How 'Many Infinities' Are There in Mathematics?
| Scientific Paper | |
|---|---|
| Title | How 'Many Infinities' Are There in Mathematics? |
| Read in full | Link to paper |
| Author(s) | Velimir Abramovic |
| Keywords | Cantor, diagonal argument, infinity, zero, synchronicity, ontology of number, science of time |
| Published | 2008 |
| No. of pages | 40 |
Read the full paper here
Abstract
(From "The Basics of the Science of Time") While reading Cantor's "Diagonalization Argument", I realized that it contains nothing which can be taken for granted, but that this proof must be analyzed in a classical manner, statement by statement, symbol by symbol, walking through it on foot, using small steps. This manner is necessary, among other reasons, because the essence of every trick, particularly an intellectual one - lies in the illusion of the apparent.
I have accepted the verification of Cantor's proof (theorem) as something entirely personal because, if it is true that there is more than one infinite in arithmetic, then my effort is pointless, my theory of time incorrect, and mathematics and physics will forever remain two fundamentally unrelated sciences.
For the sake of continuity, Cantor's proof will first be presented here in its entirety, and then analyzed in detail, and finally we will present our own conclusion to the "counting of all decimal numerals" which is in accordance with Mellis' sound minded principle by which "infinities cannot coexist".
Overview
This chapter, drawn from Abramovic's larger project The Basics of the Science of Time, is an attempt to refute Cantor's diagonal argument and with it the entire hierarchy of transfinite cardinals. Both versions of the argument are quoted in full — the decimal-expansion version and the version using infinite binary sequences — and then dismantled clause by clause.
The stakes are personal and explicitly stated: Abramovic's theory of time requires that there be exactly one infinity in arithmetic. If Cantor is right that the real numbers outnumber the naturals, then, in his words, "my effort is pointless, my theory of time incorrect, and mathematics and physics will forever remain two fundamentally unrelated sciences." He therefore attacks the proof rather than accommodating it, invoking what he calls Mellis' principle that "infinities cannot coexist," and Duns Scotus on the limiting of true infinity.
The departure from the mainstream is total on this point. Set theory since Cantor treats the uncountability of the reals as a theorem — one of the most-verified results in mathematics, with several independent proofs. Abramovic treats it as a "trick," an "extreme example of incomplete deduction," and holds that the only genuine infinity in mathematics is zero. His replacement is a scheme he calls the synchronic list, built on an added premise he says mathematics has wrongly discarded: that numbers have ontology and temporality, and that only numbers which coexist "in the present" may be compared.
The argument
What is wrong with the notion of infinity in the proof
The first objection is definitional. "'Infinity of decimal numbers'; what kind of infinity is this if it has zero and one as its outer limits? 'Infinity of counting numbers'; what kind of infinity is this if it has zero and n as its outer limits? Infinity cannot have any outer limits. An unspecified many is not infinity, or rather, an 'infinite number' is a contradictory concept in itself unless it refers to zero."
The identification of the true infinite with zero is developed later. Zero, he argues, uniquely satisfies Cantor's own criterion for an infinite quantity — that it "contain at least one component as great as itself," a part equal to the whole — and satisfies it more completely than any infinite set: "zero cannot be altered in mathematical operations because the parts of zero are mutually equal and every part of zero is equal to the whole zero; (0 + 0 + 0 + … 0 × n … + 0 × 0 = 0)." His verdict: "Enthralled by the constant division of one, Cantor forgot about zero."
The resolution objection to Cantor's list
The technical heart of the paper is an argument about indices. In Cantor's array, the first index of dnm labels which decimal number is meant, while the second labels a decimal place. Abramovic argues these are not commensurable: "the first index number signifies the exact number expressed, while the second index number does not signify with which it is expressed, but is the place number of the decimal place and simultaneously a symbol representing a group of ten numbers." Because each decimal place stands proxy for the whole interval 0–9, the list has a built-in "compression of 10:1," and "the number of decimal numbers in it does not correspond to the number of decimal places and the number of places does not correspond to the number of actual decimals to be listed." Hence, he concludes, "there is no 1:1 correspondence at all and the listing of all decimals is not even attempted."
He makes much of a notational point: Cantor writes the n-th row as dn = 0.dn1dn2dn3… and trails off, never writing dnn as a terminus. "He does this because it would undermine, or rather disprove his proof; it would show that, in the 'listing', he identifies 10 decimals with each decimal place, which would be the same as claiming that eleven ones is only one."
The diagonal number x is attacked on the same ground. Its subscripts have, he says, a triple and partly hidden meaning, and the clause "x1 is any digit other than d11" is "subjective intervention" — the one stipulation that rules out the coincidence x = d1 which the list's own resolution would otherwise permit. "Every sensible person will then inquire 'alright, x1 is not d11, than which digit is it?'"
The synchronic list
The proposed replacement adds a sub-index to Cantor's second index, so that each decimal place is written out over its whole range of ten possible values — producing, for a single number d1, a rectangular block of rows 0 through 9. Counting is then done by decomposing every decimal number into elements: the number, the places, and the decimals filling them. "In our synchronic list, the principle of 1:1 is applied to the full interval of decimals by one decimal place and achieves triple univocal correspondence, 1:1:1, or rather that 'one decimal place means one decimal which means one decimal number.'"
Because a decimal number is "a complex numeral system of three elements," it takes three units of the natural sequence N to count one. Abramovic anticipates the obvious objection — does that not make the decimals three times as numerous? — and answers by analogy: the natural number 3 is likewise a complex system that cannot be counted by a single 1 without losing what makes it a 3, so the correspondence 3 elements ↔ 3 ones is itself 1:1. "If the number of elements of d and N have a growth rate of one each, then d is equivalent to N."
