Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth root of w, etc....
Although a lot say that this is not the traditional approach and doubt it but it can be done.
I will give here an informal method to how that can be done.
It can be done using Gap-fill methodology.
Subcardinality is concerned with subsets of Omega.
Omega "w" is the set of exactly all finite ordinals.
w={x|x is a finite ordinal}
a gap in s subset "s" of w can be defined as a non empty set of elements of w that are missing in s and preceeding an element in s.
Example: s={1,2,3,4,....} here we have only one gap in s that is {0} and it preceeds 1.
Example: s= {0,2,4,6,..........} , i.e. s is the set of all evens
here the set of all gaps in s is { {1} , {3} , {5} ,.....}
the set of "fills" of s is { {0}, {2},{4},{6},.... }
Example s= { 1,4,7,10,.... }
the set of gaps of s is { {0} , {2,3} , {5,6} ,{8,9},...... }
the set of fills is {{1} , {4}, {7} , .....}
The gap-fill size sequence is the sequence of the gap sizes of the set of all gaps in a set and the sizes of elements between the gaps
Example: for the set of evens the gap-fill size sequence of
f1, g1, f1, g1 ,.......
Now we can define subcardinality of s were s is a subset of w in the following manner
total number of gaps in s > total number of gaps in r -> subcard s < subcard r
total number of gaps in s= total number of gaps in r -> subcard s = subcard r
If there is an infinitly repeatable gap-fill size moity in s then
if the ratio of total number of fills /(total number of fills + total number of gaps) in bigger in s than in r then subcard s bigger than subcard r
it this ratio is equal then both have the same subcardinality.
if there is infinity increasing gap size such that the i-th gap in s is always bigger than the i-th gap in r, then s is smaller in subcardinality than r.
I like to represent the gaps by symbole o and the fills by I
Now the gap fill structure of the set of all evens is
IoIoIoIo.........
the repeatable moity is Io
the ratio is 1/2
the gap fill structure of the set of all triplets is
IooIooIooI....
the repeatable moity is Ioo
the ratio is 1/3
so the set of all evens have bigger subcardinality than the set of all triplets
We can prove that the following sets are equal.
{0,3,6,.....} {1,4,7,.....} {2,5,8,.....}
Prove: the repeatable moity in all these three sets are respectively:
Ioo oIo ooI
all have the same gap fill ratio of 1/3
so all are of the same size.
Example: the set 0,2,5,9,.......
here the gap fill structure is
I(ith) oo...(i-th) o
so the first element which is 0 is followed by a gap of size one the second element followed by gap of size two ,etc...
Now this set will have bigger subcardinality than the set of
I(i-th) ooo...(i+1 th) o
like the set {0,3,7,12,....}.
(Of course the set s should be ordered according to membership i.e any two elements in s (which is a subset of w) x and y x preceeds y iff x in y )
So as far as subsets of Omega are concerned we can find a notion of subcardinality that can detect finer differences than Cantor's cardinality can.
The idea behind that post, is to say that this thing is workable.
> Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth > root of w, etc....
> Although a lot say that this is not the traditional approach and doubt > it but it can be done.
> I will give here an informal method to how that can be done.
> It can be done using Gap-fill methodology.
> Subcardinality is concerned with subsets of Omega.
> Omega "w" is the set of exactly all finite ordinals.
> w={x|x is a finite ordinal}
> a gap in s subset "s" of w can be defined as a non empty set of > elements of w that are missing in s and preceeding an element in s.
> Example: s={1,2,3,4,....} here we have only one gap in s that is {0} > and it preceeds 1.
> Example: s= {0,2,4,6,..........} , i.e. s is the set of all evens
> here the set of all gaps in s is { {1} , {3} , {5} ,.....}
> the set of "fills" of s is { {0}, {2},{4},{6},.... }
> Example s= { 1,4,7,10,.... }
> the set of gaps of s is { {0} , {2,3} , {5,6} ,{8,9},...... }
> the set of fills is {{1} , {4}, {7} , .....}
> The gap-fill size sequence is the sequence of the gap sizes of the set > of all gaps in a set and the sizes of elements between the gaps
> Example: for the set of evens the gap-fill size sequence of
> f1, g1, f1, g1 ,.......
> Now we can define subcardinality of s were s is a subset of w in the > following manner
> total number of gaps in s > total number of gaps in r -> > subcard s < subcard r
> total number of gaps in s= total number of gaps in r -> > subcard s = subcard r
> If there is an infinitly repeatable gap-fill size moity in s then
> if the ratio of total number of fills /(total number of fills + > total number of gaps) in bigger in s than in r then subcard s bigger > than subcard r
> it this ratio is equal then both have the same subcardinality.
> if there is infinity increasing gap size such that the i-th gap in s > is always bigger than the i-th gap in r, then s is smaller in > subcardinality than r.
> I like to represent the gaps by symbole o and the fills by I
> Now the gap fill structure of the set of all evens is
> IoIoIoIo.........
> the repeatable moity is Io
> the ratio is 1/2
> the gap fill structure of the set of all triplets is
> IooIooIooI....
> the repeatable moity is Ioo
> the ratio is 1/3
> so the set of all evens have bigger subcardinality than the set of all > triplets
> We can prove that the following sets are equal.
> {0,3,6,.....} > {1,4,7,.....} > {2,5,8,.....}
> Prove: the repeatable moity in all these three sets are respectively:
> Ioo > oIo > ooI
> all have the same gap fill ratio of 1/3
> so all are of the same size.
> Example: the set 0,2,5,9,.......
> here the gap fill structure is
> I(ith) oo...(i-th) o
> so the first element which is 0 is followed by a gap of size one > the second element followed by gap of size two ,etc...
> Now this set will have bigger subcardinality than the set of
> I(i-th) ooo...(i+1 th) o
> like the set {0,3,7,12,....}.
> (Of course the set s should be ordered according to membership > i.e any two elements in s (which is a subset of w) x and y > x preceeds y iff x in y )
> So as far as subsets of Omega are concerned we can find a notion of > subcardinality that can detect finer differences than Cantor's > cardinality can.
> The idea behind that post, is to say that this thing is workable.
> Zuhair
Very interesting, thank you.
Take, for instance, the sequences of sets:
A(n) := { k | k < n } B(n) := { k | k < n+1 } C(n) := { k | k < 2*n } D(n) := { k | k < 2^n }
Take the limit sets (in extended notation):
A := A(oo) = { k | k < oo } B := B(oo) = { k | k < oo+1 } C := C(oo) = { k | k < 2*oo } D := D(oo) = { k | k < 2^oo }
I wander what you'd say in this system is the cardinality of the sets A to D...?
> Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth > root of w, etc....
> Although a lot say that this is not the traditional approach and doubt > it but it can be done.
> I will give here an informal method to how that can be done.
> It can be done using Gap-fill methodology.
> Subcardinality is concerned with subsets of Omega.
> Omega "w" is the set of exactly all finite ordinals.
> w={x|x is a finite ordinal}
> a gap in s subset "s" of w can be defined as a non empty set of > elements of w that are missing in s and preceeding an element in s.
> Example: s={1,2,3,4,....} here we have only one gap in s that is {0} > and it preceeds 1.
> Example: s= {0,2,4,6,..........} , i.e. s is the set of all evens
> here the set of all gaps in s is { {1} , {3} , {5} ,.....}
> the set of "fills" of s is { {0}, {2},{4},{6},.... }
> Example s= { 1,4,7,10,.... }
> the set of gaps of s is { {0} , {2,3} , {5,6} ,{8,9},...... }
> the set of fills is {{1} , {4}, {7} , .....}
> The gap-fill size sequence is the sequence of the gap sizes of the set > of all gaps in a set and the sizes of elements between the gaps
> Example: for the set of evens the gap-fill size sequence of
> f1, g1, f1, g1 ,.......
> Now we can define subcardinality of s were s is a subset of w in the > following manner
> total number of gaps in s > total number of gaps in r -> > subcard s < subcard r
> total number of gaps in s= total number of gaps in r -> > subcard s = subcard r
> If there is an infinitly repeatable gap-fill size moity in s then
> if the ratio of total number of fills /(total number of fills + > total number of gaps) in bigger in s than in r then subcard s bigger > than subcard r
> it this ratio is equal then both have the same subcardinality.
> if there is infinity increasing gap size such that the i-th gap in s > is always bigger than the i-th gap in r, then s is smaller in > subcardinality than r.
> I like to represent the gaps by symbole o and the fills by I
> Now the gap fill structure of the set of all evens is
> IoIoIoIo.........
> the repeatable moity is Io
> the ratio is 1/2
> the gap fill structure of the set of all triplets is
> IooIooIooI....
> the repeatable moity is Ioo
> the ratio is 1/3
> so the set of all evens have bigger subcardinality than the set of all > triplets
> We can prove that the following sets are equal.
> {0,3,6,.....} > {1,4,7,.....} > {2,5,8,.....}
> Prove: the repeatable moity in all these three sets are respectively:
> Ioo > oIo > ooI
> all have the same gap fill ratio of 1/3
> so all are of the same size.
> Example: the set 0,2,5,9,.......
> here the gap fill structure is
> I(ith) oo...(i-th) o
> so the first element which is 0 is followed by a gap of size one > the second element followed by gap of size two ,etc...
> Now this set will have bigger subcardinality than the set of
> I(i-th) ooo...(i+1 th) o
> like the set {0,3,7,12,....}.
> (Of course the set s should be ordered according to membership > i.e any two elements in s (which is a subset of w) x and y > x preceeds y iff x in y )
> So as far as subsets of Omega are concerned we can find a notion of > subcardinality that can detect finer differences than Cantor's > cardinality can.
> The idea behind that post, is to say that this thing is workable.
> Zuhair
That half of the integers are even integers would correspond with such a notion. That concept is quite well described by the "density" of the subset within the set. The asymptotic density of the evens in the integers is 1/2. That number-theoretic truth (which is quite standard) would apply to the sets of those numbers.
To formulate, and formalize, that construct about the relative sizes of the one subset to the superset, would have that among all the possible subsets of the integers, on average half of the elements are even and half odd. Then, that might induce a notion, quite less standard, of various uniform distributions, or for simplicity of introduction mathematical constructs with many of the useful properties of distributions, over sets like the naturals and other continua, for example the unit interval of reals.
That there are "as many" even integers as integers is well known since the time of Galileo, who noted how the nonnegative integers can very simply be mapped 1-1, onto, and preserving natural order from n to the n'th square, cube, etcetera, each of the powers, to tetration, and etcetera, in terms of arithmetic progressions widely applicable for those perfect ties to geometry, for example. Besides that their infinitude is trivial, in the modern day there are actually very many useful applications of asymptotics, for example in the analysis of complexity of algorithms, generally with regards to the length of input in terms of computational time and space. In that realm, O(n) is quite definitely less than O(n log n), O(n^2) less than O(n^n), and etcetera. The products of these values are very similar to those of cardinals, although O(2^n) is still always countable, in that among various levels of the complexity, in the asymptotic, sums differences and products generally preserve the values, similarly as to polynomials.
Zuhair, half of the integers are even, it's a fact of standard number theory. It's called "asymptotic density" in number theory, the asymptotic density of the even integers within the integers is 1/2. That number theory has it as a fact suggests that the foundations, in set theory or a more plain number theory, should most definitely support that interpretation.
> Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth > root of w, etc....
> Although a lot say that this is not the traditional approach and doubt > it but it can be done.
It has already been done. Read Conway's "On Numbers & Games' Chapter 0, 'All Numbers Great and Small. Not only w - n, w/n and w^(1/n) but also infinitesimals 1/w^n and yet more of the same using aleph_xi instead of mere w = aleph_0, were constructed to form a field of transfinite numbers.
> > Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth > > root of w, etc....
> > Although a lot say that this is not the traditional approach and doubt > > it but it can be done.
> It has already been done. Read Conway's "On Numbers & Games' > Chapter 0, 'All Numbers Great and Small. Not only w - n, w/n and > w^(1/n) but also infinitesimals 1/w^n and yet more of the same using > aleph_xi instead of mere w = aleph_0, were constructed to form a > field of transfinite numbers.
> > I will give here an informal method to how that can be done.
> > It can be done using Gap-fill methodology.
> > Subcardinality is concerned with subsets of Omega.
> > Omega "w" is the set of exactly all finite ordinals.
> > w={x|x is a finite ordinal}
> > a gap in s subset "s" of w can be defined as a non empty set of > > elements of w that are missing in s and preceeding an element in s.
> > Example: s={1,2,3,4,....} here we have only one gap in s that is {0} > > and it preceeds 1.
> > Example: s= {0,2,4,6,..........} , i.e. s is the set of all evens
> > here the set of all gaps in s is { {1} , {3} , {5} ,.....}- Hide quoted text -
> > > Actually one can define subcardinalities like w-1, w-2 ,..., w/2 , nth > > > root of w, etc....
> > > Although a lot say that this is not the traditional approach and doubt > > > it but it can be done.
> > It has already been done. Read Conway's "On Numbers & Games' > > Chapter 0, 'All Numbers Great and Small. Not only w - n, w/n and > > w^(1/n) but also infinitesimals 1/w^n and yet more of the same using > > aleph_xi instead of mere w = aleph_0, were constructed to form a > > field of transfinite numbers.
> > > I will give here an informal method to how that can be done.
> > > It can be done using Gap-fill methodology.
> > > Subcardinality is concerned with subsets of Omega.
> > > Omega "w" is the set of exactly all finite ordinals.
> > > w={x|x is a finite ordinal}
> > > a gap in s subset "s" of w can be defined as a non empty set of > > > elements of w that are missing in s and preceeding an element in s.
> > > Example: s={1,2,3,4,....} here we have only one gap in s that is {0} > > > and it preceeds 1.
> > > Example: s= {0,2,4,6,..........} , i.e. s is the set of all evens
> > > here the set of all gaps in s is { {1} , {3} , {5} ,.....}- Hide quoted text -
> > - Show quoted text -
> Yea, when did conway did that.
> Zuhair- Hide quoted text -
> - Show quoted text -
For your information if you read my full system, you will see a descripition of infentismals also.
