Galileo's Paradox
in [Math]
|
From: Virgil on 6 Dec 2006 14:57 In article <4576EF9A.80408(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/5/2006 11:51 PM, Virgil wrote: > > In article <45759B2C.1030500(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/4/2006 9:56 PM, Bob Kolker wrote: > >> > Eckard Blumschein wrote: > >> > > >> >> > >> >> > >> >> 2*oo is not larger than oo. Infinity is not a quantum but a quality. > >> > > >> > But aleph-0 is a quantity. > >> > > >> > Bob Kolker > >> > >> > >> To those who belive in the usefulness of that illusion. > > > > Despite the naysaying of those like EB who have the illusion of their > > beliefs. > > The neys will have it on condition they do not adhere any illusory belief. If EB is any representative, then his adhering to as many illusions as he does represents their utter faliure.
From: Virgil on 6 Dec 2006 15:08 In article <4576F7E0.3090102(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 12:01 AM, Virgil wrote: > > In article <45759E91.309(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > >> Yes. The power set algorithm does not change what mathematicians still > >> used to call cardinality. 2^oo=oo. > > > > What "algorithm does EB refer to? Power sets come from axioms, not > > algorithms. > > 2^aleph_0 does not come from axioms. Look for Pascal's triangle. > http://en.wikipedia.org/wiki/ZFC#The_axioms 8) Axiom of power set: For any set x there is a set y that contains every subset of x. > > > > And oo is NaN, 2^oo has no meaning. > > You are a knowing-all. oo is, at best, ambiguous. Aleph_0 is not. > > > >> >> In case of an infinite set, there are nor all elements available. > >> > > >> > "Available"? > >> > >> Yes. You cannot apply the algorithm until you have all numbers. > > > > What "algorithm? > > For calculating binominal coefficients. What does one need that for in accepting the axiom of the power set? > >> >> Fictions are uncountable. > >> > > >> > "Uncountable"? > > > > What do you mean by the "uncountability" of one element? How can one > > element sometimes be countable and sometimes not? > > Your question is justified. In order to be part of a counted or at lest > countable plurality, each element has to be discrete and addressable. So > the number 1 may belong to a finite set as well as to a cuntable > infinite set. However 1.000... with perfectly oo much of significant > nils is not immediately a countable element. If it can exist in isolation, as you seem to be assuming by referring to it, then it can be counted in isolation as one object. > Fortunately, I was never trained in set-quasi-religion. No mathematicians have been either, but mathematicians also have not been trained in your anti-set-actual-religion either.
From: Virgil on 6 Dec 2006 15:10 In article <4576F816.5060809(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 12:03 AM, Virgil wrote: > > DA2 does not define anything. But if they were to b e defined by a > > theorem, they would already be defined by what I will call DA1. > > DA1 dealt with rationals. By DA1 I was referring to Cantor's first proof of the uncountability of the reals, and it deals with far more than the rationals.
From: Virgil on 6 Dec 2006 15:16 In article <4576F944.2000607(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 12:06 AM, Virgil wrote: > > In article <4575B119.2050709(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/4/2006 11:32 AM, Bob Kolker wrote: > >> > Eckard Blumschein wrote: > >> > > >> >> Notice, there is not even a valid definition of a set which includes > >> >> infinite sets. Cantor's definition has been declared untennable for > >> >> decades. > >> > > >> > That is simply not so. For example the set of integers. There is is. > >> > >> Perhaps, you are honestly bold. Believe me that Fraenkel admitted that > >> Cantor's definition is untennable. > > > > So Fraenkel is wrong! > > This time definitely not. That is your opinion, but since yours is the opinion of a non-mathematician, its relevance to anything mathematical is negligible. > > > >> The question is e.g. in case of the > >> naturals whether they are considered one by one or altogether like an > >> entity. While a set is usually imagined like something set for good, > >> this point of view is unrealistic. > > > > It is essential. And since all numbers are unrealistic in that they are > > only imagined, that is no handicap. > > Unrealistic means self-contradictory. I fail to find self-contradictory anywhere in the following definition of "unrealistic". Can EB give a citation which does include that meaning? <quote> Proximity/Merriam-Webster U.S. English Thesaurus 1 meaning(s) for Unrealistic 1. (adj) incapable of dealing prudently with practical matters (synonym) Impractical, Ivory-tower, Ivory-towered, Ivory-towerish, Nonrealistic, Unpractical, Viewy (related) Idealistic, Otherworldly, Quixotic, Romantic, Starry-eyed, Visionary (contrast) Commonsensible, Commonsensical, Realistic, Sensible, Worldly-wise (antonym) Practical <unquote>
From: Virgil on 6 Dec 2006 15:19
In article <4576FA19.1010208(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 12:08 AM, Virgil wrote: > > In article <4575B16C.6050508(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/4/2006 9:47 AM, Virgil wrote: > >> > In article <4573D4DA.4040709(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> On 12/3/2006 8:22 PM, cbrown(a)cbrownsystems.com wrote: > >> >> > Tony Orlow wrote: > >> >> > > >> >> > Well, you used the term "set" four times in your above definition of > >> >> > what we mean by a "set". That's why I said "this begs the question, > >> >> > what do we mean, exactly, by a set of properties?". > >> >> > > >> >> > There's something that we intuitively seem to think of as a "set"; but > >> >> > unless such a thing is carefully defined, we end up with the > >> >> > contradictions of naive set theory: > >> >> > > >> >> > http://en.wikipedia.org/wiki/Naive_set_theory > >> >> > >> >> Is it really justified to blame an allegedly insufficient definition of > >> >> the term set for obvious antinomies of set theory? > >> > > >> > > >> > As "set" and "is a member of" are primitives in axiomatic set theory, > >> > any "definition" of them is outside of set theory and irrelevant to it. > >> > >> Yes. The problems are shifted outside. > > > > Nothing outside of an axiom system can be a problem inside that system. > > As long as one merely intends to perform sandpit-mathematics. If by that EB means what is generally called pure mathematics, as contrasted with applied mathematics, that is what pure mathematicians do. They create the mathematics that may eventually trickle down to engineers like EB. |