Cantor Confusion
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From: David Marcus on 23 Oct 2006 22:54 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > 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'. > > Does the study of formal languages really make incapable of > understanding plain text? What is written above means: "INFINITY" IN > SET THEORY IS ALWAYS "ACTUAL INFINITY". Exactly. There is only one kind of infinity in modern set theory (and modern mathematics). We no longer distinguish between "potential infinity" and "actual infinity. These distinct notions were important historically until the concept of infinity became better understood. > This could be translated as > completed or finished infinity but usually is not, because that would > deter new students from studying this matter. And most of them never > get a grasp of that fact. -- David Marcus
From: David Marcus on 23 Oct 2006 23:02 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > Binary Tree > > Unfortunately, it was described in a way that I can't understand it. A > > wild guess on my part is that you mean to set up a correspondence > > between edges and sets of paths. > > I am sorry, but if you need a wild guess to understand this text, then > we should better finish discussion. Observe just how the discussion > runs with all those who understood it, like Han, William, jpale. Han doesn't understand it (although he probably thinks he does). William and jpale simply pick the mathematically meaningful statement that is closest to what you write and go from there. I could do that too, but I suspect that what you are actually thinking is rather far from their guesses. I'd rather wait until you say something coherent before I comment on it. -- David Marcus
From: David Marcus on 23 Oct 2006 23:14 Dik T. Winter wrote: > In article <1161517636.934369.301190(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > > > > > It proclaims nothing of the sort. You want to read more in notation than > > > is present. > > > > You want to read less than present and necessary. > > lim{k --> oo} a_k = pi means omega reached. Or would you state that pi > > belongs to Q like every a_k for natural k? > > Of course not, and of course I do not state that oo is reached. Stating > (for the first limit above) that omega is reached is tantamount to stating > that there is an n such that 1/n = 0. There *is* no such n. And in the > second statement it would mean that there is a k such that a_k = pi. > Both are nonsense. What do you think Mueckenheim means by "omega reached"? I really have no clue what he means. It really is remarkable how much stuff he makes up. Obviously, he couldn't have gotten it from any math book. -- David Marcus
From: David Marcus on 23 Oct 2006 23:17 Han de Bruijn wrote: > Ha! Mathematicians can't even define their most frequently used symbol, > which is the equality ' = '. Why do you say that? -- David Marcus
From: David Marcus on 23 Oct 2006 23:21
MoeBlee wrote: > Han de Bruijn wrote: > > > x = y :<-> Az(z e x <-> z e y) > > > > Of course. Because all members are sets. But I think this is an infinite > > recursion with the equality definition. Does it end somewhere? > > "Infinite recursion with the equality definition". You're amazing in > your sheer ignorance and arrogance to opine on what you are completely > ignorant. > > Read chapter one of virtually any textbook in set theory. I think Han would need to start with a book on mathematical logic. -- David Marcus |