A Realistic Division by Zero
Abridged draft of my 2005 (Metaphysica 6, 91-109).
1.
Introduction
References
The following (which is a bit simpler than my 2005 because it omits various details) is an informal look at the algebraic coherence of a neglected hypothesis about continuity—that the cardinality of physical continua, were there any, would equal the (arbitrary unit) length of a line of points divided by the (zero) length of an individual point.
Clearly, were physical continua (e.g. space-time) to exist there would be an objective fact of the matter about the truth of our hypotheses about continuity, which might eventually become scientifically important. One hypothesis that has shaped mathematics (and thence logic, metaphysics and physics) to a great extent is that the (primitive) geometrical line and the (standard) real number line are isomorphic, which I will call C-D (since it is due to Cantor and Dedekind). C-D is assumed by most scientists nowadays, but nonetheless we may one day benefit from our having available the widest possible range of alternative hypotheses, if only to properly justify our confidence in C-D. So in what follows I take an informal (pre-theoretic) look at the neglected hypothesis mentioned above, which develops (in §6 below) into two novel possibilities for the structure of physical continua, C-I (which resembles Peirce’s continuum) and C-II (which resembles Weyl’s continuum), of which I prefer the latter (for such reasons as those in the unpublished third of this tryptich).
My hypothesis can be introduced informally as an extrapolation from the familiarly finite. If we consider sand grains to be cubic millimetres of silicate, to keep things simple, then a sandstone mountain composed entirely of such grains and occupying exactly one cubic kilometre, would contain 109 m3 ÷ 10–9 m3 = 1018 grains. By extrapolating, it is not hard to imagine that, if lines were composed of points, each of length 0, then a line of arbitrary unit length u might contain 1·u ÷ 0·u = 1/0 points. Of course such extrapolations, from finite cases to an infinite case, are naturally unreliable but they do not necessarily fail (e.g. such extrapolations are also at the heart of set theory, cf. principle b of Hallett 1984: 7) and that one is prima facie coherent, as you will see.
But you may well believe that division by 0 is impossible. You may think that, for example, were we to allow an arithmetical division by 0, then from 1 × 0 = 2 × 0 we would be able to deduce the contradiction 1 = 2. But although that 1 = 2 does follow from assuming 0/0 = 1 (together with associativity), we need not assume that 0/0 = 1.
One reason why we might have thought that we should, is that a/a = 1 whenever a/a is defined at present (for finite a), and that additionally defining 0/0 is like allowing a to be 0. Nonetheless such extrapolations are, as mentioned above, notoriously unreliable, and in fact it need only follow, from dividing 1 × 0 = 2 × 0 by 0, that 0/0 includes x iff (if and only if) it contains 2x, which would allow an arithmetical division by 0 (although not as a function) were 0/0 a suitable collection of numbers, and such a collection—which I call a ‘mere-sum’—will be specified at the end of §2.
Another reason might be that ‘division by x’ means multiplication by the multiplicative inverse of x within a number field, where the multiplicative inverse of x is whatever yields 1 (the multiplicative identity) when multiplied by x (within the field). Nonetheless, from 0/0 ¹ 1 it need only follow that division by 0 is not allowed within a number field. You will see (in §5) that it is allowed within a number pitch (which contains little more than the field that it very neatly rounds-out).
Note that I am not suggesting that functions and fields are not useful—they are of course extremely important, but we may certainly extend that repertoire, e.g. for internal rounding-out, or for increased applicability. (Cf. how, although there is an obvious utility to having the cardinality of a collection increase by 1 whenever a new object is added to it, we can nonetheless consider infinite cardinals.)
I ought first to define my terms a little more precisely, so:
Let the primitive line be the line that would be physically instantiated (e.g. as time, were time infinitely divisible)—note that the idea of such a pre-theoretic geometrical line makes sense whether or not any physical continua actually exist.
