Galileo's Paradox
in [Math]
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From: Virgil on 11 Dec 2006 16:35 In article <457D8056.6080909(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 8:01 PM, Bob Kolker wrote: > > Eckard Blumschein wrote: > >> > >> I did not say this. Please quote me carefully. > >> The set of existing Dedekind cuts is finite. The set of feasible cuts is > >> countable. > > > > The set of Dedikind cuts is not finite. There is a Dedikind cut for each > > rational number for starters. Why do you say such stupid things? > > Because Dedekind himself admitted that he cannot provide any evidence > substantiating his basic assumption. There is not a Dedekind cut for > each rational number. A Dedekind cut of the rationals, Q, is a partition of the rationals into two sets, L and U with the properties: (1) for every x in L and y in U, l < u (2) for every x in L and y in Q, if y < x then y in L (3) for every x in U and y in Q, if y > x then y in U. In addition, one arbitrarily chooses either that (a) the GLB of U, if in Q is in U, OR (b) the GLB of U, if in Q is in L. Given any rational, it is easy to construct the set of all larger rationals, and by its complement the set of all smaller ones plus the number itself, using option (b) above. .. How is such a pair of sets NOT a Dedekind cut for that rational?
From: Virgil on 11 Dec 2006 16:43 In article <457D846A.3080208(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 1:20 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> On 12/6/2006 5:19 AM, David Marcus wrote: > >> > Eckard Blumschein wrote: > >> > >> >> >> Why do you think that the diagonal argument defines the reals? > >> >> > >> >> You all know that DA2 shows by contradiction that real numbers are > >> >> uncountable. I carefully read how Cantor made sure that the numbers > >> >> under test are real numbers. He did not use Dedekind cuts, nested > >> >> intervals or anything else. > >> > > >> > Well, of course he did't use Dedeking cuts, etc. > >> > >> Cantor explained why he preferred his own definition. > >> Read how he made sure that the numbers under test actually were real > >> numbers. > > > > I just got through telling you that it is irrelevant how the real > > numbers are defined. All that matters is that they are a complete > > ordered field. > > Cantor himself has shown with DA2 that they are not such field. False! EB apparently has no idea what DA2 says. What part of DA2 does EB allege shows that the reals are "not a complete ordered field"? > > >> The set of existing Dedekind cuts is finite. The set of feasible cuts is > >> countable. > > > > There you go again, making up words ("existing", "feasible"). In > > mathematics, we are allowed to make up words, but only if we define > > them. > > > > You said that "sofar nobody was able to show that the numbers allegedly > > defined by Dedekind cuts are uncountable". The natural translation of > > this into normal language is "no one has showed that the set of Dedekind > > cuts is uncountable". Clearly, this statement is false. So, you now say > > that this is not what you meant. > > I clearly said that Dedekind cuts did not define any new number. He just > declared the irrational sqrt(2) an irrational number. > Where can I find "the set of Dedekind cuts"? In many texts on foundations of mathematics. > > Fine. Try saying what you mean in > > normal mathematical English, so we have a hope of understanding it. > > Who does not intend to understand may hide behind formalisms. It is those who do not intend to be understood, like EB, who hide behind words and phrases that they will not clarify.
From: Virgil on 11 Dec 2006 16:45 In article <457D87C1.3050805(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 8:48 PM, Virgil wrote: > > In article <4576EB55.4040803(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/5/2006 11:41 PM, Virgil wrote: > >> > In article <457593EB.9030809(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> > >> >> Rationals are p/q. This system cannot be improved by adding genuine > >> >> (i.e. rational) numbers. One can merely move to the fictitious > >> >> continuous alternative. > >> > > >> > All numbers are equally genuine in any meaning of "genuine" other than > >> > EB's improper meaning of "irrational". > >> > >> Kronecker was correct in that irrationals are no genunine numbers. > > > > That may have been thought to be the case in the ninteenth century, but > > this is the twenty first century. > > > > > > Kronecker died in 1891. Mathematics has progressed since then. > > Concerning its putative fundamentals obviously with no avail. On the > contrary. Sounds like sour rapes, form someone who is still stuck in the 19th century.
From: Virgil on 11 Dec 2006 16:56 In article <457D8AC9.5060306(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/7/2006 1:27 AM, David Marcus wrote: > >> And which role has been envisioned for aleph_1? > > > > Kind of a silly question. aleph_1 is the first cardinal after aleph_0. > > That's its "role". > > So far I was told aleph_1 means the continuum of the reals. Given the continuum hypothesis, it is. But that is not how it is defined, and absent the continuum hypothesis, one cannot say for certain.
From: Virgil on 11 Dec 2006 17:00
In article <457d945e(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Tony Orlow wrote: > >> Of course, he has no need for omega. It's illegitimate schlock, like I > >> said. > > > > Do you really think Robinson is talking about ordinals? > > > > Did you even read what I said? Of course he's not talking about omega > and the ordinals, he's talking about a sensible approach to the infinite > and infinitesimal for a change. Sheesh! Then why do you conflate the two distinct issues, TO? If Robinson is not dealing with ordinals, then what he has to say is not relevant to ordinals. |