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Using Kurt Godël's Incompleteness Theorem and Kenneth Arrow's Impossibility Theorem, I will explore the possibility of finding a 'perfect' distribution model for a commodity within a society

Using Mathematics to determine an optimal method of vaccine distribution

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## There are two mathematical theorems that are crucial to this study: Godel's Incompleteness Theorem and Arrow's Impossibility Theorem. The former describes the self-destructive nature of mathematics and how the truth of a mathematical system can never be proved within that same system. The latter discusses the incompatibility between certain conditions of voting in a democratic society. The combination of these two theorems as well as some discussion of the ethical implications of this distribution serve as the tools to solving one of the most pressing problems in our world at the moment.

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## Godel's Incompleteness Theorem Mathematics: Image

## “This statement is false”

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## A theorem is composed of a well-formed sequence of formulae that can be constructed using these symbols. The proof is centered around Godel Numbering: each symbol in the list above is represented by a natural number and the formulas are now represented by a finite list of these numbers that are called “Godel Numbers”, created specifically for the proof of this theorem. For example, the number 3 can be represented as SSS0 = 4. In Godel Numbering, we can write: 27 * 37 *57 * 76 = …. You can write many of the arithmetic laws used today and their proofs using solely Godel Numbers:

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## Pr(s) = P(A) = P(g(A))    P is a predicate, meaning it expresses a relationship between the godel numbers of the proof and the godel numbers of the statements

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## In this statement, X is not only being contradicted by the negation of itself but is stating its Godel number before the proof for the Godel number. This self-referential statement states its own contradiction, showing that although this statement is true, it cannot be proven within the system it is created. In a way, even if the perfect vaccine distribution method is able to be represented in Peano arithmetic, it would not be able to be proven. Mathematics: Image

## Proposition of a Different Model

The most effective method to translate this problem into a mathematical model would be to approach it using concepts in engineering of dynamic systems. By taking into account the amount of healthy, infected, and deceased people as well as the rates of infection and death, I will construct a series of first order linear differential equations that model the effects of vaccination rates over time. Because it is a time-dependent function, it could become an anomaly in Godel’s Theorem.

In an engineering course I took over the summer, I realized I could create and analyze the system of the United States using a core formula:

Input + production = output + accumulation                                           (1)

It’s a general formula that can be used to evaluate any dynamic system. For our purposes, we will be using humans as the currency in this formula as we can find, for example, the input of vaccinated people, output of deceased, and accumulation of infected people. We will have three systems to analyze, the systems of the healthy, infected, and deceased.

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## The three systems are outlined where dNHdt , dNIdt , and dNDdt represent the accumulations of healthy people, infected people, and deceased people respectively. The arrows moving from one system to another represent the inputs and outputs where Ri, Rv, and Rd are the rates of infection, vaccination, and death respectively. For our purposes, because we are focusing our system within the topic of an epidemic, we will take the production of humans to be 0. Mathematics: Image

## Using the computer program Matlab, the set of differential equations can be solved and plotted.

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3. ## Thinner or short-lasting spike in infections, indicating less time taken to control infections

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## In the North Dakota graphs, the spike of infection lies around 900 days, almost triple the time of California yet are only marginally worse than the plots for the United States. For the trade-off between a significantly better result for California with a slightly worse for North Dakota and the long recovery time for the United States as a whole, the plots suggest that the vaccination and safety protocols should lie in the hands of the states themselves.

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## While Godel’s Incompleteness theorem disproves the possibility of a “perfect” vaccination method, the graphs and transient solutions give an alternative solution. The results per state are overall better than a general solution for the United States. If each state could create their own plan for vaccination rates catered to their priorities, the outcome would benefit those states.

