Cantor Confusion
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From: MoeBlee on 20 Oct 2006 17:07 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > I've never seen "potential infinity" or "actual infinity" in any > > > > textbook I've used. > > > > > > So you read not the right books or too few. > > > > 'potentially infinite' and 'actually infinite' are mentioned often in > > philosophy and history of mathematics. But would you please just refer > > to a single textbook of set theory, analysis, or calculus that gives > > mathematical definitions of 'potentially infinite' and 'actually > > infinite'? > > In > fact there are few modern books mentioning the difference. One is > Fraenkel, Abraham A., Levy, Azriel: "Abstract Set Theory" (1976) > p. 6 "the statement lim 1/n = 0 asserts nothing about infinity (as the > ominous sign oo seems to suggest) but is just an abbreviation for the > sentence: 1/n can be made to approach zero as closely as desired by > sufficiently increasing the integer n. In contrast herewith the set of > all integers is infinite (infinitely comprehensive) in a sense which is > "actual" (proper) and not "potential". (It would, however, be a > fundamental mistake to deem this set infinite because the integers 1, > 2, 3, ..., n, ... increase infinitely, or better, indefinitely.)" > > and later: "Thus the conquest of actual infinity may be considered an > expansion of our scientific horizon no less revolutionary than the > Copernican system or than the theory of relativity, or even of quantum > and nuclear physics." Yes, just as I said, the discussion is about the philosophy of mathematics and set theory (and, I should add, about informal concerns and motivations), but there is not, WITHIN the set theory discussed there, a definition of 'actually infinite' and 'potentially infinite'. MoeBlee
From: Virgil on 20 Oct 2006 19:07 In article <1161376969.925821.157640(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > It is true that every set can be well ordered and that any two sets can > be compared. Both are equivalent or one is equivalent to a sequence > (ordered subset) of the other. Without the axiom of choice, or some equivalent, it need NOT be true that every two sets can be compared. It is true that any two well ordered sets can be compared, but without an axiom of choice it need not be the case that every set can be well ordered.
From: Virgil on 20 Oct 2006 19:17 In article <1161377207.110077.302900(a)i3g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Consider a finite path with n edges. This path carries a load of (1 - > (1/2)^n)/(1 - 1/2) own edges. What is a "load" that such a path doesn't it carry a "load" of n edges? Is there some edge which is not completely included in the path running through it? > For n --> oo this yields a load of 2 > edges. It is very simple, it derived from current mathematics, which is > derived from your axioms, therefore it must not further be confused by > retranslating it with your axioms. Until we have some unambiguous definition of "load", it does not derive from anything at all. In a finite binary tree, the number of paths always equals exactly the number of leaf (terminal) nodes. Since an infinite binary tree has no leaf nodes, "Mueckenh" should be arguing that it does not have any paths either. > > The diagonal argument does not contradict that 1=.999... > > No, it is even unaware of this fact, because the necessary convergence > is missing. The diagonal contruction avoids the ambiguity of double representations, like 1.000... = 0.999..., by having a rule which avoids the use of 0 or 1 or 9 in the constructed numbers, but like all endless decimals, it always does converge to a real value.
From: Virgil on 20 Oct 2006 19:26 In article <1161377432.036002.62580(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > Definition (by me): A number which can be constructed like pi, sqrt(2) > or the diagonal of a list is that what I call constructible. If you > dislike that name, you may call these numbers oomflyties. Anyhow that > set is countable. And that set cannt be listed. Therefore the diagonal > proof shows that a set of countable numbers is uncountable. "Mueckenh" mistakes, as usual. A list of reals, suitable for construction of a Cantor diagonal need not be a list of computable numbers, it only needs that the nth number be computable to the nth digit, but not necessarily any further. And for each n, the set of reals computable at least to the nth decimal place, but not necessarily further, is not be countable.
From: Virgil on 20 Oct 2006 19:35
In article <1161377635.539998.278340(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > > The distance between any two edges of one path is infinite? > > > > The distance of any edge from the end of an infinite path is infinite. > > Therefore we have an infinite series with value 2. Such "series" are irrelevant to the issue of whether there is an injection from the set of edges to the set of paths or vice versa. Every edge is the terminal edge of a finite path, so that there is a natural bijection between terminal edges and terminating paths. But given any terminating path, including any single edge, there are uncountably many non-terminating paths containing it, just as there are uncountably many non-terminating binary strings.. |