Infinite mathematical series of numbers are of much interest to philosophically inclined people. Do they really exist and what value are they since it is possible only to use a finite portion?
Greog Cantor developed the idea of trans-finite numbers. That means that not all infinite series are created equal, though we may be biased to believe that all truly infinite series must be eternal since if they are simply in progress of increase they are not really finite with some sort of beginning and present terminus even if only for a fraction of an instant of time.
The series of natural numbers (positive integers) are infinite in one direction; to the right, and I believe that Cantor would regard that as a trans-finite series. What does that mean? In regard to the infinite set between the numbers 2 and 3 for example where smaller and smaller rational numbers can be expressed in fractions, the natural numbers breezes beyond it.
Infinite number series though seem to be limited and finite in some respect simply because they are not a set of all sets of any kind of number that can be represented. Even number series do not include odd numbers, and positive numbers don't include negative numbers, in fact any series of numbers or geometry is a finite slice of the infinite continuum of numbers in all forms and size.
One of the interesting characteristics of infinite sets of rational numbers is that they seem to have the potential for being implictly contradictory in time as they are expressed. Because they cannot seemingly exist a priori in infinite numbers without continuing to diminish in proportionate size retroactively from the beginning, with a kind of reverse causality, they might need to exist just in a finite temporaly context.
Numbers seem be nominal finite, proportional sets of true infinity encompassing all directions, dimensions, ordinal and cardinal sets and so forth that is inexpressible. The relationship of that infinity non-contingent One to the expression of forms in the 'real world' studied by physicists makes one wonder of the expressions of possible cosmologies in space-time by The One.
The experience of a finite element of space-time or a given Universe is a provision of from the realm of Absolute Spirit evidently, at least as a point of faith and logical inference.
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P.S.
If one defines 'ontology' as a weltanshuung, or world view of explanation of existence comprehensively, then an ontology today seems to be conditional since their are so many of them (although there is only supposed to be one true ontology). Deontology in philosophical history meant 'duty' as one might infer from ontology what one's duty's were.
W.V.O. Quine adapted the use of the word 'ontology' to linguistic sets and logical parameters as if it were a modal universe. In constructing a theoretical ontology I suppose one can make up a set of all valid ideas, put them in a set of words that can be tested for logical validity and then call it an ontology.
One might find that sort of thing in contemporary cosmology with the requirement implicitly that the math adds up for the values and constants integrated with observable facts etc.
In modal logic one may construct a 'universe' with theoretical values that are self-consistent even if not at all like the 'real world'. Some computer programs comprise a kind of example of modal logic-for instance gaming worlds where a left turn for an avatar walking down a street always results in transporting to another galaxy in a slum before this universe existed.
Trans-finite series were the result of Georg Cantor's working with irrational numbers and comparing the scale of one set of an infinite series to another. Some infinities were thought to be larger than others.
There were thought to be two kinds of infinite number series-those with irrational numbers than can be made infinitely smaller incrementally such as between the numbers 4 and five, and those oars-in-the-water number series such as positive integers that were believed to be able to increase forever. For that trans-infinite series the puny infinity between 4 and 5 would be laughable I suppose.
Trans-finite series though do seem to be just another way of adding prior numbers to a next one in a series with a certain value, and of course they all seem in a way to be the same number representing an arbitrary concept of always being able to add one more to the end of an infinite line in one or both directions. Trans-finite numbers do not seem to be as impressive or actual as one initially thinks they might unfortunately. They exist in theory yet not in the real Universe, and are representative of quantity and location within a given relativistic frame of reference if one cares to associate the cleverness of number systems with spatio-temporal coordinates.
http://en.wikipedia.org/wiki/Transfinite_number
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