Let primitive cardinality be what collections must have in common, if they can be related by bijections (one-to-one correlations), and what collections that cannot be so related must differ in.
Let ‘k’ denote the primitive cardinality of the continuum—in other words, the number of (n – 1)-dimensional spaces that would exist within any physical n-dimensional space (were there any).
E.g., the cardinality of the standard continuum is 2^À0, where I use ‘^’ to denote standard cardinal exponentiation (since the more familiar superscript notation will be used for a more familiar form of exponentiation in §4) and where À0 is the cardinality of the standard natural numbers.
Given C-D, k = 2^À0 then, but while standard set theory provides mathematics with a definite subject matter it is hardly a comprehensive theory of cardinality, such as would be required for deciding the propriety of other kinds of cardinal numbers. E.g. it is obviously coherent to regard the totality of the standard cardinals and the totality of the standard ordinals as 2 totalities, despite the impossibility of doing so within standard set theory.
Furthermore we know that the cardinality of the totality of all the standard sets, say W, is not the size of a standard set, but nonetheless it appears to be a primitive cardinal number, because Àa denotes a transfinite cardinal iff a denotes an ordinal (whence those 2 classes correlate one-to-one).
And after all, the objectivity of the natural numbers (which we ourselves instantiate) means that we can hardly define them to be standard sets (cf. Hamming 1998). It is only their supposed emulation by some standard sets (the finite von Neumann ordinals) that justifies those sets being called ‘natural numbers’ within standard set theory—and yet the totality of the natural numbers may well be potentially infinite, for all we really know (cf. my unpublished).
That possibility (present within current mathematics as the persistence of constructivism) will be accommodated below (in §6) by considering lines of points in two cases, C-I, in which N (the positive integers) is an actual (or a completed, finitesque, combinatorial, etc.) infinitude, and C-II, in which N is a potential infinitude.
So in order to avoid unnecessarily prejudging such things, my collections of numbers will have to be more primitive than standard sets of sets, and more like atomic mereological sums (see Simons 1987: 14) of numbers. I shall call such a collection, one that has at least the following 4 informal properties (cf. plural quantification), a mere-sum, and denote it by square brackets, e.g. N =df [1, 2, 3, …].
The first property of mere-sums of numbers is that the individual numbers are regarded as atoms—i.e. mere-sums of numbers are not also mere-sums of whatever comprises those numbers (in a different way), if anything does (e.g. their elements or members, if numbers really are sets or classes).
Secondly, because the mereological sum of x and y is just x and y, internal brackets can be eliminated, e.g. [[1, 2], 3] = [1, 2, 3]; and similarly, the mere-sum of a single number is merely that number, e.g. [1] = 1.
Thirdly, two mere-sums are naturally defined to be equal iff a bijection between them may relate each atom with an equal atom. Consequently, [x, y] = y iff each atom of x is also an atom of y, so that x is a part of y (formal mereologies being part-whole theories), which I shall abbreviate to x Ð y below; e.g., [1, [1, 2]] = [1, 2], whence 1 Ð [1, 2]. Furthermore, if x Ð y, and also z Ð x implies x Ð z, then x is an atom of y, abbreviated to x @ y below, e.g. 1 @ [1, 2].
And finally, arithmetical operations naturally distribute over mere-sums of numbers, e.g. adding 1 to both 1 and 2 yields 2 and 3, whence 1 + [1, 2] = [(1 + 1), (1 + 2)] = [2, 3].
So finally I’m ready to begin! The most direct way to examine the possibility that k resembles 1/0 is to try to maximise the algebraic coherence of the adjunction of an undefined symbol ‘#’ to N, where the informal properties of # derive from two heuristic assumptions, which together assert that 1/k equals the length of a point.
