Now before we proceed further, just a little bit of clarification!
When I refer to cardinal and ordinal with respect to number, I am doing so from a Type 3 mathematical perspective (where complementary quantitative and qualitative aspects are thereby implied).
Now, it is important to bear in mind that we cannot in this context identify cardinal - strictly - with the quantitative - and ordinal - strictly - with the qualitative aspect of number respectively.
Just as the wave aspect of a particle entails its complementary particle aspect (and wave its wave aspect), likewise it is similar in number terms (from a dynamic interactive perspective).
So in Type 3 terms we can only define independence and interdependence (from a phenomenal standpoint) in a relative manner that depends crucially on context.
Therefore when we temporarily fix the frame of reference to consider the cardinal aspect of number as quantitative (i.e. as relatively independent), this equally implies in the context of the opposite frame, relative interdependence (which is qualitative).
Likewise when we temporally fix the frame of reference to consider the ordinal aspect of number as qualitative (i.e. as relatively interdependent), this implies again in the context of the opposite frame, relative independence (which is of a quantitative nature).
Thus in the Type 1 approach (that is ultimately compatible with Type 3), we do indeed temporally fix the frame of reference, so as to deal with the quantitative features of number in isolation. However implicit in this approach is the realisation of an equally valid qualitative treatment of number of an utterly distinctive nature.
Likewise in the Type 2 approach (that is ultimately compatible with Type 3) again we temporarily fix the frame of reference so as to deal with the qualitative features of number in isolation. However, once again implicit in this approach is the realisation of the equally valid quantitative aspect (using a distinctive approach).
Only when both aspects (Type 1 and Type 2) are explicitly brought together in a dynamic interactive manner (Type 3) can we see clearly how the quantitative (cardinal) necessarily has a qualitative (ordinal) aspect, and that likewise the qualitative (ordinal) necessarily has a quantitative (cardinal) aspect.
However I cannot state strongly enough that the conventional Type 1 version of Mathematics (which is misleadingly identified in our culture as "Mathematics") does not lend itself at all to comprehensive Type 3 interpretation.
The simple reason for this is that Conventional Mathematics is based on an absolute - rather than relative - Type 1 approach (where a distinctive qualitative aspect to Mathematics is not formally recognised).
Before writing this blog entry, I spent some time examining the manner that the ordinal nature of Mathematics is conventionally explained in Mathematics.
Though I found much mention of ordering and ranking, nowhere did I find its most obvious attribute clearly emphasised i.e. that ordinal refers to a qualitative rather than strict quantitative notion.
You see, the very fact of clearly admitting this would imply that the study of number - and by extension all Mathematics - cannot be viewed in a solely quantitative manner. And as a strong unconscious resistance exists to facing up to this most fundamental limitation of Conventional Mathematics, the true meaning of ordinal is masked behind an abstract web of quantitative type reasoning.
This reaches its zenith in the work of Cantor which in many ways - for all its ingenuity - represents the reductio ad absurdum of the merely quantitative type approach to Mathematics.
Initially in an apparently different context, I spent some time several decades ago attempting to come up with a satisfactory modern explanation of what lay behind the strong medieval belief in hierarchies of angels. For example, this played a very important part in the theological system of Thomas Aquinas who probably still remains the most influential of all (Roman) Catholic theologians.
It gradually dawned on me that these hierarchies actually served as ways of giving expression to different notions of the infinite (when viewed through a somewhat rigid rational lens of interpretation).
It then struck me that no essential difference existed as between this pursuit of the infinite (in the context of theology) and Cantor's later attempt to clarify the nature of the infinite (in the context of Mathematics). Indeed I later discovered - not to my surprise - that Cantor had been deeply influenced by medieval theology!
So the medieval preoccupation with the theological question of how many angels can dance on a pinhead (which might seem as laughably quaint from a modern perspective) is equivalent in mathematical terms with the question of how many numbers exist within a small interval on the real number line! And Cantor's answers are in fact the modern equivalent to the medieval manner of creating hierarchies of angels as an infinite succession of bridges - as it were - between God and man.
Thus the conclusion that we can can have a whole series of infinite (i.e. transfinite) sets is really just a reduced way of concluding that the infinite notion is qualitatively distinct from the finite. So properly understood, in dynamic interactive terms, a variety of transfinite sets, really refers to the fact that the infinite (which is of a qualitatively distinct nature) can indeed interact with the finite in an - ultimately unlimited - variety of ways.
But exploration of these varying - relative - notions of the infinite, requires the alternative (hidden) aspect of Mathematics i.e. Type 2. So when seen from this perspective, every number as dimension has a qualitative significance that mirrors in a unique manner the infinite.
Now the clue to the unrecognised qualitative nature of Mathematics in the context of Cantor's development, comes from that the fact that the famed Continuum Hypothesis cannot be proved or disproved from within its axioms. In this sense, number thereby transcends the attempt to understand its nature in merely quantitative terms.
And Cantor having attempted to "quantisise" the cardinal numbers in infinite terms, then attempted to do the same from an ordinal perspective.
However let us state once again the obvious point that quickly gets lost through all such convoluted abstraction.
If cardinal numbers are treated as absolute (in a merely quantitative manner), then strictly this rules out their interdependence, through relationship with other numbers (which is qualitative in nature).
So the qualitative aspect of number - as revealed through its ordinal nature - cannot be attributed to its quantitative identity.
Conventional Mathematics attempts simply to deal with this issue through fundamental reductionism i.e. where the qualitative aspect is directly reduced to quantitative interpretation.
And over and over again, we have seen that it is such reductionism that defines Conventional Mathematics (and its associated 1-dimensional qualitative interpretation).
Now, this does indeed define an extremely important limiting special case. However it should only be understood - rather like Newtonian Physics - as a convenient approximation that works well within a very restricted range of interpretation (which is merely quantitative in nature).
So, in truth an unlimited variety of alternative approaches to Mathematics (with each defined to a dimensional number other than 1) exist, with all entailing unique configurations of both quantitative and qualitative type meaning.