Why Different Infinities Are Really Equal

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.

We can easily show that infinity = infinity + 1, but if that's true, then infinity + 1 = infinity + 2 ... and so on all the way up to infinity^infinity and beyond. Every time we create a new real number, we are adding 1 to infinity which gives us back infinity. Using the diagonal method, we can create a new real number as many times as we like, to a point where we imagine a value greater than infinity. Given that infinity has no upper bound, it should be sufficient to cover any set with infinite members. New members can be added to a an infinite-member set and infinity should still be large enough to cover all the members.

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

Debunking Ramanujan's Finite Solutions to Divergent Series

ABSTRACT:

This paper shows how an infinite series yields both an infinite solution and a finite solution and how to determine which one is correct and which one is just plain crazy!

Imagine the following geometric series:

Is s convergent or divergent? Does it blow up to infinity or have a finite value? So far, we can't say. However, according to Ramanujan and his apologists, even if s is divergent, it has a finite value. Let's see if this is really the case. With a little algebra we derive equation 5 below:

What solutions will work for equation 5? Here are two:

One solution is infinity. If k is finite, the other solution is finite. This means, if we want, we can always extract a finite solution from a divergent series. It also means we can always extract an infinite solution from a convergent series! Ramanujan claimed that divergent series have finite solutions. He could have (but didn't) claim that convergent series have infinite solutions. If such claims seem counter-intuitive and just plain crazy, it's because they are. Extracting finite solutions from divergent series (or infinity from convergent series) violates the geometric series rules. Here are the rules:

At 9 we have a system of equations that show that the value of x matters. We don't just get to choose willy-nilly which equation we like best. If we choose the appropriate equation for a convergent series, we get a finite solution as we should. If we choose the appropriate equation for a divergent series, we get infinity as we should when x is greater than or equal to 1. Albeit, our system of equations doesn't cover x when it is equal to or less than -1. Let's start with the case where x equals -1:

Equation 10 doesn't converge, nor does it blow up to infinity. The partial sums oscillate between 1 and 0 forever. Mathematical physicists like to take the average of .5. However, it is obvious there are two solutions: 1 and 0. In the case where x is less than -1, the partial sums oscillate toward infinity and minus infinity. Again, there are two solutions. For negative numbers less than or equal to -1 we need two equations:

The complete system of equations is as follows:

Where x is less than or equal to -1, we can take advantage of a peculiar property of infinity: Infinity minus infinity doesn't have to equal zero, since infinity equals any number plus infinity:

Thus when x is less than -1 we can take the average of the two solutions s1 and s2 and get a finite solution. In fact, since n can be any finite number, there can be an infinite number of finite solutions.

Where x equals -1, there is one finite solution: .5.

Now, given what we know about infinity, finite numbers and the geometric series rules, what kind of solution(s) can we extract from the following series:

This is a tricky one, since there is no x variable to tell us which equation to use. So let's go with the usual method. We first take the average of the "a" series:

Then we do some fancy Ramanujan algebra to get the average of the "b" series:

Now that we have the b-series average, we can find solutions to the c-series:

At 38 and 39 above, we have two solutions: infinity and -1/12. One solution seems correct and the other seems counter-intuitive. Are they both valid? There is a test: simply sum up the series. The fastest way to do that is find the average number in the series, then multiply it by the number of terms. The average number and the number of terms are both infinity:

As expected, c = infinity. For any series, a good rule of thumb is if the series is divergent, a finite solution is invalid and just plain crazy! We can tell if a series equals infinity by looking at the last term. If the last term is infinite, then the series sum is infinite because infinity plus a bunch of other numbers equals infinity. We get a finite solution because the series itself does not know if it is convergent or divergent. Both convergent and divergent series are structured the same way: a sum of numbers--so the algebra is the same for both and yields more than one solution. Thus it is up to us to take a further step and test each solution.

Proof that Various Infinities Have a Finite Value

In the late 19th century, German mathematician Georg Cantor demonstrated that there are a variety of infinities of various sizes. Equations 1 and 2 below compare two different arbitrary infinities:

The infinity at equation 2 is n times bigger than the infinity at equation 1.

Srinivasa Ramanujan, an Indian mathematician (1887–1920), was cleverly able to extract finite values from various divergent infinite series. If having various infinities isn't weird enough, imagine various infinities having a finite value! Normally, the process of extracting a finite value from an infinity requires tackling that infinity with a unique set of steps. Thus it is not clear if all infinities have a finite value. Perhaps there are some that do and some that don't. Can mathematics provide any clues? Let's begin with an arbitrary function f that diverges to infinity:

What makes this particular infinity unique is not that variable N tends to infinity, but that N is multiplied by a coefficient alpha. This infinity is alpha times the size of a benchmark infinity that is simply N with an infinite limit. Below is another infinity (function s) with a coefficient of beta, where beta may not equal alpha:

The relationship between f and s is as follows:

Using some Ramanujanian algebra, we can solve s:

We can now solve f by making a substitution:

If coefficient k is finite and x doesn't equal 1, the solution to any arbitrary infinite function is finite. Otherwise it is infinite.

