Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 6 Dec 2006 11:24 On 12/5/2006 11:50 PM, Virgil wrote: > In article <45759AD4.3040505(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/4/2006 9:58 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote: >> > >> >> No wonder. Exactly this selfdelusion was the intention of dedekind. >> > >> > 'Twas no delusion. >> > >> > The Dedikind cut defining the square root of 2 is just as well defined >> > as the successor to the integer 2. >> > >> > Bob Kolker >> >> Dedekind's cuts did not create any new number. The square root of two is >> still what it was before. > > The cuts provided a construction of sets having the properties of those > numbers, i.e., a model of the reals within set theory. They supported the illusion that there are genuine numbers which complete the rational ones. Actually there are merely indirect definitions of irrational positions between the rationals.
From: Bob Kolker on 6 Dec 2006 11:27 Eckard Blumschein wrote: > > They supported the illusion that there are genuine numbers which > complete the rational ones. Actually there are merely indirect > definitions of irrational positions between the rationals. There are no "genuine" numbers. There never were any "genuine" numbers. Any numbers that there are live only in our imaginations and are no more genuine than unicorns and fire breathing dragons. To characterize a number as "not genuine" is to say nothing beyond that the number is a number. All numbers are products of the imagination. Bob Kolker
From: Eckard Blumschein on 6 Dec 2006 11:28 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.
From: Eckard Blumschein on 6 Dec 2006 11:46 On 12/5/2006 11:55 PM, Virgil wrote: > In article <45759C29.7040003(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/4/2006 9:54 PM, Bob Kolker wrote: >> > Eckard Blumschein wrote: >> >> >> >> You are right if you consider each time the complete set. Neither the >> >> integers not the rationals are actually complete. So both complete sets >> > >> > Really? Tell me an integer or rational that is not in the set of >> > integers or rationals? Which ones did we miss? >> > >> > Bob Kolker >> >> Read carefully: I did not write set of integers, I just wrote the >> integers. The complete set of integers is something quite different. > > When you speak of "the integers" do you mean all of them on not. > If you do mean "all, then the set of all of them is all of them. Maybe the answer is as pivotal as tricky. There is the ordinary Archimedean point of view: One may count 1, 2, 3, .... without any limit (potentially) Sometimes, I need the sum of a converging series indexed 1, 2, 3, ... I this case, the contribution of the elements is declining with growing number. Theoretically I would need (actually) all nines in order to equate 0.999... with 1. This is the opposite (fictitious) point of view. I would like to avoid the term set because Cantor's defition was ambiguous. It claimed to see the fictitious entity of all element at a time with each single element. This ambivalence caused obvious antinomies. In axiomatic set theory the possible ambivalence still persists, but it cannot be made obvious.
From: Mike Kelly on 6 Dec 2006 11:51
Bob Kolker wrote: > Mike Kelly wrote: > > > > > > > The set of finite binary strings is a subset of the set of finite > > decimal strings. > > > > Then b) precludes them being the same size. > > > > They are also both the same size as the set of natural numbers. > > > > Thus they are the same size as each other. > > > > Contradiction. > > What contradiction? Item b above is simly a mapping of the binary > sequences (not beginning with 0) to a subset of the integers. Where is > the problem? > > Bob Kolker By "b)" I was referring to the statement "(b) proper subsets are smaller than their supersets " that was made several posts earlier in the thread. SixLetters doesn't see why this leads to a contradiction. I tried to explain it. -- mike. |