(ha1): # is the value of k
(ha2): 1/# is the additive identity for magnitudes
Because that approach is relatively direct—being analytically metaphysical (i.e. pre-theoretic) rather than axiomatically mathematical (e.g. set-theoretic)—we may be able to avoid prejudging the nature of number. And note that, although it would be simpler, if less direct, to define # to be 1/0, such an approach would be undesirable because # would, were it coherent, be a cardinal number, which is clearly a more fundamental kind of number than a ratio of magnitudes such as 1/0.
Similarly although my exclusion of 0 from N is fairly unconventional (it being standard to regard the natural numbers as the finite cardinals), it is desirable to begin with the positive integers because the informal properties of # are to be obtained via the intuitive concept of a line of k points, and in that context 0 is primarily the length of a point, which is an undirected magnitude (an answer to ‘How much?’) rather than a multitude (an answer to ‘How many?’). Introducing 0 as an abbreviation for 1/# (at the end of §3) should therefore make it easier to see the coherence of ha1 and ha2 (although it would of course be possible to start with the finite cardinals).
But before adjoining # to N (at the start of §3) the possibility that primitive lines might be full of points will be defended briefly (since only then would we have any real use for #). Fortunately, points are hardly inconceivable; e.g., an imaginary black square on a white background is easily envisaged with points at its corners, where its edges intersect (whence my choice of the name ‘#’ for #).
If points do exist then, as there is nowhere in a line where it cannot be intersected by another (0-width) line, lines are clearly full of points. And although planes do seem more like glass panes than sandpaper, that intuition cannot imply that they are not full of points, because points (with size 0) are infinitely smaller than sand grains, so that planes of points would be infinitely smoother than sandpaper. Furthermore, lines in planes are not like scratches put onto glass panes, because the positions of such scratches would make much better analogies for primitive lines, and they were clearly there already.
In Aristotle’s (spatial) line, a point had a potential existence that was actualised only if something happened there, but it is perfectly reasonable to think that if something could happen there, then an actual position (or point) must have been there already, so that it could happen there. Incidentally, although Aristotle introduced the distinction between actual and potential infinity (i.e. an infinite being and an infinite potential to be), whence those who have thought that the natural numbers do not form an actually infinite totality have also tended to deny that lines are full of points, the relationship between N and k is really more subtle than that (see §6, and my 2007).
I shall now adjoin # to the natural numbers, N, to make what I call the notional numbers, N# =df [N, #], in order to see how strong the informal arithmetic of N# can be, given ha1 and ha2.
To begin with (where n is, as usual, a natural number variable), (# + n), (# + #), #·n and #·# all equal #. The first three of those equations follow (via ha1) from how the points of a line of length n correlate one-to-one with the points of a unit line, while the fourth equation follows fairly obviously from ha2. Addition and multiplication may clearly both remain both associative and commutative (since any finite expression containing # just equals #), while it is trivial to show that multiplication distributes over addition, via a few typical equations such as #·(# + n) = #·# = # = (# + #) = (#·# + #·n).
Although N# (like N) is clearly closed under both operations, where a mere-sum S is closed under a (binary) commutative operation o if (x o y) @ S whenever x and y are atoms of S (which is isomorphic to the familiar definition of closure, because a set S is closed, under o, if (x o y) Î S whenever x and y are elements of S), a weaker generalisation (which includes closure as a special case) called ‘mere-closure’ is going to be more useful (within the rounded-out number systems that follow the adjunction of #):
S is mere-closed, under o, if (x o y) Ð
S whenever x and y are parts (e.g. atoms) of S.
Note that mere-closure fits the intuitive meaning of algebraic closure no worse than the familiar concept does (since if a mere-sum is mere-closed, then operating within it cannot generate anything that is not there already) and that it can also cope with pluralities (or indeterminate forms) such as 0/0.
