Musatov responds to WM on Cantor: intelligence, real , judge for yourself: Cantor’s idea seems to me to assume because two sets converge to infinity the nature of infinity and the number of elements in each set must become equal.
Jun 17, 2010 5:37 AM
On Jun 17, 2:06 am, WM <mueck...@rz.fh-augsburg.de> wrote: > On 17 Jun., 09:54, Virgil <Vir...@home.esc> wrote: > > > In article <4c19cd2c$0$316$afc38...@news.optusnet.com.au>, > > "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > > > Cantor's proof applied to computable numbers proves you cannot form a > > > computable list of computable numbers. Cantor's proof applied to Reals > > > proves you cannot form a computable list of Reals. > > Cantors proof is nonsense from the beginning, because a real number > can never be defined by an infinite sequence alone. A definition > defines something, but an infinite sequence does not define a number > before the last digit is known. > > > > > To be correct, there is no computable list of ALL of the computable > > numbers, even though the set of computable numbers is e countable, but > > there are lots of possible computable lists of computable numbers. > > And there are lots of lists of more than all computable numbers, > namely lists of all finite expressions. > > Regards, WM To discuss infinities further as regard Cantor?s infinite sets 2; where it is postulated the set of whole numbers and the set of even numbers is equal because there is a one to one correspondence as; Whole numbers 1 2 3 4 5 (set 1) etc. Even numbers 2 4 6 8 10 (set 2)
For any interval of, for example, 1 (0-1) there are an infinite number of points and for 2 (0-2) there are 2x the number of points in 0-1 or; As Ratios 0-1 infinite number of points 0-2 2 x infinite number of points 0-3 3 x infinite number of points etc.
Now if Cantor?s sets concern the number of numbers (symbols themselves) in each set, then for a given interval, like 1-10, they are not equal as there are 10 symbols (numbers) in set 1 and only 5 in set 2 (see also Addendum 1). Therefore extending the interval to infinity there are 2x the number of elements (symbols/numbers) in set 1 and only 5 in set 23. As concerns not number of numbers but units for each correspondence in number (as each even number is twice as large as its corresponding whole number). It might be considered there are 2x as many even numbers, as many whole numbers per interval and 2x as many even numbers per correspondence, therefore they cancel out to a 1:1 ratio. However both are wrong as it neglects the need to add the units over an equivalent interval. Doing so gives a fluctuating answer for each interval chosen, which (intuitively figured) tends toward a 2:1 ratio (as previously figured) of whole numbers: even numbers, as each set continues infinitely (see also Addendum 1). CHART I-1 Examples of infinite ratios
Likewise with other sets a ratio of their infinite quantities is arrived at by these methods. For example: Whole numbers : multiples of 3 Whole numbers added 1+2+3= =4+5+6= +7+8+9= Multiples of three added 0+3= +6= +9=
Or like whole numbers: perfect squares x a billion. Breaking this down into: Whole numbers: a billion Whole numbers added 1+2+3 ? 109 = 109
The question becomes one of how is a ?set? defined. In the set of whole numbers the first element of the set is 1 and represents one complete unit. 2 represents two units, etc. Now in the set 1-4, does this mean only the #4 (four units) or does it mean the value of each element added? I am talking about apples at 1, there is 1 apple, at 2, 2 apples, etc? The set 1-4 implies to me all apples from each element, not just the #4, for if we wanted to talk about the number 4 we would just say, ?the number 4?, but here we are saying ?the set 1 to 4?. Likewise the set of all even numbers would exclude all odd number elements or at 2, 2 units of 1, 4, 4 units of 1, etc. Therefore there would be 6 units in the set 1-4 of even numbers. How are sets to be compared? To the point, how are infinite sets to be compared, regardless of whether we compare the quantities or just the number of symbols? Either way I propose this idea: Infinite sets must be compared in a like manner as finite sets. For if we compare infinite and finite sets differently, then the consistency of mathematical operations is compromised. If we apply Cantor?s hypothesis to this dictum as regards finite sets, we may compare single elements of the same or different quantities. Or we may compare finite groups or ranges of the same number of elements, of equal or varying quantities. But we may not compare groups, or ranges of numbers with a different numbers of elements. For example, if we compare the set 1-4 of whole numbers and the set 1-4 of even numbers, by Cantor?s hypothesis there ?is? a one-to-one correspondence, therefore for each element in the first set there must be an element in the second set, which there is not. To compare such finite sets is actually excluded by Cantor?s definition (which I claim is incorrect). Cantor?s definition is a self-defining one is by definition a need to compare element to element, thereby Cator excludes the possibility for anything other than a one-to-one ratio for the ?total? set. Approaching the problem, first for finite sets compare element to element, same or different. Group to group, same or different, and special case, range to range, same or different (for example for two sets of whole numbers compare a different range 10 to 20 with the range 5 to 35; or the same ranges 10 to 20 and 10 to 20). Compare the set of whole numbers 1 to infinity and even numbers 1 to infinity: in comparing finite amounts variable sets can be compared, but no such liberty exists in infinite sets, as by definition they contain total amounts. These must then be both groups and ranges, and since the expressions 1 to infinity are equivalent, they must be equivalent ranges. In the finite ranges, 1 to 10 and 1 to 10 of even numbers, there are 2x the number of elements in set 1 as in set 2. Therefore in the infinite ranges there must be a like ratio is the ratio of elements of two infinite sets is equal to the ration of any finite equivalent range of each set, as the definition of infinite sets is of an infinite range, and infinite sets must be compared in a like manner as finite sets. The quantities are not equal over equivalent intervals because set 2 is a subset of set 1 and the progressive nature of the odd numbers to be counted are left in set 1. But this quantity can be figured easily, as the ratio of the infinite quantities is the same as the ratio of elements, therefore; Ratio of elements set 1 : set 2 is 2 : 1 Ratio of quantities set 1 : set 2 is 2 : 1 Likewise a definition for all ratios of infinite sets is: The ratio of elements (number of symbols) of any two infinite sets is the same as the ratio among any finite subsets of equivalent intervals. The ratio of quantities of any two infinite sets is equal to the ratio of the elements.
COMMENTARY To contrast my conception to my perception of Cantor?s conception of the infinite. Cantor?s idea seems to me to assume because two sets converge to infinity the nature of infinity and the number of elements in each set must become equal. Requiring one to one ?mapping? anything other than a one to one correspondence is excluded. My conception is more as infinite quantities are not mathematically expressible, in the sense we can only conceive of infinity as something consisting forever. You start with something finite and then multiply or divide it forever. In this way two infinite sets are not equal but, as shown in they can be ratio-ed by the finite expression defines the set, but both extend infinitely. Cantor?s reversal of conceptions, for some sets (non-d enumerable) uses conception #2. This is a contradiction to conception #1, and it would seem to me it can?t be both ways.