Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 12 Dec 2006 08:41 On 12/11/2006 10:07 PM, Virgil wrote: > In article <457D7067.7030006(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 8:02 PM, David Marcus wrote: >> > Eckard Blumschein wrote: > >> >> I did not consider measures. Let's get concrete. Are there more naturals >> >> than odd naturals? This question could easily be answered if there was a >> >> measure of size. >> > >> > It can easily be answered once you say what you mean by "more". Why do >> > you think that common English words have unambiguous mathematical >> > meanings? >> >> More is just inappropriate as to describe something infinite. There are >> not more naturals than rationals. There are not equaly many of them, >> there are not less naturals than rationals. > > > That EB chooses not to consider measures of sizes of sets which can be > applied to Dedekind infinite sets does not mean that no one else is > allowed to do so. Not even Dedekind himself was legitimated to do so. He even confessed that. > The Cantor definition of cardinality, at least in ZFC or NBG, Not a single axiom in ZFC provides Cantors definition of cardinality. EoD is > well-defined and self-consistent, and as such is a valid measure of set > size.
From: Eckard Blumschein on 12 Dec 2006 08:43 On 12/11/2006 10:09 PM, Virgil wrote: > In article <457D70D8.9080902(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > >> Can you reveal just one illusion of mine? > > That you know more about mathematics than the thousands upon thousands > of those who have studied it much harder and longer then you have done. Maybe, it needs some distance in order to overlook large things.
From: Eckard Blumschein on 12 Dec 2006 08:47 On 12/11/2006 6:32 PM, Bob Kolker wrote: > Eckard Blumschein wrote:> >> No. Cantor again merely showed by contradiction that the power set is >> not countable. The reason is: Already the entity of all natural numbers >> is an uncountable fiction. > > By definition, the set of integers is countable. I know this definition, and it is even quite plausible if one only takes the ordinary (Archimedean) point of view. > A countable infinite > set is a set which can be put in one to one correspondence with the set > of integers. I cannot imagine any reason why an intelligent person may reiterate this. > > Bob Kolker >
From: Eckard Blumschein on 12 Dec 2006 08:50 On 12/11/2006 10:16 PM, Virgil wrote: >> No. Cantor again merely showed by contradiction that the power set is >> not countable. The reason is: Already the entity of all natural numbers >> is an uncountable fiction. > > Not in ZFC or NBG. Do they claim that?
From: Eckard Blumschein on 12 Dec 2006 09:08
On 12/11/2006 10:19 PM, Virgil wrote: > In article <457D75A0.1060201(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/6/2006 9:10 PM, Virgil wrote: >> > 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. >> >> ??? What paper? Do you mean Cauchy's zig-zag diagonalization? > > As anyone but a fool would have known I meant: > > http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof DA means diagonal argument. You did not refer to the first diagonal argument but to an argument that leans on Dedekind's illusion. This early attempt did not get famous. |