Similarly, although the familiar definition of the inverse of o is that it is (something like) an operation i such that (x i y) = z iff x = (y o z) (e.g. 3 – 2 = 1 because 3 = 2 + 1 and nothing else yields 3 when 2 is added to it), it so happens that, because # – # and #/# (which equals 0/0) will be collections of numbers, a more appropriate definition of i (in terms of o, and within a domain containing atoms x, y and z) is:
(In): z @ (x
i y) iff x @ (y o z)
In includes the usual definition as a special case (e.g. # – n = # follows from In, since # + n = # and N is closed under addition), but from In we can also obtain, e.g., # – # = N# (since # + # = # = n + #) and #/# = N# (since #·# = # = n·#). Subtraction and division are not closed in N so it should be no surprise that they are not mere-closed in N#, and similarly they are also neither associative nor commutative.
What is prima facie unfortunate is that multiplication cannot distribute over subtraction within N#, e.g. (2 – 1)·# = # does not equal 2·# – # = N#, because we naturally expect it to, e.g. 3·(2 – 1) = 6 – 3.
However, consider the informal set, NÈ{0, À0}, where N (given by n Î N iff n @ N) is informal because it contains natural numbers rather than sets (although it is quite adequate because NÈ{0} is arithmetically isomorphic to the standard set of finite von Neumann ordinals). Cardinal multiplication cannot distribute over subtraction within NÈ{0, À0} lest À0 = (2 – 1)·À0 = 2·À0 – À0 = À0 – À0 = (1 – 1)·À0 = 0·À0 = 0. But that does not, of course, imply that À0 is an impossible number. And note that defining À0 – À0 might even be useful (e.g. removing À0 objects from À0 objects would leave m objects, where m Î NÈ{0, À0}), and that N# would be relatively strong anyway, even were À0 – À0 undefined, because at least # – # is defined.
So, it may well be that a loss of distributivity (to be replaced in §5 by mere-distributivity) is as natural, when cardinals can be infinite, as a loss of commutativity is when ordinals can be infinite. Note that although consequently multiplication will no longer distribute over addition when negative numbers are adjoined (in §4), it is not totally unheard of for a commutative multiplication to fail to distribute over a commutative addition (e.g. it may do so within category theory).
To press on, within the motivating context of a line of # points we may consider # points, and n points for any n, and also n line intervals, whence the ratios of the notional numbers will be considered next—the continuity of the line makes it possible (in principle) to continue to subdivide intervals endlessly, so although we usually close subtraction before adjoining the rest of the rationals, in the current context it is perfectly natural to extend N# to a domain that is mere-closed under division, before adjoining the negatives (in §4).
I say that a ratio of two notional numbers, if it is not #/# (which is a plurality), is a fractional number, an atom of F#. The elementary arithmetic of F# subsumes that of N# Ð F# of course, and includes that of 1/#. The other atoms of F# have the form r = n/m (where n and m are relatively prime natural numbers with m > 1), and their arithmetic follows straightforwardly from ha1 and ha2 (details are not included as they are unsurprising, and anyway they are encapsulated by the rational number pitch of §5). Addition and multiplication may remain commutative and associative, with multiplication distributing over addition (it is trivial, if tedious, to show the consistency of retaining those algebraic strengths). And since it follows from ha2 that 1/# is isomorphic to the familiar magnitude 0 within the finite part of F# (for details see my 2005: 98), hence 1/# will now be called ‘0’.
Incidentally (to pick up a point from §1), dividing 1 × 0 = 2 × 0 by 0 within F# just yields F# = F#, rather than a contradiction, because (via In) 0/0 = #/# = F# (and a similar resolution applies within all the following number systems).
The most natural way to extend F# would be by adjoining irrationals (and infinitesimals), because #·0 should include all such lengths (of lines of points), and in fact (as you will see in §5) such extensions retain the algebraic strengths of F#, but for brevity the negative numbers will be adjoined next. Surprisingly little is overlooked with that shortcut, because a resulting algebraic structure—the number pitch (defined in §5)—extends all the number fields just as it extends the rationals.