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## Arrow's Impossibility Theorem Mathematics: Image

## Arrow’s Impossibility Theorem is mainly centered around voting systems, which could present itself as a plausible way of solving this problem. The definition of his theorem goes as follows: “Suppose that there are at elast 3 candidates and finitely many voters. Any social welfare function that satisfies universal domain, independence of irrelevant alternatives and unanimity is a dictatorship.” In a democratic society such as the United States, voting seems like a possible route. Arrow’s Theorem applies to ranked voting systems in which there are more than 2 options to be voted on. This process of collective-decision making will concern the distribution of a certain commodity: vaccines. To introduce the system of voting, we can imagine that there are 3 different methods of vaccine distribution (there are many more in reality) that we can name A, B, and C. Voters a, b, and c rank their preferences from 1st to 3rd choice and their preferences can be modeled in the table below:

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## To introduce the concept of a dictator within voting systems, we can first examine the p-voter dilemma. This problem is centered around a pivotal voter who can alter the social preferences in some way, thus becoming a dictator in the context of the voting system. In other words, this voter has the ability to move a preference A from the bottom of the social ranking to the top by switching his personal preference from A as his third preference to his first preference. This can be modeled in the tables below where p represents the pivotal voter who is assumed to be in the middle of the group of voters. In this system it can be assumed that all voters before voter p choose A as their first preference and all voters after voter p choose A as their last preference.

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## In this voting system, since more than half of the voters put A as their first preference (including voter p), according to independence of irrelevant alternatives, society puts A at the top of social ordering. In short, choice A wins.

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## In this voting system, since voter p changed his vote to C, more than half of the voters put C as their first preference. According to independence of irrelevant alternatives, society puts A at the top of the social ordering. Because this change in result is caused by one voter and that voter p’s personal preference will always be the same as the social preference, that voter is then characterized as a dictator within the system.

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2. ## If an alternative x rises (or does not fall) in an individual ordering without any changes in those ordering, and if x was preferred to another alternative y before any change, then x is still preferred to y. In other words, the relationship between a previously preferred alternative A and a lower preferred alternative B will not change if A rises

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## 5. The social welfare function is non-dictatorial, meaning that one voter cannot dictate the results of the vote (xP1y implies xPy regardless of individual preferences of voters other than voter 1 where P represents the true social ordering

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## In other words, if voter 1 and voter 2 prefer x to y (assuming the vote takes place solely between these two voters), then society prefers x to y. We can prove this using condition 4 by suggesting that there are individual preferences R1 and R2 for voters 1 and 2 and that these preferences rank three alternatives x, y, z such that the resulting overall social preference is xPy. This can be modeled in the table below:

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## Similar to the proof of Godel’s Theorem, theories cannot be effective while being complete and consistent (and include elementary arithmetic). The different possibilities offered in the voting system are similar to the columns and rows in the diagonalization table in Godel’s Incompleteness Theorem. If certians conditions hold, others will inevitably be violated. No matter how many possibilities are presented to the public of the United States, certain conditions will hold while others will inevitably be violated.

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## The first attempts at applying a strictly theoretical mathematical approach led to an initial dead-end as it was proven that any system that can be written using arithmetic is inherently unprovable. But in transitioning to changing systems over time, while the theorem still holds, results showed that while there is not a single perfect solution, if each state dealt with vaccination procedures their own way, the general outcome would be better. In that sense, there is no single solution, but many. By transitioning to one of the core characteristics of the United States, democracy, there was still hope in finding a perfect method of vaccination distribution through voting. Yet, Arrow’s impossibility theorem shows that not only are voting systems (having three or more options to vote on) in the U.S. unjust, but so are any voting systems in general. Although these theorems have disproved any formal distribution method, it shows that extremely biased opinions about a perfect vaccine distribution method are false. There can be educated guesses and estimations, but there is no sure distribution method that will eradicate the epidemic. However, these are the types of results that are achieved in the world of abstract and complex math. In this realm, nothing is 100% definitive and it becomes the process of logical thinking that allows one to evaluate what works and what does not. The results of this research are not without value and represent only the first step to understanding and analyzing problems similar to this one.

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