On the comment thread below CAB (whose moniker perfectly mimics a Goedelian joke, but leave that aside for now) raises a series of questions about the nature of diagonal arguments, their relation to Cantor’s classical example, and also their centrality to hyperstitional thinking. Because these issues have never be systematically explored within the confines of this blog or its obvious precursors, it seems prudent to address them in an exploratory spirit, without rushing to premature conclusions.
No discussion of diagonal argument can bypass the Cantorian example, so I will very briefly rehearse it here, with minimal topical context.
Georg Cantor employs diagonal argument to rigorously consolidate a specific mathematical discovery: that of infinite sets higher than the lowest order of infinity (denoted “aleph-null” coded here “A0”). A0, the smallest transfinite cardinality or lowest of actual infinities, is equal in size to the sets of Rationals, Integers, Naturals and Primes, as well as those of cubes, squares or triangular numbers and (very (very ( ))) many others besides. It is the size of every countable but nontermninating series, each of which can be mapped onto any other, so that – for instance and counter-intuitively – the set of all even numbers is exactly equal to the set of all Naturals, with both equalling A0.
A0 + 1 = A0
2 x A0 = A0.
A0 x A0 = A0. And this is scarcely to begin (since A0 is not only infinitely larger than itself, included within itself infinitely, but this “infinitude” must itself be comprehended recursively (as A0)).
However, enough of that for now, since infinity is only indirectly the matter at stake.
Cantor innovated diagonal argument as a procedure to test denumerability (‘countability’) among infinite series, seeking to demonstrate that a nondenumerable realm of infinities existed above the scale of A0. By proving that even the most exhaustive matrix of countable numbers misses rigorously (if abstractly) identifiable numbers, diagonalism punctures the outer limit of A0, opening it onto vistas beyond. Cantorian diagonalism is an abstract procedure, meaning that, although its concrete execution is impractical, it is evidently realizable in conception and can therefore be considered operative in a domain of pure theory.
Arithmeticians have long been confident that every number can be expressed through an infinite decimal expansion. Perhaps most obviously, this might be a string of zeroes (0 = 0.000…), but a slightly more elaborate example is also available. Since 1/3 x 3 = 0.999..., the arithmetical case for the perfect equality of such an infinite recurrence and its summarization as unity (0.999… = 1) has seemed incontestible (qabbalists must of course remain unconvinced, but that is a matter for another occasion). This equality makes even the most thoroughly domesticated integer equivalent in principle to a ragged-ended fractional series without term. From this it follows that the matrix of an infinite set of cardinality A0 should be considered no greater than the segmentarity of each item in the matrix. Irrespective of modulus, there is a place-value slot at least implicitly available in each of the infinite numbers in an infinite countable series to echo the scale of the series, making the set of elements 2-dimensional (with equal cardinalities for each dimension (macrocosm = microcosm)).
Any segment of the number line has a cardinality of A0, so the series of Rationals 0.000… to 0.999… can be considered an adequate (indeed ((( ) hyper-)extravagantly) ample) map of any denumerable infinity. Selecting this segment technically simplifies the diagonal operation.
Finally, selecting modulus-2 for the demonstration – modulus being an entirely arbitrary aspect of diagonal procedure – minimizes semiotic distraction.
Everything is now in place to execute an abstract diagonalism and make intelligible contact with a higher infinity.
1) Construct the A0 matrix as an infinite series of numbers from 0.000… to 0.111… each with infinite fractional expansion.
2) Manifest the ordinal (compressive) diagonals within the matrix, whereby the nth place of each number correlates to the nth number in the series. Each number thus provides an abstract map or isomorphic (microcosmic) fractalization of the whole.
3) Re-trace the compressive diagonal to systematically produce an anomaly exceeding the denumerated infinity. In the anomalous number, the first digit differs from that of the first number, the second digit differs from that of the second number … the nth digit differs from that of the nth number (through simple alternation in a binary modulus). The resulting diagonal monstrosity must necessarily be distinct from any existing member of the A0 matrix – however comprehensively the matrix has been constructed. Even God – of whatever transcendental sublimity – is incapable of denumerating a set that can resist diagonalization.
Lest the power of this method escape comprehension (an inevitability), permit me to reiterate: Diagonalization methodically produces monsters that elude the recognition of God. This is a matter of perfect mathematical rigour, and thus lies beyond reasonable controversy.
[Whilst CAB’s questions have yet to be seriously addressed, further development of this discussion must be postponed beyond an interval of sleep. To be continued (no pun intended) … For original CAB comments see especially Sore Losers tangents thread 12:11:06 1:24 am and 3:36 am]
Posted by Old Nick at December 11, 2006 03:39 PM | TrackBack