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From: Simplane Simple Plane Simulate Plain Simple on 17 Jun 2010 05:37
On Jun 17, 2:06 am, WM <mueck...(a)rz.fh-augsburg.de> wrote:
> On 17 Jun., 09:54, Virgil <Vir...(a)home.esc> wrote:
>
> > In article <4c19cd2c$0$316$afc38...(a)news.optusnet.com.au>,
> >  "Peter Webb" <webbfam...(a)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

- ratio of values after each interval.


Whole numbers added : Even numbers added
Whole numbers 1+2= +3+4= +5+6= +7+8= +9+10= +11+12=
added
Even numbers 0+2= +4= +6= +8= +10= +12=
added


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

Billions added 0+109 =1

And whole numbers : perfect squares
Whole numbers added +2+3+4= 5+6+7+8+9=
Perfect squares added +4 = +9 =

and combining again
109 ∞ 109 ∞
1 1 1

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.


From: Transfer Principle on 17 Jun 2010 20:50
On Jun 17, 2:37 am, Simplane Simple Plane Simulate Plain Simple
<marty.musa...(a)gmail.com> wrote:
> 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.

Musatov might be interested in knowing about Tony Orlow,
who is working on a set size that is very similar to
Conception #1 above.

In particular, Musatov describes how he uses ratios to
determine that the size of {2,4,6,8,...} is exactly half
that of {1,2,3,4,...}, since the elements of the sets
are in the ratio 2:1. TO uses a similar argument to
conclude that if {1,2,3,4,...} has the set size (or
"Bigulosity") tav, then {2,4,6,8,...} would have a
Bigulosity of tav/2.

TO uses the name "Post-Cantorian" to describe those who
use what Musatov calls "Conception #1," as opposed to the
"Cantorian" Conception #2. Thus, Musatov and TO are
natural allies wrt set size.

On the other hand, Herc and WM are "Anti-Cantorian" in
that they don't believe in different sizes of infinity. So
I'm glad that Herc and WM are in this thread together. So
these two reject both Conceptions #1 and #2.

If Musatov would like to learn more about TO's set size,
he can click on the current TO thread (warning -- this
thread now exceeds 500 posts).
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