Infinity as a real physical thing is still often treated with skepticism. The definite nature of infinity is very much an unresolved mystery in both physics and mathematics, but there are a few tolerated infinities, such as electrons and black holes. And there are infinite series equations which mathematicians say are defined, because they express Convergence.
For example, 1.999... is said to equal 2, because the value of the number converges to the number two. Convergence requires that the difference or remainder moves ever nearer toward zero. The difference between the two numbers becomes so small that some mathematicians consider it too minute to be a relevant value. Others accept the leap to a definite sum by saying the infinite series equation would equal the finite value in an infinity of time.
The sequence: 1/1, 1/2, 1/3, 1/4 ... is converging toward 0.
The equation: 4 - 2 - 1 - 1/2 - 1/4 - 1/8 - ... is converging to 0 also.
The equation: 1/2 + 1/4 + 1/8 + 1/16 ... is converging toward a limit of one.
And the equation: 1 + 1/2 + 1/4 + 1/8 + 1/16 + ... is converging toward 2.
Converging equations are very different than 1 + 1 + 1 + ..., which has an increasing value, so there is never a completed sum. There is no convergence. In ordinary math we think of the set of positive or negative numbers as continuous and indefinite. If we add 1 + 1 + 1 + ..., there is always a next greater number for the sum to equal and there is never a last number to the series. The value expands yet is somehow never nearer to an ultimate end to the series.
In Symmetry Math there is one ultimate value which contains all other numbers and on either side of this Omega Zero other values decrease. As we count into ever larger numerics, the true value of ever greater or lesser numbers diminishes, decreasing toward an infinitely small value. Surprisingly, the value of larger numerics converge toward two outer extreme points on this new numeric plane.
Unlike our present finite system of values which does not treat a positive or negative infinity as a number, in the true value system we plainly discover a final number to an infinite series. In symmetry math, if we add 1 + 1 + 1 + ..., there is still always a next greater number, but the value of the sum is decreasing and also converging toward a point of infinitely small value. The summation is consequently converging toward an identifiable point on the plane and thus the value becomes a definitive number. In symmetry math, there are three extreme definite values, or three ultimate numbers. There is of course zero, the sum of all numbers. The convergent sum in the series 1+1+1+... I call Proto, which means first in time. Proto is a numeric of positive infinity, here written +oo. The ultimate negative number, the point where (-1) + (-1) + (-1) + ... converges toward, I call Eleat.
Notice that this entire spectrum is definitive and whole, the three infinite extremes are not processes, but complete and static values, virtually identical to the state space model explained in earlier essays.
In treating a positive and a negative infinity as numbers, we then can write this simple equation:
+oo + (-oo) = 0 or oo
Identical to what we found as we explored the universe's state space, here we find the symmetry plane is infinite but bordered by positive and negative extremes. So just as we cannot count through the infinite decimals between zero and one in regular math, we cannot linearly count to Proto from Omega zero, although we can count away from these two ultimate numbers, away from Proto or Eleat, toward a zero whole.
In symmetry math the whole mathematical number line is a spectrum of infinities extending out from a whole infinity toward two half infinities. In the modern complex plane a nothing zero cannot relate to positive or negative infinity in the way convergence allows the extreme values of this symmetry plane to relate. In symmetry math we can say both sides of the plane are balanced around zero and not balanced around any other number.
One of the elegant features of this system is that although the two smallest numbers, Proto and Eleat, are points of infinitely small value, each number represents half of the whole. The two smallest values of symmetry math still include all of the positive or all of the negative numbers. Where the true value of one includes all numbers except (-1), the true value of Proto equals all numbers except all the negative numbers. In order to create Proto all the negative numbers must be made separate, just as density order separates or groups together density to create definition.
Note that just as flat space extends infinitely in all directions, Omega zero is infinitely large. And just as a physical positive or negative density becomes an infinitely small point, the values of +oo and -oo are the smallest values possible in symmetry math. Each is an infinitely small point of value yet each is still half of the whole, since values here are defined not by what they are but rather by what they are not.
I shall refer to the positive and negative infinities as Polar Infinities. The two polar infinities summed equal zero, just as the properly aligned sum of all real numbers sums to zero.
Three Different Answers
Now we return to the problem of three different answers to the sum of all reals, or integers as was described in the last page. How do we resolve that issue in the symmetry plane? Having now entered into a new domain of values, and now with a final number to an infinite series, we again consider the equation:
0 + 1 + (2 + (-1)) + (3 + (-2)) + (4 + (-3)... = 1 + 1 + 1 + ...
If there is a final number to an infinite series, we can count backward from that number toward zero. Just as we add and subtract from zero, we can subtract from positive infinity, and add to negative infinity. In symmetry math, the equation above ends with the same displaced pattern that it began with. As we observe that ending, we discover there is a remainder of (-oo), which can be combined with the positive infinity of ones (+oo), and thus the true sum returns to zero, as written:
Note that +oo can be subtracted from (reduced) but not added to, and -oo can be added to but not subtracted from. Also, in symmetry math the addition of -oo and +oo is actually a true combination of both sets rather than a cancellation of positive and negative as occurs in ordinary math.
The numbers of the plane cannot be said to be derived from an elementary first thing somehow emerging from nothing or an empty set. Here positive and negative infinity are the two fundamental numbers, from which all definite numbers are derived. They exist as we would imagine possibilities to exist, permanently. In my own thoughts I relate the two basic polar infinities to something and nothing, the simplest of any two meanings, although the the meaning of nothing is changed in my own philosophy.
Symmetry math relates far more gracefully to the evolution of our spacetime universe than do ordinary finite values. In the same way that Proto (alpha state) has an infinitely small value and yet has an area that includes all positive numbers, the infinitely dense point that was our 'first in time', from which the Big Bang arose, can be thought of as spatially infinite. What appears to be an infinitely small point, at the beginning of time, is actually half of the infinite universe. This is why I call the first moment a dual singularity. Relative to Proto there is a negative singularity within the primordial particle which births our universe.
When we imagine that Proto contains all positive numbers, it seems like we should be able to count them. It is possible to count finite numbers without negative numbers, but in symmetry math, without using the negative half to combine with the positive, there is only one number; Proto. Without the negative half, Proto is not further defined than itself, much like the word something. So although Proto is an infinity of numbers, and infinite density is an infinite of space, we can relate to them physically only as singularities.
We can further apply symmetry values to mass and density. Again the switch to values is uncomfortable at first, but seeing the world from this perspective, objective things are not more than nothing or empty space, rather they are less than. Mass and density seems convincingly to be a value that is more than the transparent space that surrounds us, when in fact objects cannot exist unless their opposite is removed from that seeming emptiness. As David Bohm suggested, space is an implicate order. The infinite surrounds us disguised as the world we confront everyday.
We have seen that a symmetry mathematical plane is complete and whole. Because we now see the known world around us as finite, an infinite sequence has not been treated as real or fully defined. We have yet to fully make the leap out of our position in time to consider that the ultimate universe may not change in time in the way we observe spacetime to change. We have also not yet integrated a somewhat inevitable principle into science, that ultimately there is but one universe of existence. As we study a bit further, symmetry math becomes ever more related to physics and cosmology, and it becomes increasingly more meaningful.