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From: Arthur Fischer on 13 Apr 2005 12:56 Eckard Blumschein wrote: > According to my reasoning, definition of the first element does not at > all make the reals a well-ordered set. The real problem is the > impossiblitity to numerically distinguish any real number from its > immediate successor. I consider a choice function inside the reals a > self-requiring assumption. You seem to agree that rationals are countable, while dithering on the meaning of the uncountability of the reals. Part of your arguments seem to be about the inability to precisely define a well-orderring on the set of real numbers which somehow agrees with the natural ordering; and thus leads you to ask "what is the successor of the real number x?" However, this objection can be immediately turned on its head when I ask you what the successor of 1/2 is in the rational numbers. __ Arthur
From: Eckard Blumschein on 13 Apr 2005 13:33 On 4/13/2005 3:33 PM, Matt Gutting wrote: > Okay, but I was intending to be able to refer both to finite and to infinite > sets. The statements I made, although they didn't specifically refer to finite > or to infinite sets, are applicable to both. On costs of violation to basic principles. >>>results as a quantitative comparison does for those sets to which "quantitative >>>comparison" applies. Having found such a comparison, >> >> >> I wonder if you really found it. Concerning infinite sets I am only >> aware of the possiblity do decide whether or not there is a bijection >> being synonymous to countable. > > Very specifically, here is my method of comparison: Given two (for the moment, > let's say non-empty) sets A and B, if one can find an injection, but not a > surjection, from A to B, say that A "comes before" B or that A is "smaller > than" B. Similarly, if one can find an injection, but not a surjection, from B > to A, say that A "comes after" B or that A is "bigger than" B. Finally, if > one can find both an injection and a surjection (that is, a bijection) between > A and B, say that A "occupies the same position as" B or that A is "as big as" > B. > > (I put "smaller than","bigger than", and "as big as" in quotes in the paragraph > above to make it clear that these concepts are not to be literally interpreted > in the usual sense. This is also why I provided equivalent phrasings not as > laden with specifically mathematical meaning.) > > Since injections and surjections can be found between infinite sets as well as > between finite ones, Are you sure about that? > To extend some of Will Twentyman's work here, the existence of a bijection > between sets demonstrates that the two sets occupy the same position in an > order of cardinalities. The lack of a bijection indicates that the two sets > are in different positions in this order. One can find two infinite sets which > occupy different positions in this order. I will relpy tomorrow. >> Just an example. Children at school must not be taught Cantor's nonsense >> infinite whole numbers. There are no infinite numbers. > > No one is saying that there are numbers which are infinite. Except for Cantor. > There are > cardinalities that don't describe finite sets. And there are different > cardinalities which describe different infinite sets. Originally Cantor replaced oo by omege. Later on he veiled the traces of his thinking. Eckard
From: Eckard Blumschein on 13 Apr 2005 13:35 On 4/13/2005 3:35 PM, Matt Gutting wrote: > That's what keeps me reading, anyway. Hopefully reading of M280, too. E.
From: Eckard Blumschein on 13 Apr 2005 13:42 On 4/13/2005 5:53 PM, worldsofsolution(a)yahoo.com wrote: > There is something I've been wondering about cantor's proof: the > decimal number generated to prove the contradiction, was taken to be a > real number. There is a tacit assumption that all decimal numbers > represent a real number. Does that not require a proof? I apologize for not even reading your contributions. I stumbled about the same question and came to the conclusion that inside the system of real numbers all embedded natural and rational ones require an infinite sequence of numerals in order to be addressed. In other words, they are loosing their identity too. E.
From: Arthur Fischer on 13 Apr 2005 13:27
Eckard Blumschein wrote: > On 4/13/2005 2:22 PM, Arthur Fischer wrote: > >>Eckard: >> >>Just out of curiosity, could you provide your definitions for the >>following concepts: >> >>- finite set >>- infinite set >>- countable set >>- uncountable set >>- non-countable set >>- enlarging a set >> >> >>Of course, mathematically precise definitions would be preferable, and >>dictionaries do not, in general, provide for such definitions. > > > Regrettably my time is limited. The pertaining definitions are easily > available except for the last one. I would just like to try and comment > on "enlarging a set" in case of infinite sets: This simply does not work. If you do not say how one "enlarges" any set, how can you hope to have any discussion here? This seems to be one of your important points about the nature of "infinity", but yet you keep all of your intuitions about this concept to yourself. Many people here will explicitly define for you concepts such as "having larger cardinality", and give you various facts and theorems based on these concepts. You seem to like to take dictionary definitions that do not match up to mathematical terminology, and then say how _mathematicians_ used their own particular language incorrectly. This is a solipsism of the highest order. __ Arthur |