Cantor Confusion
in [Math]
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From: Virgil on 20 Feb 2007 16:53 In article <1171981192.821547.278230(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 19 Feb., 15:19, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1171890516.474583.210...(a)p10g2000cwp.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > On 19 Feb., 01:31, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > > > > > Can you tell me a form of set theory where 0 is *not* the > > > > > > > > first > > > > > > > > ordinal or cardinal number? If so, how many elements does > > > > > > > > the > > > > > > > > empty set have in such a system? > > ... > > > > > In all set theory 0 is called the first ordinal number, but in fact > > > > > it > > > > > is the zeroth one. Why do you start counting ordinals with 0 but > > > > > start > > > > > counting ordinally with 1? > > > > > > > > So your "natural number" above was a red herring? A human being in > > > > its > > > > first year has the age 0. > > > > > > Before completing his first year, the being has the number of years > > > which comes before 1. > > > The first number drawn in lottery may be the 7. > > > > > > The first is that ordinal number which we start with. > > > > So your statement above: "In all set theory 0 is called the first ordinal > > number, but in fact it is the zeroth one" was nonsense? > > No. The first is the first. The first is not the zeroest. It is > nonsense to start counting by zero as is done in set theory. > > > > > > > Indeed. There is only a definition of set. > > > > > > Not even that. > > > > <http://en.wikipedia.org/wiki/Set> > > LOL. Now, good old Cantor is good enough? He objected to numbers which > are not finitely definable. Every user of his definition must agree > with him because uncomputable reals are not "distinct objects of our > perception or of our thought". By what means can two uncomputable > numbers be distinguished? > > No, Dik. Either there is no definition of a set, then set theory may > continue to exist as a matter of dreamwork. > Or there is a firm definition of a set, then set theory crashes. Various set theories work quite well as long as one does not require them to satisfy WM's extra axioms. When one demands a system satisfy more axioms than it is designed to handle, as WM keeps doing, only then are there problems.
From: Virgil on 20 Feb 2007 16:54 In article <1171980305.458947.244550(a)p10g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > On 19 Feb., 15:06, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > In article <1171890209.831371.70...(a)h3g2000cwc.googlegroups.com> > > mueck...(a)rz.fh-augsburg.de writes: > > > > > On 19 Feb., 01:02, "Dik T. Winter" <Dik.Win...(a)cwi.nl> wrote: > > ... > > > > Not in my opinion, and I think not in mathematics. If the meaning of > > > > 3 is "three of something" how than do we calculate "three of > > > > something" > > > > times "three of something"? Or "nine of something" divided by > > > > "three of something"? Or more concrete, if I divide nine apples by > > > > three apples what is the result? > > > > > > three of something, where the "something" here means the unit. > > > > What unit? > > I > > One of something. Like one orange, when one is working with apples? Makes as much sense as WM ever does.
From: MoeBlee on 20 Feb 2007 17:04 On Feb 20, 1:45 pm, mueck...(a)rz.fh-augsburg.de wrote: > On 20 Feb., 20:37, "MoeBlee" <jazzm...(a)hotmail.com> wrote: > > > On Feb 17, 12:52 am, mueck...(a)rz.fh-augsburg.de wrote: > > > > 0 may be the first (or better the zeroest) ordinal or cardinal number > > > (if you wish to have the empty set in the theory). Nevertheless it is > > > not the first natural number and not a natural number at all. > > > Natural numbers are counting the elements of natural sets, i.e., of > > > sets which exist in reality (in nature, as Cantor woud have said). > > > So if we call 0 and the positive whole numbers 'mamtural numbers' > > It is enough to call them ordinal numbers. It will not change > mathematics but will help to avoid confusion about the term natural > number. When I say that every set of even natural numbers contains > numbers larger than its cardinality, then I am right with my favourite > meaning of natural numbers but wrong with Bourbaki's. Different authors use 'natural numbers' in different ways, so that some authors regard 0 as a natural number and some authors regard 1 as the least natural numer. However, it seems to me that 0 as a natural number has become a convention very much more widely held than the convention that 1 is the least natural number. Especially in a thread such as this, you are talking with people almost all of whom take the convention that 0 is a natural number. In this instance, there is no gain in clarity or mathematical knowledge by insisting that a different convention be used. MoeBlee
From: Franziska Neugebauer on 20 Feb 2007 17:04 mueckenh(a)rz.fh-augsburg.de wrote: > Note that the number of natural numbers can be mapped one-to-one on > the natural numbers: > > 1,2,3,...,n <---> n Applied brace elimination? > It is easy to see that natural nunmbers alone do not bring together an > infinite number of numbers. M�ckenheim Axiom 1 X is not finite -> there must be an x in X which is infinite. F. N. -- xyz
From: G. Frege on 20 Feb 2007 17:02
On 20 Feb 2007 13:40:47 -0800, mueckenh(a)rz.fh-augsburg.de wrote: > > Note that the number of natural numbers can be mapped one-to-one on > the natural numbers: > > 1,2,3,...,n <---> n > > It is easy to see that natural numbers alone do not bring together an > infinite number of numbers. > Since WM claims this to be "easy to see" (for him), it must be false. And right, it is false. Since there are infinitely many natural numbers, they in fact "bring together an infinite number of numbers". With other words, the set of _all_ natural numbers is infinite. F. -- E-mail: info<at>simple-line<dot>de |