Just because mathematics is logical at its root does not ensure that all possible terminations of its logical reasoning chains will fall within the bounds of typical human comprehension. When mathematics reaches beyond intuition, it seems to operate in the realm of religion.
Historically, concepts with no seeming real basis such as negative integers or imaginary numbers, or even the number zero, have subsequently become instinctive and necessary to even our simplest mathematical constructs. No special axiomatic extensions are required to accept them into our logic. Our intuition, now trained, admits them as well.
Currently challenging intuition (at least mine) are the mathematical concepts of infinite set, the continuum, and infinity (a number larger than all numbers). The words infinite, continuum, and infinity are language metaphors for an existential beyond, for states more distant/large or more closely connected than reality can provide or mind can grasp.
In the mathematical world, these concepts have utility and hence demand and find mathematical definition. These definitions, however, require special axioms to enable their admissibility into our mathematical realm. And these axioms are admitted to mathematics by necessity alone, since our intuition does not extend to the infinite or infinitessimal. Hopefully, mathematics will some day offer typical human minds the promised x-ray specs to access the infinite via intuition.
What Do We Mean By ‘Set’? By Infinite Set?
The infinite enters mathematics as an extension of the concept of finite set. Here set is a mathematical concept, referring to a collection of mathematically-defined objects: numbers, curves, points, other sets, etc.
One should best exemplify a set with purely mathematically-defined elements. Sets of cabbages or kings would be bad form. While perhaps the gaffe would only rise to the order of a mixed metaphor when referring to a finite set, it becomes gibberish when an infinite set is being exemplified.
A finite set is countable in the sense that there is a bijection from the set to some subset of the natural numbers. All finite sets are countable, and hence share the well-ordering of their image under the bijection above. Their size, the count of the number of elements, is defined as an ordinal number, ordinal implying the well-ordering of its image within the natural numbers.
There is also the notion of cardinal set size, which applies to any set, whether ordered or not. For finite sets, ordinal and cardinal share a common semantic and arithmetic, since all finite sets are well-orderable.
Below we will see that for infinite sets, ordinal and cardinal have very different semantics and arithmetic.
Does Infinity Exist?
The natural numbers are naively defined by an algorithm: begin with 1; then just keep counting (adding 1) to get all the successors. In formal set theory, there is defined a successor function that produces the set of all the natural numbers in the form of their set-theoretical definition. This is our smallest inductive set. The definition and construction method are highly intuitive.
Yet although the construction method is explicit, the axioms of set theory cannot prove that such a completed set actually exists without an added Axiom of Infinity, which declares that there exists the infinite set of natural numbers. Thus we are on shaky ground, because the other axioms seem intuitive to us based on experience of our reality. This axiom, however, expresses a hypothesis beyond our intuition.
We understand an infinite set mathematically as an abstraction, comprehensible only via its mathematical definition. Infinite sets are metaphysical abstractions with no real counterparts; they are not an ideal realization of some characteristic form in our reality, nor in any reality we could intuit.
Let’s quote Gauss, whose intuition matches mine:
I object above all the use of an infinite magnitude as if it were complete, which is never permitted in mathematics. The infinite is only a façon de parler, when we are properly speaking about limits that certain relations approach as much as one wishes, while others are allowed to increase without limits.
Gauss was not speaking set-theoretically, and many mathematicians since have disagreed that Gauss was stating a general mathematical principle. Yet my intuition persuades that there is only one ‘kind’ of infinity, which one may call mathematical infinity. My limited intuition admits no real world counterpart.
It is a convenience in mathematics to define such a concept, which definition enhances the usability of our mathematics. For example, whether speaking set-theoretically of the concept of a countably infinite set, or speaking topologically of the point at infinity of the complex plane, the structure of mathematics is logically completed by such defined constructs. And there is no price to pay, for our intuition of set and topology readily admit these concepts, if not also the axiom required to prove their mathematical existence.
What Do We Mean By ‘Infinite’ and ‘Infinity’?
The word boundless means a limitless supply. You can have as much/many as you need of something, and you can always go back for more. Note boundless has only metaphorical real world meaning. We have no need for a truly boundless quantity, nor could we fathom a real world construction of same.
In mathematics, however, boundless is assigned meaning via a transfinite (infinite) set (with a little help from the Axioms of Infinity and of Choice). Cantor defined use of transfinite rather than infinite in such a context, although the terms are now interchangeable.
We must give up some of our finitary sense that equinumerosity is somehow representable by a number, such as the maximal element; rather transfinite size is now expressed as the order type of the set. As we will see, there are various transfinite order types; and not all transfinite sets are equinumerous, but many will prove to be.
Why do we (and Gauss) speak of a single in common discourse; we reserve the concept of this common infinity, , for specifying a value of a real number larger than any other real number on the unbounded real number line (aka the ‘point at infinity’). This point bounds the largest well-ordered transfinite set, as well as all its ordered subsets. We, and mathematics, will it into existence.
The set of natural numbers is the basic mathematical instance of a transfinite set; it is axiomatized as the inductive set. Omega, (), is defined as the order type of the inductive set. The inductive set is also characterized by the cardinality identified by . So how can we think of this? As in all things beyond our intuition, we need to let the logic of mathematics guide us on our path forward.
Beyond ‘real’ , mathematics provides notions of transfinite sets, via two parallel definitions that extend the concepts of cardinal and ordinal to transfinite sets. Special infinity symbols represent these various orders of numerosity.
An ordinal represents an equivalence class of well-ordered sets between which there is an order-preserving bijection. The ordinal represents the numerosity of elements in any member of the equivalence class. is the smallest transfinite ordinal, the order type of the natural numbers.
is like no finite ordinal, in that it is a metaphorical number even in its mathematical sense; it is not a number at all, but the order type of the natural numbers. It represents the size of a set, not by counting, but by size order. is the size of all sets for which there is a bijection onto the inductive set. Such a set is called countably infinite.