Consider two signed collections +F# and –F#, defined by: +x @ +F# iff x @ F# iff –x @ –F#. In the current context of geometry, the adjunction of negatives may be thought of as the introduction of two directions, ±1. The familiar properties of signs follow from the intuitively obvious properties of directions, and it is trivial to show that no inconsistency results from allowing addition and multiplication to remain commutative and associative, so the arithmetic of +# and –# follows straightforwardly from that of #.
E.g., what is +# – +#? Well, given any rational, or +#, adding it to +# results in +#, so (from In) the values of (+# – +#) include all the rationals and +#. And because +# is an atom of (+# – +#) = (+# + –#), therefore –# also yields a mere-sum that includes +# when added to +#. There are no other atoms in our current domain, so (+# – +#) = [+F#, –F#].
There is one odd-looking result however, because (+#)·(+0) = +F#, whereas (+#)·(–0) = –F#, which means that +0, which is +(1/#), is not quite the same as –0. But again (cf. distributivity) this is not too bad, because the rational equation 0 = (–1)·0 can be obtained by replacing [+0, –0] with a new object (not necessarily a pair-set) that relates to the other numbers just like the mere-sum [+0, –0] does, and which will be called ‘0’ when the positive numbers are called by their previous (unsigned) names.
That is not an unreasonable way to proceed, for the following reasons. To begin with, were there serious objections to the introduction of such new objects, there would be equally serious objections to almost all the sets of the standard approach.
Furthermore, the rational 0 is not, intuitively, an undirected quantity, not in the way that the fractionals are undirected. So it does not make less sense to think of it as having all the directions (of the domain) rather than none. After all, the mainstream approach is much less intuitive, its integers being equivalence classes of pair-sets of finite ordinals. The main thing is that the mere-sum [+0, –0] is indeed isomorphic to the rational 0 within the current domain (for details see my 2005: 99-100).
So with [+0, –0] replaced by an isomorphic atom called ‘0’, and the positive numbers called by their old names, the new domain consists of #, –# and the rationals. Division by # and –# are still multiplication by +0 and (respectively) –0, by definition, so multiplication by 0 is now division by [#, –#], and 1/0 = [#, –#]. Although it was the case within F# that 1/0 = #, such differences between domains are not too unusual, even within school mathematics; e.g., square numbers (e.g. 1, 4, 9) each have one square root in N, but two in Z, where x @ Z iff x Î Z (the informal set of integers).
What is more of a problem is that, although # – # now equals the whole domain, #/# is only the non-negative part of it. A neater structure therefore results from replacing not just [+0, –0] but also [#, –#] by new atoms. That structure is the rational number pitch, in which 1/0 = QR (defined in §5)—the symbol ‘Q’ was chosen because an ideal point at infinity turns an infinite line into an infinite circle, within projective geometry (see below), while the subscript ‘R’ indicates the rationals (see below).
But before looking at that structure, note that although Z has been bypassed, in the above development of the number systems, that was not because of any inconsistency between Z and #. In fact, because 1/0 + 1/0 = [#, –#] + [#, –#] = # – # = 0/0, the familiar rules for adding and multiplying ratios of integers, i.e. (w/x) + (y/z) = (w·z + x·y)/(x·z) and (w/x)·(y/z) = (w·y)/(x·z), may now remain valid when w, x, y and z are any integers; and of course, being able to extend the validity of familiar rules, in such a way, is an indication of coherence.
Furthermore, since that particular example of rounding-out was only possible because multiplication does not distribute over addition, now that the subtraction of a number is the addition of its negative, we also have some compensation for (and hence an indication of the underlying coherence of) that algebraic weakness.
