, other than by implicitly excluding them. In fact, rather than explaining Cantor's results, Computability Theory when considered on its own seems to imply that the set of real numbers actually is countably infinite using some ordering of the formulas of the Lambda Calculus to drive the bijective function. That such an ordering is possible is another consequence of Gödel's work. Generations of mathematicians and philosophers have accepted that Cantor's argument is valid, but the very success of Computability Theory begs the question: how "real" is a number that cannot be computed?

Taken strictly in its own terms, Cantor's argument is air tight. But from the historically more traditional, empiricist view of mathematics it is deeply circular. Cantor begins by stipulating that concepts like the "set of natural numbers" and "set of real numbers" make sense and that such sets have cardinalities. Only on that basis can you even pose the question as to whether and how their cardinalities differ. Even accepting those premises, you can never literally carry out his procedure for showing that \(|\mathbb{R}| > |\mathbb{N}|\) since you would first have to produce a "completed infinity" of representations of real numbers, each of which was a "completed infinity" of digits and finally produce yet another "completed infinity" of digits using the "slash" procedure. So, while Cantor's results and all the results that followed from it are logically valid, the only meta-mathematical realities to which they lead are represented by the countably many real numbers described by Computability Theory. Consider the opposite interpretation, for a moment. Viewing Cantor's argument as showing not that the cardinality of the continuum is greater than that of the set of natural numbers but, rather, that the continuum does not have a cardinality in any meaningful sense helps address quite a number of problems, including the so-called "tail problem" of Quantum Physics. More fundamentally, the continuum cannot be real in the same sense as physical reality since there is no such thing as an infinitely subdivisible physical substance. Subdivide any real substance into smaller segments enough and you reach a point at which you must start pulling molecules apart, then atoms, subatomic particles, quarks and eventually, if some popular theories hold, you reach the level of the "strings" of "string theory" which could then not be subdivided any further. Long before then, of course, the structural integrity of whatever object you began with would have broken down so far that the pieces into which it had been split would no longer be meaningfully understood to be smaller segments of a formerly single object made of a single substance. E.g. cut an actual macroscopic string in half, then each of the halves in half and so on. Long before you reach the level of molecules the fibers out of which the string was made would stop being at all string-like and so you would no longer be cutting strings in half as you continued to "subdivide a piece of string". At the sub-microscopic scale, the whole point of Quantum Physics is that physical matter is an agglomeration of discrete constituents, each of which can be observed to be in a specific, computable state when observed under the correct conditions. The "real number line" with its infinitely large number of points between any two points can only be understood as a mathematical object with no direct meta-mathematical correspondence to anything in the real world. Unless almost everything contemporary Physicists believe is wrong, the number of possible states of all the subatomic particles in the real world is not even infinite, much less uncountably so even as they use mathematics involving the continuum to reason about it. As already pointed out, the risk is in mistaking features of a mathematical model for features of the thing being modeled by some meta-mathematical theory. That there are these sorts of tensions between "pure" and "applied" mathematics became inevitable once mathematicians embraced validity over truth. What is remarkable, despite that tension, is how often what begins as an exploration of a pure mathematical abstraction, e.g. Computability Theory, yields results with powerful and even world-changing applications, e.g. Computer Science. Similar examples can be found in other scientific domains. Non-Euclidean Geometries have found practical (for some definition of "practical") use in various branches of Physics rather than remaining solely the province of abstract considerations of Euclid's axioms.

Programmers use the term "lambda" to refer to an anonymous function because of Church's Lambda Calculus and, more importantly, the Lambda Calculus is the original model for all so-called "Turing complete" programming languages. Turing Machines were the original model for the Von Neumann Architecture on which all current electronic programmable computing devices are based. Conversely, Cantor's non-computable real numbers can only ever remain the stuff of abstract considerations of the theoretical properties of an infinitely subdivisable number line. But the same embrace of infinite quantities Cantor pioneered has been useful in both abstract and applied math in quite concrete ways. Division by zero, which once was regarded as an anomalous discontinuity, now is understood to have a definite result: \(\pm\infty\):

\[ \lim_{x \to \infty}{1 \over x} = 0 \iff \lim_{x \to 0}{1 \over x} = \infty \]

The preceding is baked into the IEEE 754 standard for floating-point arithmetic, along with the special NaN value for remaining cases of undefined operations such as \(0 \over 0\) or attempting to calculate the square root of a negative number.

\[ \frac x {\pm 0} = \begin{cases} \mathtt{NaN} & \text{where } x = \pm 0 \\ \pm \mathtt{INFINITY} & \text{otherwise} \end{cases} \]

where NaN is the IEEE 754 constant meaning "Not a Number" and INFINITY is the IEEE 754 constant meaning \(\infty\). Also, 0 and INFINITY are signed in the IEEE 754 specification and follow the usual rules for sign agreement in division.