Consider the following purpose: modeling an infinity of space.
Define modeling as follows: a camera is placed in the infinity of space that is the model, and the image it produces is displayed on a screen.
Define infinity as follows: First, define the model as being a description of a space. Why is it a space and not simply space? Because the space that is being described is defined by the model, that is, by the description. Consider this: space, by definition, is infinite. If, then, we assert the existence of two spaces, these, by definition, overlap, since they are both infinite. They overlap completely. Two spaces is therefore a contradiction in terms, if they are simply defined as being expanses. The expanse of space that is defined by one model is necessarily the same expanse of space defined by another model. And yet, we can speak of two spaces, defined by two models. What differentiates one from the other is not the space each model describes but the objects which each model describes as being in the space the model describes, versus the objects described in the other model as being in the space that model describes. The fundamental object which can be described in a model is a point. A point has no dimensions, so it cannot, by itself, define space, but two points do begin to define space. We can construct a model, then, using two points, and, in a model defined by two points, the space the model describes is defined by a single number, which we name "distance." With reference to modeling two point spaces, then, the term infinity refers to the allowable size of the number named distance. A non-infinite model places upper and lower limits on the size of the number named distance. An infinite model places no upper or lower limits on the size of the number named distance.
A variation: A number is a symbol, and symbols occupy space. The space a symbol occupies is defined by a medium. For example, a number may be recorded on a page, and, if we add a rule stating that the entire number must be recorded on one page, whose dimensions are also given, and, additionally, if we assert that, first, to qualify, the number must be decipherable, and, next, that numbers are composed of digits, and then that a digit must be larger than some minimum size to be decipherable, and that digits may not overlap, we will arrive at some maximum number of digits which can be recorded on the specified page, and this will define upper and lower limits on the size of the number which can be recorded, given the system as described. We are presented with a kind of predicament. Given a specified system for recording numbers, we cannot describe an infinite space. If, however, we wish to describe a space beyond the limits imposed by a system for recording numbers, we can extend the system. So the space we can describe is infinite, but there are limits, but a given system for recording descriptions of space will define limits which do not limit the space we can define, but do limit the space we can define without extending the system. We could further refine this terminology by saying that a particular infinity is defined by a system defined for describing that infinity, and that it is defined by the limits of what can be defined by that system and without extending that system.
We can now say that the infinity of a two point model is defined in one sense by the capacity of the system we are using to build the model, and, having said that, we can add another problem, which is the possibility of adding additional points to our model, and then we can observe that each point added to the model presumably reduces the upper and lower limits on the distances between the points in the model which can be described. We can say, then, that the size of the infinity of a model is proportional to the capacity of the medium, and inversely proportional to the number of points in the model.
Next, thinking about one medium we might be using to model a space, taking, as an example, a computer, we find that there are at least two spaces which comprise the medium. This is a reiteration of the idea that numbers occupy space, which is to say that, while numbers may be used to describe spaces - that is perhaps their defining function - they themselves occupy space, and are recorded in or on spaces. Defining a medium as being a system, this system we are investigating consists of what we call memory, which is a space in which the numbers which describe a space are recorded, and a separate space which in some way displays those numbers, which is to say, some process is applied to the numbers in memory, and that process results in numbers which are displayed as colors on a screen. The colors displayed on a screen are numbers. The screen space, as a medium, can display a certain number of rows, each consisting of a certain number of pixels, and each pixel can display, at a given moment, one of some number of colors.