Coherence is further indicated by situations that involve exponentiation. Although 0(2 – 1) = 0 and 0(1 – 2) = [#, –#] differ, 02/0 = 0/0 = 0/02, whence the generalisation of the familiar rule z(x + y) = zx·zy to include z = 0 has to be a weaker rule, z(x + y)Ð zx·zy (cf. mere-distributivity in §5). But that weakness is useful, because it allows 00 to equal 1 instead of 0/0 and it can be useful to stipulate that 00 = 1, e.g. when algebraically manipulating polynomials (cf. Kaplan 1999: 169) or when recursively defining exponentiation.
Furthermore, a natural way to handle rational powers is via biconditionals such as x @ y½ iff x2 = y. Then y(½ + ½) = y Ð [y, –y] = y½·y½ , and the rule (y½)2 = (y2)½ can be kept even when y is negative; whereas the familiar root function, say Ö, takes only positive values, (Öx)·(Öy) = Ö(x·y) failing when x and y can be negative (e.g. it becomes –1 ¹ 1 when x = y = –1).
I call the aforementioned substructure of the arithmetic of [F#, –F#] within which # and –# only occur within the mere-sums [+0, –0] = [+(1/#), –(1/#)] and [#, –#], the rational number pitch because (i) it contains the rational number field (plus one new object) and (ii) any field may be extended to its corresponding pitch (by adjoining a similar object) as follows.
A number field F is usually a set F of numbers together with two arithmetical operations that satisfy the familiar field axioms. But an isomorphic structure is therefore possessed by a mere-sum F given by x @ F iff x Î F, when @ replaces Î in those axioms. And adjoining QF (defined by the following properties) to the field F yields the number pitch, FQ =df [QF, F]. The field operations are extended by the following six equations (where x @ F and x ¹ 0):
QF + 0 = QF
QF + x = QF
QF + QF = FQ
QF·0 = FQ
QF·x = QF
QF·QF = QF
Also, division by 0 is defined to be multiplication by QF, and vice versa, while the subtraction of QF is the same as its addition. It is easy to show that, with those definitions, the pitch operations can both remain both commutative and associative. Pitches are mere-closed under addition, subtraction, multiplication and division, and the only cost of extending a field to a pitch is rather beneficial (as mentioned above), being what I call mere-distributivity, i.e. if x, y and z are atoms of FQ, then x·(y + z) Ð x·y + x·z, with equality (distributivity) only if x ¹ QF.
Of course, unfamiliar algebra is often obscure, so the following tables illustrate the rounding-out of the field to mere-close division. They show the multiplication and division of the identity elements, 0 and 1, first for the field and then, below, for the pitch (where for clarity ‘Q’ abbreviates ‘QF’ and ‘X’ names FQ).
|
´ |
0 |
1 |
|
¸ |
1 |
|
0 |
0 |
0 |
|
0 |
0 |
|
1 |
0 |
1 |
|
1 |
1 |
|
´ |
0 |
1 |
Q |
X |
|
¸ |
0 |
1 |
Q |
X |
|
0 |
0 |
0 |
X |
X |
|
0 |
X |
0 |
0 |
X |
|
1 |
0 |
1 |
Q |
X |
|
1 |
Q |
1 |
0 |
X |
|
Q |
X |
Q |
Q |
X |
|
Q |
Q |
Q |
X |
X |
|
X |
X |
X |
X |
X |
|
X |
X |
X |
X |
X |
So, pitches are neat—whilst containing all the algebra of a field, and little else, they are much more symmetrical (which is important here, because in general, symmetrical structures are more likely, than asymmetrical ones, to be physically instantiated). Describing more precisely the increase in symmetry, caused by the adjunction of QF to a field, is easier given the concept of a team structure, áT, e, a, Mñ, as follows.
Briefly, teams are commutative generalizations of abelian groups, e.g. the fractionals form a proper multiplicative team, áF#, 1, ×, {0, #}ñ. Abelian groups are improper teams, e.g. áZ, 0, +, Æñ, teams being called ‘proper’ if Æ is a proper subset of M (whose elements are those atoms of T without inverses—with only mere-inverses—in T). So whereas a field F contains an additive abelian group áF, 0, +, Æñ and a multiplicative commutative monoid, a pitch FQ contains two proper teams, áFQ, 0, +, {QF}ñ and áFQ, 1, ×, {0, QF}ñ.