References:

1. Matson, John. 07/19/2007. Strange but True: Infinity Comes in Different Sizes. Scientific American

2. Dodds, Mark. 09/02/2018. The Ramanujan Summation: 1 + 2 + 3 + ⋯ + ∞ = -1/12? Cantor's Paradise

Introducing Stochastic Trigonometry for Quantum Physics and Statistical Mechanics

In the field of quantum physics, each eigenvalue has an eigenvector, and, when the eigenvector is normalized and squared, we get the probability for the eigenvalue. The normalized eigenvector is sometimes referred to as the probability amplitude.

When all the probability amplitudes are squared and added, the total should be 1. We can represent this with the Pythagorean theorem and the right triangle below:

The above diagram consists of two probability amplitudes: 'a' and 'b.' One is a wave function cos(theta) and the other is a wave function sin(theta).

Now, suppose there are more than two eigenvalues/eigenvectors? The diagram below shows that a and b can be broken up into smaller pieces or smaller and more numerous probability amplitudes. As before, when they are all squared and summed, they give us a total of 1.

It is possible to break up 'a' and 'b' into as many pieces as we like. Below we focus on amplitude 'a':

We can imagine breaking up amplitude 'a' into as many as an infinite number of sub-amplitudes. This can be done in both Euclidean and curved space. Equation 10 below shows how amplitude 'a' and its sub-amplitudes are invariant within flat or curved space.

With a little algebra, we can derive equation 14:

Equation 14 shows amplitude 'a' consists of an infinite number of eigenvalues (eta), each with its own probability (P(eta)). Without the probabilities, the etas would add up to infinity, and that would necessitate some sort of re-normalization technique. If we assume, however, that all quantum numbers have a probability, we will not get infinity; rather, we get the expectation value, i.e., the value actually observed.

What kind of probability values yield a finite result when eta increases linearly to infinity? Probability values that decrease exponentially. Below we derive such a probability function by using the natural-log function and converting eta to 'n':

At 16.2 we have a probability function that will reduce the probability exponentially. It gives us a number between 0 and 1, but we can derive a better function that gives us a number between 0 and 1, and, we can make a substitution. The end game is equation 16.9:

Equation 16.9 claims that if n = Q, the probability of Q (P(Q)) equals the definite integral of the probability function over a range from Q-1 to Q. We can further justify this claim with the diagram below which shows the relation between discrete values (in red) with continuous values (blue line).

Note how the area under the blue line, say, from Q-1 to Q is the same as the area of the red squares from Q-1 to Q. Equation 17 models the fact the the area under the blue line is the same as the area of the red squares over the entire range.

Now, to get a finite expectation value (amplitude 'a') we could combine equations 16.9 and 17, but the math would be more complicated than need be. To simply the math we will encounter later, let's first stretch the above diagram vertically:

Next, we draw a yellow line from zero N+1. This new line is going to make our lives easier and has the same area beneath it as the red line. Wouldn't it be nice if we could nix the red and substitute the yellow? Sure! But first we have to rotate the diagram:

Ah ... now we're in business! Below is the adjusted diagram and equation 18 with a new slope of N/(N+1):

The integral has a new range of zero to N+1, so we give the probability integral the same range:

Let's combine equations 18 and 19 to get 20:

If the limit of N is infinity, equation 20 will always give us the finite probability amplitude 'a.' No re-normalization required.

Using the diagram below and equations 21 and 22, we can derive a formula that finds probability densities:

What we've covered so far allows to find probabilities for integer values. This works fine if the value is, for example, the number of vertices in a Feynman diagram. Albeit, energy can have values of n+.5. Below is the math for that circumstance:

Notice if we divide both sides of equation 26 by Q+.5, and use summation signs, we arrive at equation 23, the formula for finding probability densities.

Now that we have the math the way we want it, let's put it to a test. Let's say we want to add up an infinite number of quantum numbers to get a finite value. Let's assume that the principle of least action applies: the most probable value will be the least action (e.g. least energy, least time, least distance, least resources required, etc.). The least probable value will be the action or event that requires the most resources, time, energy, etc. So we expect the probability to drop exponentially as the value of 'n' increases linearly--this will ensure a finite result.

Let's also assume that experiments confirm that probabilities change according to equation 27:

OK, now we only have to do some complicated math to find the expectation value 'a,' right? Wrong! At 28 and 29 below we convert the right side of 27 to a natural exponent function. If we look at equation 20, it becomes obvious that we can solve this problem by mere inspection. Looking at the exponent, everything to the right of -n is 1/a. Thus equation 31 is our final result.

Here's another test: What is the probability that a particle will travel a distance 'Q' along a pathway 'omega'? Equation 34 below can answer that. At 32 we assume that each pathway has the same probability if the distance traveled is constant, since the action is the same along each pathway (except for the direction, angles, curves, twists, turns, etc.).

Equation 35 gives us a definite answer if we want to know the probability density of a range of distances and pathways the particle can travel:

As you can see, stochastic trigonometry simplifies mathematics that can turn into a complicated, ugly, and infinite mess. It can also improve statistical mechanic's coarse-graining techinique:

Why use squares when you can use triangles?

Update: The following math shows both a convergent series and divergent series can yield a finite number 'q.' First, we start with a divergent series and make it convergent by using the probability function we derived above.

Next, we take a divergent series and assume the coefficients (the c's) don't add up to 1. Each could be any finite size; they could be a random series. The strategy is to factor out 'c' from the coefficients and use one of Ramanujan's techniques:

Another update: The following math generalizes the idea that a finite value can result from any arbitrary convergent or divergent series: