Zeta Zeros Made Simple (10)

For many years I have been fascinated with the possibility of converting the Type 2 aspect of the number system in a coherent Type 1 manner.

So once again the natural numbers in the Type 1 aspect are expressed with respect to the default dimensional number of 1, i.e.

1¹, 2¹, 3¹, 4¹, ......

Then the natural numbers in the Type 2 aspect are expressed in a complementary manner with respect to a default base number of 1, i.e.,

1¹, 1², 1³, 1⁴, ......

Whereas the natural numbers in the Type 1, carry the direct connotation of cardinal identity in a quantitative manner, by contrast in the Type 2, the "same" numbers carry the complementary connotation of ordinal identity in a qualitative type manner.

So 1¹ in the Type 2 carries the connotation of 1st (in the context of 1) which clearly is the unit 1.

Thus if there is only one unit (in cardinal terms) no ambiguity can exist as to the 1st member of this group 

So the ordinal meaning of  1¹ is thereby identical in this case with its cardinal counterpart.

However when we move on to 2nd, 3rd, 4th and so each ordinal ranking cannot be unambiguously identified with one fixed position (unless the order is predetermined according to some arbitrary rule). In other words, in such cases ordinal ranking is of a relative - rather than absolute - nature.

However to express the notion of 2nd unambiguously with the last member of 2 we obtain. 

1^2/2. So 2nd is now effectively reduced to being identified with the 1 remaining unit of 2 i.e. the 1st of the remaining one unit).

1^2/2  reduces to 1¹ unambiguously in this manner.

And we can continue on in this manner so that 1^3/3 is  identified with the last member of a group of 3 and 1^n/n  with the last member of a group of n.

In this way, each of the ordinal positions can be unambiguously identified in a linear manner with their corresponding cardinal identities i.e. 1st with 1, 2nd with 2, 3rd with 3 and ....nth with n. 

Though it might appear trivial, an enormously important transformation has taken place, whereby in each case what was originally defined ordinally in a Type 2 manner has now been successfully converted in a cardinal Type 1 manner.

However this absolute type definition of ordinal positions represents just one limiting case where the nth member of a group is always identified with the last  member of the group n.

But there are innumerable other possible cases where the nth is identified with the group of n + 1, n + 2, n + 3 members and so on.

For example in the case of a cardinal group of 3 members, as well as the (default) case of the 3rd (of 3) we could also try and define the 1st of 3 and the 2nd of 3.

In fact this problem is intimately identified with the question of obtaining the 3 roots of 3.

Thus first in the context of 3 is defined in Type 2 terms as 1^1/3; 2nd in the context of 3 is 1^2/3; and 3rd in the context of 3 is - as we have seen - 1^3/3 .

What we have established here is in fact the Type 2 definition of rational fractions.

So for example with respect to a small cake in Type 1 terms, 1/3, 2/3 and 3/3 would have the recognised quantitative meaning of 1 (of 3 equal slices), 2 (2 of 3 equal slices) and 3 (of 3 equal slices) respectively

However in Type 2 terms, 1/3, 2/3 and 3/3 have the corresponding (unrecognised) qualitative meaning of 1st in the context of 3, 2nd in the context of 3 and 3rd in the context of 3 respectively.

Thus remarkably we can use the n roots of 1 to obtain all possible ordinal positions (in the context of a group of n).

And by obtaining these roots, we are in effect obtaining  a Type 1 transformation in quantitative terms of Type 2 ordinal notions that are strictly of a qualitative nature.