Now, we have already met one example of a number pitch, because when F = Q (the familiar rational number field), adjoining QR to F = R (rho, for ‘rational’, or ‘Pythagoras’) yields the rational number pitch, RQ, which is the same structure that results from replacing [#, –#] with an isomorphic atom called ‘QR’.
Similarly, let a field D (delta, for ‘decimal’, or ‘Dedekind’) be defined by x @ D iff x Î R (the familiar real number field). The adjunction of QD to D yields the real number pitch, DQ. However, since lines of # points can occur in two possible cases, C-I and C-II, which correspond to N being, respectively, an actual or a potential infinitude, and since in C-II only reals that could (in principle) be defined by finite laws are legitimate (cf. §6), let L (lambda, for ‘legal’, or ‘Lebesgue’) be a field of such reals. The adjunction of QL to L yields a legal real number pitch, LQ. Had irrational magnitudes been adjoined to F# in §4, then QD and QL (in cases C-I and C-II respectively) would have replaced the mere-sum [#, –#].
Similarly, let a field G (gamma, for ‘complex’, or ‘Gaussian’) be defined by x @ G iff x Î C (the familiar complex number field), which is a Gaussian plane. The adjunction of QG yields the pitch GQ, which is a projection of a Riemann sphere. In C-II, adjoining the imaginary unit i to L yields a legal complex number field, I (iota, for ‘imaginary’, or ‘Intuitionistic’), while i’s adjunction to LQ yields IQ. And QG and QI would have replaced the mere-sum of all the #·eiq for –p £ q < +p (legal q in the case of QI).
That was a lot of Greek, so a brief recap may be useful. From two heuristic assumptions, (ha1) that # is the number of points in a line, and (ha2) that 1/# is the length of a point, the arithmetic of # was obtained, and then directions were given to the fractional magnitudes, whence the isomorphism between [+(1/#), –(1/#)] and the rational 0 led to the replacement of [+(#/1), –(#/1)] by a new infinite number, QR, one of a new kind that neatly rounds-out number fields.
The next stage in the analysis of the hypothesis that k = # would naturally be its analytic geometry (see the final section of my 2005, and here), whence it becomes important whether the infinitude of N is actual or potential. Clearly # must be a non-set-theoretic cardinal, because according to this neglected hypothesis the essence of continuity is that 0·# ¹ 0, whereas 0·Àa = 0 for every ordinal a. So there seems to be just two possibilities, which I call ‘Case I’ and ‘Case II’:
(C-I): # is bigger
than every Àa
(C-II): # is cardinally incomparable
with every Àa
According to C-I, lines of # points would contain all transfinite cardinalities of points, a proper class of them, which is a lot of points—but then, 0 is very small—whilst according to C-II lines of # points would not even contain À0 points, which is to say that (by comparison with #) the infinitude of N would be potential (a concept that is usually associated with constructivism, but which is also associated with proper classes).
The first thing to note about C-I is that, within any reasonable theory of (well-ordered) classes, the Cartesian product of the null-class with any class is likely to be the null-class, whence # = W seems unlikely (and given the axiom of choice, we can rule out # = 2^À0, of course). Furthermore, one heuristic principle of Cantorian set theory is that “any potential infinity presupposes a corresponding actual infinity” (principle a of Hallett 1984: 7), which seems to imply that # > W. But another Cantorian principle is that W “cannot be mathematically determined” (principle c of Hallett 1984: 7). That is, whilst being actual (in the sense of principle a), W cannot be as finitesque (in the sense of principle b) as the transfinites are, which might imply that lines of # points would not also contain W points, so that # would be cardinally incomparable with W. In short, proper classes remain quite mysterious (e.g. see Rayo and Uzquiano 2006), whence little can be said about (e.g. against) C-I at present.
