How to Conceptualize the Imagine a Square drawn on a piece of paper. Now imagine the Square shrinking smaller and smaller. It remains a Square no matter how small it shrinks. If we stop shrinking it and start magnifying it back we can bring the Square back to the original size. But now imagine the Square shrinking to Zero size. All points of the Square collapse to a single point and there is no longer a Square on the paper. The square has been transformed into a single point. We would not be able to magnify the resulting point back the the original Square. We could also shrink a Triangle in the same way and at Zero size it would be a single point just like the Square. The Square and the Triangle lose their identity when they are Zero size. They become something different. They become something less than what they were. Zero size is an unrecoverable threshold of size that changes everything. Now imagine a Square that is the smallest Square that is not equal to Zero. This thought sends your mind into an endless recursive loop of the Square getting smaller and smaller and we soon realize that it is impossible to imagine such a smallest Square. One thing we can say is that this Square is Infinitely small but is still a Square. In general mathematics this would be called a differential Square or an infinitesimal Square.
Next imagine the Square that was drawn on the paper growing larger and larger. If the Square was exactly in the center of the paper the sides of the Square would eventually move off of the paper and past the edges of the universe. It remains a Square no matter how large it grows. If we stop growing it and start shrinking it back we can bring the Square back to the original size. But now imagine the Square growing to Infinite size. The sides would all move out to infinity. No matter how far you went in the universe you would never encounter a side of the Square. The Square has effectively exited the universe. We could also grow a Triangle in the same way and at Infinite size it will no longer be found in the universe. The Square and the Triangle lose their identity when they are Infinite size. They become something different. Paradoxically they become something less than what they were. You might think that the Square and Triangle are still out there at Infinity. But there is no "there" at Infinity. The Square and Triangle are gone. If you think you can go out "there" to an edge of the Square or Triangle at Infinity then that "there" is not Infinity. Infinite size is an unrecoverable threshold of size that changes everything. Now imagine a Square that is the largest Square that is not equal to Infinity. Similar to the differential Square, this thought sends your mind into an endless recursive loop of the Square getting larger and larger and we again soon realize that it is impossible to imagine such a largest Square. We can say that this Square is Infinitely large but is still a Square that exists in the universe. I think that just as Infinite Squares are not possible it is probably true that any Infinite Physical quantity of anything is not possible. Just because an equation in Science goes to Infinity, it doesn't mean that the Physical quantity in the equation is able go to Infinity. I think this is a limitation of what we can do with Mathematics. Seems like a minor limitation but it has big consequences when equations in Science go to Infinity.
Now we will construct a number line using the above concepts. Lets call this number line the x axis. Construct the number line as individual points separated by dx which is a differential line segment on the x axis. The first point is located at x=0, the second point is located at x=dx, the third point is located at 2dx, and etc. We know that dx is the smallest line segment that does not have Zero length. If dx=0 then all the points would collapse onto a single point at x=0. There would be no number line anymore. But for dx > 0 we would like to show that the above construction creates a point at every Rational, Real, and Hyperreal number on the line. Any point P on this axis satisfies the following equation:
P = dx * (P / dx)
Where P / dx is the number of dx segments that add up to the interval from 0 to P. It will consist of a large integer number of dx segments and a fractional part of a dx segment. Writing this equation in terms of the integer and fraction parts:
P = dx * ( Integer(P / dx) + Fraction(P / dx) ) P = (dx * Integer(P / dx)) + (dx * Fraction(P / dx))
Let: FRAC(dx) = dx * Fraction(P / dx) INT(dx) = dx * Integer(P / dx)
For the graphical example below let P=pi. Let dx>0 and we get graphically:
FRAC(dx) converges to Zero with a saw tooth characteristic. The top of each saw tooth of FRAC(dx) decreases monotonically to Zero, and the bottom of each saw tooth of FRAC(dx) is Zero. INT(dx) converges to P=pi with a saw tooth characteristic. The bottom of each saw tooth of INT(dx) converges monotonically to P=pi, and the top of each saw tooth of INT(dx) is equal to P=pi. Obviously when FRAC(dx) is Zero, INT(dx) is exactly P for each saw tooth in the graph. In general this says that for any number (Rational, Real, or Hyperreal) there will be a point on this number line that consists of an integer number (probably but not necessarily infinite) of dx segments.
