3/12/2023 0 Comments The infinitesimals![]() ![]() From the modern point of view, Cantor is conflating cardinality with measure, but then the modern concept of magnitude isn't Cantor's, just as the modern concept of geometry isn't Kant's. With concepts like that, to each their own. It is hard to argue with Cantor's "concept of a linear magnitude", which excludes infenitesimals, just as it is with Kant's "pure intuition of space", which excludes multiple parallels. As Moore shows, this is essentially because Cantor's ordinals only allow well-ordered concatenations. In other words, one can not produce a finite magnitude out of infinitesimals, even concatenated transfinitely many times. In the passage, Cantor's issue seems to be that the infinitesimals, traditionally intuited as the "inverses" of infinities, can not be the "inverses" of his transfinite numbers, which, in the time honored tradition, he saw as the only "true" infinities. " in none of its incarnations is the argument particularly easy to follow, and though there are resemblances among the three versions it is not even clear that they are in fact versions of a single argument". ![]() Peano in 1892, and Russell in 1903 gave their variations on the theme, but, according to Moore, See a modern reconstruction of the argument, and how it fails for infinitesimals of the non-standard analysis, in Moore's Cantorian Argument Against Infinitesimals. Hence, the so-called Archimedean Axiom is not an axiom at all but a proposition which follows with logical necessity from the concept of linear magnitude". But then the supposition made contradicts the concept of a linear magnitude, which is such that every linear magnitude must be thought of as an integrated part of other ones, and in particular of finite ones. But this means that ζ cannot be made finite by any actually infinite multiplication of any power, and hence surely cannot be an element of finite magnitudes. ν is less than every finite magnitude however small, where ν is an arbitrarily great transfinite ordinal number (i.e., cardinal or type of a well-ordered set) from any arbitrarily high number class.n, for every finite whole number n however great is smaller than unity, and then prove, from the concept of a linear magnitude and with the help of certain propositions from transfinite number theory, that then ζ." I begin from the supposition of a linear magnitude ζ which is so small that its product by n, ζ Here is Cantor in his own words (from his influential 1887 letter to Weierstrass): ![]()
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