According to C-II, N is some sort of potentially infinite totality—by comparison with a line of points that could be physically instantiated, the totality of the natural numbers would be more like a potentially infinite totality than an actually infinite totality (a platonistic rather than an anti-realistic hypothesis), as follows.
Since it is the repeated addition of 1 that yields the primitive numbers in N, starting from 1, via 2 =df 1 + 1, and 3 =df 2 + 1, and so forth, hence their totality (all of those numbers) is defined in an endlessly hierarchical kind of way, whereas the way that primitive lines are full of points is clearly relatively non-hierarchical (for all that, within analytic geometry, the numerical labels of points would inherit the hierarchical nature of N).
Consequently there is the logical possibility that, although there are n points, for any natural number n, in a line of points, there are not À0 points. An indication of the reality of that possibility (for others see section 7 of my unpublished, and here) is that, despite a heavy emphasis upon set-theoretic foundations during the last century, good reasons why N should act finitesquely (as N does) that are not also reasons why the proper class of all the cardinals should too (which it cannot) are rather elusive (cf. Fletcher 2007: 531-51).
Suppose that, because of the actual infinitude of the natural numbers, we can indeed consider any arbitrary subset of them, and thence the totality of all such combinatorially possible subsets—the full power-set of the natural numbers (which is isomorphic to the standard set of real numbers). If that is why the transfinites are proper numbers, existing to no lesser degree than the natural numbers do, then why should we not also consider an arbitrary subclass of the transfinites, and thence the totality of all such combinatorially possible subclasses? That we cannot is Cantor’s paradox.
And note that, even when N’s infinitude is regarded as actual (as on the standard view), the concepts of cardinal and ordinal do begin to diverge with À0 and w because of N’s endlessness—that endlessness does seem able to cause some shift away from finitesque behaviour. And even on the standard view, the approach of the natural numbers to À0 does resemble the approach of the cardinal numbers to W, even though W cannot be as actually (or finitesquely) infinite as À0.
In short, note that the possibility of C-II just requires that two infinite collections (the endless sequence of primitive natural numbers, and the primitive line of points) that are even more different in kind, than À0 and W, might differ significantly. And according to my unpublished (a modern physical instantiation of Levy’s paradox), C-II is actually rather plausible (despite its conceptual obscurity, which after all it shares with the proper classes that, even on the standard approach, are unavoidable).
Admittedly neither of those two cases is prima facie compelling, but that may just be the way with lines of points (e.g. the Banach-Tarski paradox is a famous problem for standard continuity). After all, the symmetrical algebra accompanying # is mathematically attractive (and physically plausible). And the coherence of # is again indicated when numerical labels are assigned to points; e.g. (see my 2005: 105-7; also see here), were k = # an infinite space would, by containing infinitely separated points, resemble the space of projective geometry, which is the most symmetrical of the geometries in their group-theoretic classification.
Over several years my 2005 benefited from Peter Fletcher’s
interest in it.
Cooke, M. C. (unpublished) ‘Infinite Probes: A Problem with Probability’—here.
Fletcher,
P. (2007) ‘Infinity’, in D. Jacquette (ed.) Philosophy of Logic,
Amsterdam: North-Holland, 523-85.
Hallett,
M. (1984) Cantorian set theory and limitation of size, Oxford: Clarendon
Press.
Hamming,
R. W. (1998) ‘Mathematics on a Distant Planet’, American Mathematical
Monthly 105, 640-50.
Kaplan,
R. (1999) The Nothing That Is: A Natural History of Zero, London: Allen
Lane.
Rayo, A. and G. Uzquiano (eds) (2006) Absolute Generality, Oxford: Clarendon Press.
Simons, P. (1987) Parts: A Study in Ontology, Oxford: Clarendon Press.