The arithmetic of transfinite ordinals is unusual. Addition is not commutative. Thus , but . Further, the cardinality of sets with order type . Transfinite ordinal arithmetic is currently beyond the scope of this discussion.
Each cardinal symbol represents an equivalence class of sets between which there is a bijection (one-to-one correspondence). It represents the numerosity of any member of the equivalence class.
Although we will not be in counting land any more, we can still maintain a concept of equinumerosity via definable 1-1 mappings. Mathematics posits an ordered list of infinite set numerosities (order types) that are represented by transfinite cardinals and denoted by the aleph symbols.
Any set that exhibits a 1-1 correspondence with the inductive set has cardinality . Such sets are said to be countable. Examples are the countable number sets , , , (natural numbers, integers, rationals, algebraic numbers; each an superset of the preceding set).
Aleph-null () is the first transfinite cardinal. is thought to be the second transfinite cardinal (assuming CH is true). The cardinal numbers then form a sequence: . Thus each cardinal in sequence represents the power set of the prior cardinal. But here’s the rub: , but they all compare ordinally to 'real' .
The cardinal numbers are actually ordinals, because the set of cardinals form a well-ordered set. But this transfinite land is strange:
- The only transfinite set that can be demonstrated (within ZF) to be larger than is
The cardinal symbols have a defined arithmetic (e.g. a + b = max(a,b)), and so exhibit a somewhat number-like behavior. But it is clear we are no longer in counting land when considering the countable number sets above. Each is a superset of the prior set, yet they all have the same transfinite cardinality. In fact, all countable unions of countable sets are countable (equinumerous with the inductive set).
My comprehension struggles with any flavor of the infinite; concerning transfinite cardinality, even mathematics struggles, but it can carry on by isolating itself from the worst pathologies.
What Do We Mean By ‘Continuum’?
The real number line is called the continuum. There are no end points, but the value beyond all the points is called the point at infinity. These ‘beyond’ values are not included in , but there are extensions of that include either +, –, or both.
The set of reals has greater cardinality than the rationals (there is no 1-1 mapping of the reals to the rationals). How much greater is the question that currently eludes mathematics’ logical ability. Since mathematics today cannot say where the cardinality of the real numbers falls compared to the sequence of transfinite cardinals defined above, mathematics uses a different symbol for it, often or (for continuum). The continuum hypothesis states that .
To get at the concept of a real number, we need the concept of infinite sequence and of limit, the basic tools of real analysis.
What Do We Mean By ‘Limit’?
The common meaning for limit is a boundary. In one quantitative sense, a limit is a stopping point or end point for a process. In mathematics, a limit is a numerical value that a function or sequence approaches as its argument or index respectively increases.
We can intuit the two concepts of infinite and infinitessimal as corresponding to two cases of a degenerate triangle process, in which we flatten a triangle incrementally until, at the limit of the process, the triangle becomes just the line that forms its base. Since we still want to call our figure a triangle, we insist that the one dimensional degenerate state never quite be reached.
Begin with a plane triangle of area A, height of length H, and base of length B. Begin reducing the value of H and consider two cases. Hold A fixed and B will grow in an unbounded fashion. Hold B fixed, and A must diminish to zero. By the former case, we mean that B->∞ as H->0, or the limit of B is ∞ at H=0. By the latter, we mean that A->0 as H->0, or the limit of A is 0 at H=0.
The unbounded limit in case one is called infinity. The process in case two, of two quantities A and 0 getting endlessly closer together, results at its limit in the concept of a continuum, in which case numbers will get increasingly close. We have neither concept nor intuition regarding an endless number of things, nor of what it would mean for quantities to have no gaps, yet remain distinct. In what sense do distinct numbers touch? Here, we just have to believe in the math. For example, uncountably many reals are sandwiched between any two distinct rationals, no matter how close together the rationals become.
Does A Continuum Exist?
The irrational numbers are abstractions that exist only in our minds, conceived as the result of a limit process. We intuit with some difficulty a number resulting from such a process. While the natural numbers and the rationals are sensed as somehow atomic (our intuition about countability), an uncountable continuum has no such natural counterpart. We can describe how to construct an irrational number to any desired exactitude (as an unique infinite continued fraction whose successive rational convergents form a Cauchy sequence), although we intuit that at most, countably many such could ever be constructed.
We have algorithmic representation, in the form of polynomials with integer coefficients, for the irrational roots of polynomials. These irrationals thus are constructed from the integers via standard arithmetical operations. All irrationals which are not polynomial roots are called transcendental. We have symbols for the most common and useful transcendentals, defined typically as ratios, such as the ratio of the circumference to the diameter of a circle (π), or the ratio of the diagonal of a square to its side (√2). But even these most useful of irrationals do not exist in the real world. It is the symbols we assign to these ‘atoms from the continuum’ that allows our intuition to accept them as numbers. Being able to uniquely specify an irrational number to any exactitude does not yield utility. The mathematical utility of an irrational number derives from its assigned semantic within our abstract construction. Only algorithmic/symbolic real numbers have such utility.
A real number is an entity magically plucked from the continuum (Platonically an idealized curve). Each unique real number has no construction other than as a process with an infinite number of steps, for example a unique continued fraction whose rational convergents form a Cauchy sequence converging to the desired quantity.
Our intuition and our reality prevents us from ever realizing a distinct real number from an infinite process. We can know as much about any unique real number as we will ever need to know, but we can never know it completely. Further, only countably many could ever be realized and named, and of these, only the ones for which we have algorithmic evidence (geometrically or algebraically defined) will ever find utility in our construction.