The temporal apparatus supports this. Numbers are "actualized" in sequence: the first decimal written down is 1d1, the second 2d2, "which especially is true in the case of equal numbers" — two occurrences of 0.27451401 are not one number but two, since "mathematical reproduction does not mean that the numbers physically coincide, but only that they are of equal value." The generic symbols a, b, c, x, y are held to be temporal contradictions, presenting "the unknown future 0,1,2,3,4,5…" as "the known present." And the succession n, n+1 is denied simultaneity: "while we have an actual n, we do not have a real physical correspondence for the actual n+1, so we should not imagine it in mathematics either."
The binary-sequence version
The second half applies the same treatment to Cantor's original argument on sequences of 0s and 1s. Abramovic reads the condition "each xi is either 0 or 1" as smuggling in time: "for one place 1S1x1 (space) has two times 2T'x1", geometrically "one same space has two different time coordinates." He then builds "synchronicity tables" — for length 1 the two sequences, for length 2 the four, for length 3 the eight, for length n the 2n — and enumerates them explicitly up to length 5.
The claimed resolution is that every finite sequence of n members occurs in the table Txn(0;1) = 2n, that "all vertical sequences are the same as some horizontal sequences," and that the diagonal sequence s0 of seven members is simply found in the table 27. "Finally, every sequence in the form 0101…1…0… is included in the table nT'xn(0;1), which further excludes the possibility of the existence of Cantor's uncountable set."
The general charge
The closing complaint is methodological: Cantor "by avoiding two-way equivalency … succeeds in that exploration by his method by definition have negative results," and then converts a negative — the failure of one counting method — into the sweeping positive claim that a larger infinity exists. "From certain negative examples and the inability to apply a method successfully, we cannot follow a certain positive conclusion." Abramovic ends generously, though: Cantor's real service to science was that "he inadvertently brought attention to the fact that the main problems in mathematics cannot be solved without physics and that the temporality of mathematics is absolutely necessary."
Assessment
There is a serious philosophical impulse under this paper, and it deserves stating plainly. Abramovic is pressing the question of what mathematical objects are — whether a numeral written twice is one entity or two, whether the completed infinite totality presupposed by the diagonal argument is a legitimate object or a figure of speech. That is a real question, and he is in respectable company in asking it. Finitists and constructivists from Kronecker to Brouwer to Wittgenstein have refused actual infinity, and Wittgenstein in particular pressed something close to Abramovic's complaint that the diagonal argument trades on a picture. His observation that Cantor's x is defined only by what it is not, and his instinct that the rows and columns of the array are doing different kinds of work, are the right places to look for a flaw. The reading of zero as the only quantity all of whose parts equal the whole is elegant, and the closing point — that mathematics has an underdeveloped ontology — is one many philosophers of mathematics would concede.
But as a refutation the paper does not work, and the reasons are specific.
The resolution objection misreads the proof. Cantor's diagonal argument does not require that the second index count the same things as the first, and does not claim a bijection between decimal places and decimal numbers. It is a reductio with one moving part: assume any function f from ℕ onto the reals in (0,1); construct a real differing from f(n) in the n-th digit; conclude f is not onto. The "compression of 10:1" that Abramovic identifies is simply the fact that a digit position can hold any of ten values, which is what makes the construction of x possible, not what invalidates it. Nothing in the proof depends on writing dnn as a last term; there is no last term, and the ellipsis is not an evasion.
The complaint that "x1 is any digit other than d11" is "subjective intervention" mistakes a definition for an assumption. One is free to define a number by specifying each of its digits; the stipulation is a construction rule, and it constructs a perfectly determinate real. Abramovic's own question — "than which digit is it?" — has an answer: pick 5 unless dnn = 5, in which case pick 4. That standard version of the construction is not addressed.
The synchronic list does not do what it claims. Its two counting claims are internally inconsistent with each other. To count each decimal number "as several decimal numbers" — 0.975 counted as three units, a number with 17 places as 17 units — is to enumerate the terminating decimals, which are indeed countable; every argument he gives works, and works uncontroversially, for finite decimal expansions. The uncountability claim concerns the non-terminating expansions, and the paper's device of counting a number with unlimited places as (n+1) = (n+1) never confronts them. The same gap is starker in the binary section: the synchronic tables enumerate, correctly, all 2n finite sequences of each length n — the set of finite binary strings, which is countable and which no one disputes. Cantor's set T is the set of infinite sequences, and no infinite sequence appears in any table 2n. When Abramovic locates s0 = (1,0,1,1,1,0,1) in the table 27, he has located a seven-element truncation, not the diagonal sequence, which by construction has infinitely many terms. The conclusion "which further excludes the possibility of the existence of Cantor's uncountable set" does not follow from anything demonstrated.
Finally, the paper never engages the independent routes to the same result. Uncountability follows equally from Cantor's earlier nested-intervals argument, from the theorem that no set injects onto its power set (which is a purely combinatorial fact with no decimal notation involved), and from measure-theoretic arguments in which a countable set of reals has measure zero while [0,1] has measure one. A refutation of the diagonal argument alone, even a successful one, would leave these standing. And the temporal premises — that n and n+1 are not simultaneous, that two equal numerals are two numbers — are stipulated rather than argued, and are used to license conclusions about a formal system that does not contain them.
Read as philosophy of mathematics, the paper is provocative and occasionally acute about what mathematicians take for granted. Read as the mathematical result its title advertises, it demonstrates the countability of the finite decimal expansions and the finite binary strings, and stops short of the claim it needs.