ABSTRACT:
Assuming different infinities are unequal leads to strange and counter-intuitive mathematical results such as Ramanujan's finite solutions to infinite divergent series. On the flip side, assuming different infinities are equal leads to infinite solutions to divergent infinite series. This last assumption, however, flies in the face of the current dogma: that different infinities are unequal. This paper offers proofs that show that different infinities are in fact equal.
Cantor's diagonal argument is used to prove that bijection fails between the set of non-negative integers (Z) and the set of real numbers (R). Even though both sets have an infinite number of members, the number of members of Z are not considered equal to the number of members of R. However, the implicit assumption is that infinities behave like finite numbers. If we assume infinity has no upper limit, or fixed value, Cantor's diagonal argument can be turned on its head.
In the diagrams below, we have the diagonal argument:
The top diagram is a list of real numbers paired with non-negative integers to the left. At the bottom diagram, the numbers along the diagonal are changed (in red). This creates a real number that is not on the list. We place that number in red below the list.
As you can see, we started with a list of infinite real numbers, so allegedly there are no positive integers beyond infinity we can assign to this new real number. Thus this new real number is considered uncountable. In fact, real numbers are considered uncountable because they are a continuum, and not discrete, but Cantor's diagonal process shows a way to count real numbers discretely, because if it is repeated, it produces one real number at a time. We can just simply add each newly produced real number to the total. But does this imply that R's infinity is greater than N's? Let's assume, arguendo, this is true. We can then justify counting the new real number with a so-called larger infinity, so we pair it with infinity + 1.
Now, assuming infinity has no upper limit, is infinity + 1 really larger than infinity? Equations 1 through 5 below prove that different infinities are equal.
Now here's the rub: if different infinities are equal, why do two lines appear to be unequal? Below, line n and line k each have an infinite number of points; yet, line n is longer than line k. Surely there are more points along line n than line k; albeit, the proof beneath the lines tells a different story:
At 12 we see that an infinite number of zero-points (0 * infinity) can equal either line n or line k. However, at 16 we see that infinity equals n * infinity equals k * infinity. This means that each point along n can be mapped to a unique point along k. This seems incredulous, so let's further test our conclusion. Consider the divergent infinite geometric series below. We derive two different infinities: s and sx. Let's assume they are unequal.
At 23 we end up with an inequality. Infinity simply does not equal a finite number! Albeit, Ramanujan would disagree. As a side note, we can see how Ramanujan got finite solutions to divergent series. He treated different infinities as though they were unequal. Now, watch what happens when we assume different infinities are equal (s = sx):
At 29 we get infinity like we should. This result adds further support to the conclusion that different infinities are equal. Based on what we know so far, the following two equations seem valid:
References:
1. Cantor, Georg. Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Foundations of a General Theory of Sets). jamesrmeyer.com.
2. Cantor, Georg. Uber eine elemtare Frage de Mannigfaltigketslehre (On an Elementary Question of Set Theory). jamesmeyer.com. 3. Cantor's Theorem. Wikipedia
4. 2020. SP20:Lecture 9 Diagonalization. courses.cs.cornell.edu
5. Cantor's Diagonal Argument. Wikipedia
6. Cardinality of the Continuum. Wikipedia
7. Continuum Hypothesis. Wikipedia
