For many years the term ‘infinity or infinite’ has been used without much thought given to it. As a result it has been misused and applied incorrectly. Many get confused when working with a system that is endless, it does not matter if it is unique or repeating, and they label it infinite. Endlessness is a necessary condition for infinity but it is not sufficient. Mathematicians have a very limited definition and understanding of infinity in many of the systems they study. They hardly ever use the concept of ‘eternity or eternal’ which would be the correct fit for the systems they are dealing with. Because of this mismatch in concepts they are lead to all types of problems for which they have no resolve. The idea that some infinity is larger then others and some will fit inside others makes no sense at all. If infinity is the grandest, biggest, largest thing we can imagine how can you have one bigger then another??? For that matter how can you have more then one??? I know there is some systems when studied appear to lead to the conclusion that there are different sizes of infinities. Some of these systems appear as proofs and others as paradoxes but keep in mind the simple statements above: can anything be bigger then the biggest, can there be more then one of the largest, grandest, biggest.
Below I have listed the current ideas and understandings I have about infinity, eternity, endlessness and our number system. All of these conditions in each case are necessary and I hope sufficient but I am not completely sure as this is a work in progress and there may be some conditions I have overlooked.
There can be and there is only one. Everything is contained within it and there is nothing within the multiverse that affects it. It is independent of time, space and dimensions. There is no beginning and no end, no starting or stopping points and no center. There are no boundaries or limits and it is endless, all points are equal and equivalent from the infinite point of view.
Infinity is open, not closed: this is a very important point to understand. It means that if you travel in a straight line in any direction, or dimension you will never return to the point you started from or touch any point you have already passed through.
There is no center, no points of reference; all points are equal and equivalent. Looking from the infinite point of view one point can not be distinguished from another.
They share the same characteristic of endlessness as infinity but that’s where the similarity stops. There can be many eternities; the major difference between them and infinity is that eternities can have a center, boundaries, limits, reference points and in most cases are closed.
There are at least two types or kinds of eternities that I know of, cyclic and linear. The easiest to see is cyclic; circle, sphere or torus. Take the simple case the circle. How many times can you go around a circle, forever, endlessly, and eternally? If you choose a point on a circle or any closed system and move through the system sooner or later you will reach that point again. If this action is applied to a sphere or torus not only will the starting point be reached but also the path traveled will be repeated over and over again. In fact all the paths traveled will be repeated. This is the nature of closed cyclic systems. They are eternal and endless.
Most eternities have boundaries and limits from which a center point can be established. Starting and beginning points that also act as stopping and ending points can be established. There are different size eternities and the characteristics used to determine the size is unimportant from the system point of view. Most eternities are linked or associated with time and or space. The nature of the relationship is dependent on the characteristics of the eternity. Eternities can be entered or exited. A case in point is the atom. If undisturbed an electron will orbit the nucleus of the atom eternally. Until time or space run out. But if enough energy was applied to the atom the electron would leave orbit and go out into space. After the energy is dissipated the electron will once again enter another eternity orbiting around another atom.
Linear eternities are endless and have a center point and reference points like the cyclic eternities but they do not have boundaries or limits. An example of a linear eternity is our number system. Its center is zero. It is endless in any direction with no boundaries or limits but has many reference points. Each number within the system is a reference point to any other number in the system. That is the very nature of our number system and why it exists.
It is however not infinite.
Our use of this term is one of the roots of the problems and paradoxes we encounter. Endlessness is a process not a state of being. It is a verb, not a noun. A system that is in the process of endlessness is never ending and continues forever but this alone does not mean that it is infinite. Endlessness is just one of the necessary conditions for infinity but it is not sufficient. Many have applied the concept of infinity to systems incorrectly. Many systems have been judged infinite when they are really just endless.
For years mathematicians thought our number system was infinite, it is not, it is eternal. Example: How many numbers are there in the universe, most would say endless and infinite number of numbers? Well then how many numbers are there between 0 and 100, 0 and 10, 0 and 1. The answer is the same in all three cases - endless, infinite. (For any number you choose in any of the intervals I will just add 1 to It. 1.0 1.1 1.11 1.111 etc) But wait we know that all the numbers between 0 and 1 will fit between 0 and 10 and all of them will fit between 0 and 100 and all of them will fit in all the numbers in the universe. Does this mean that some infinities are bigger then others??? And if so what does it mean to fit one infinity inside another??? The answer is NO the number system is not infinite, it is eternal. In the case of the intervals: they have boundaries, 0&1 0&10 0&100 that act as starting and stopping, beginning and ending points. Because of the boundaries you can be sure there is a center point although you may not be able to reach it. Now take the case of all the numbers in the universe. True it is endless and has no boundaries or limits but it does have a center point, zero. This also acts as a reference point from which a change in direction can be determined and all other numbers can be referenced. This makes all other points unequal and distinguishable. The number system is also closed in the sense that any two numbers can be combined and will form another number that will be unique and different then any other number. The number system is a unique type of eternity, what I view as a linear eternity.