But as dx>0, Fraction(P / dx) seems to forever toggle between 0 and 1 and Integer(P / dx) seems to increase without bound. But what if instead of saying dx>0 we say that dx is the smallest number that is greater than Zero. Saying dx>0 is a process that never really gets there. Even though we can't really imagine it, what if there is an actual number that is the smallest number and the only smaller number is Zero? The Theory of Hyperreal Numbers (THN) proposes that there are an Infinite number of extremely small numbers near Zero, and Smooth Infinitesimal Analysis (SIA) proposes that there is a smallest number near Zero which has some unusual properties. We can make some statements about dx if it is the smallest number near Zero:
Even though dx > 0: Zero = dx * dx because Zero is the only next possible smallest number. Since these are seemingly conflicting properties, we will have to be careful applying them to any analysis.
So Fraction(P / dx) must be identically Zero because if dx is the smallest number then there cannot be any smaller fraction of dx so the Fraction part must be Zero. This also says that there will be a point exactly at P when dx is the smallest number. The value of Integer(P / dx) will be the number of dx segments that add up to get to P on the number line. So every Rational, Real and Hyperreal number is represented on the number line when dx is that smallest number.
Now lets consider the infinite side of the number line. Even though we can't really imagine it, call the segment from 0 to Gx the largest segment not equal to Infinity. Where Gx means the most Gigantic segment possible. If Gx were Infinite it would not be a segment because one end could never be found in the universe. THN proposes that there are an Infinite number of extremely large numbers near Infinity. In analogy with SIA we will propose that there is a largest number near Infinity and it has some unusual properties. We can make some statements about Gx if it is the largest number near Infinity:
Even though Gx < Infinity: Infinity = Gx * Gx because Infinity is the only next possible largest number. Since these are seemingly conflicting properties, we will have to be careful applying them to any analysis.
Note that I understand and have sympathy for why mathematicians might not like the concept of an actual smallest and largest number, but if there are such things then we can ask if there is a relationship between dx and Gx?
I think so but first I'm going to state some Definitions involving Zero and Infinity. These Definitions are for Zero and Infinity themselves and not for a limit situation where something is approaching Zero or approaching Infinity. Note that I also understand and have sympathy for why mathematicians might not like these Definitions.
First Definition: Zero = Zero * Infinity This is like saying you add Zero to itself an Infinite amount of times. It has to be Zero.
Second Definition: Infinity = 1 / Zero Division is like saying for A / B, how many times can you subtract B from A until there's nothing remaining. So 1 / Zero is like saying how many times can you subtract Zero from 1. This is obviously Infinity.
Third Definition: Zero = 1 / Infinity Multiply both sides by Infinity: Infinity * Zero = Infinity * (1 / Infinity) Using the First Definition: Zero = Infinity * (1 / Infinity) For the right side to be Zero, 1 / Infinity must be Zero.
Recapping: 1) Zero = Zero * Infinity 2) Infinity = 1 / Zero 3) Zero = 1 / Infinity
Now for some Properties of the Smallest and Largest Numbers:
Using the Second Definition we get: Infinity = 1 / Zero Infinity = 1 / (dx * dx) Infinity = Gx * Gx = 1 / (dx * dx) Gx * Gx = 1 / (dx * dx) The above equation is probably the most fundamental relationship between these Extreme small and large numbers.
I'm a little more skeptical about doing these next things but lets do them anyway: Infinity = (Gx * Gx) * 1 / (dx * dx) = Infinity * Infinity Infinity = (Gx * Gx) / (dx * dx) Infinity = Gx / dx * Gx / dx Infinity = Gx / dx
Then: Gx = dx * Infinity = Infinity But: Gx < Infinity So this is where we must be careful and stop at: Infinity = Gx / dx
Using the Third Definition we get: Zero = 1 / Infinity Zero = 1 / (Gx * Gx) Zero = dx * dx = 1 / (Gx * Gx) dx * dx = 1 / (Gx * Gx)
Again I'm a little more skeptical about doing these next things but lets do them anyway: Zero = (dx * dx) * 1 / (Gx * Gx) = Zero * Zero Zero = (dx * dx) / (Gx * Gx) Zero = dx / Gx * dx / Gx Zero = dx / Gx
Then: dx = Gx * Zero = Zero But: dx > Zero So this is where we must be careful and stop at: Zero = dx / Gx
