Philosophy The Incompleteness Theorem and Ethics

[Doomsought]

Let us assume that a valid moral system exists.
We define Moral systems are made of rules that can be used to judge what should be.
Because the moral system is made of rules, the moral system is equivalent to set of moral axioms.
According to the Disproof by contradiction, a system that is not consistent is not valid, therefor a valid system is consistent.
According to the Incompleteness Theorem, a set of axioms must be either inconsistent or uncountable.
Therefore a valid moral system must have a cardinality of uncountable.
The state of the current world is flawed and does not satisfy a valid moral system.
We may alter the state of the world. Therefore with knowledge of morality, we may alter the world to being it in line with a valid moral system.
Because of the limits of linear time and causality, we may only alter the world iteratively.
Because the valid moral system is uncountable, we may iteratively implement moral rules onto the world infinitely, and will never completely satisfy the valid moral system with the status of the world.

Conclusions:
What is and what should be will never be the same. We cannot bring about a perfect world. Nor may we justify extreme actions by believing that we will solve all problems in the world with them, because the problems are not only infinite but also uncountable. We can only achieve a finite amount of good with our actions.

 

[Navarro]

There seems another obvious implication to this, which is that with humans also being flawed, it would be impossible to live up to any valid moral system.

 

[Abhorsen]

According to the Incompleteness Theorem, a set of axioms must be either inconsistent or uncountable.

Right here you hit a few problems. First, it's not uncountable, it's incomplete you are looking for, that's the first incompleteness theorem. Completeness basically means that any statement able to be formalized is either provably true or false from the set of axioms. Second, there's a major limitation on the theory that you didn't account for, namely that the axiom set needs to be able to do arithmetic.

The first theorem is roughly: Any consistent set of axioms that can do (very) basic arithmetic cannot be complete.

For a counter example of a consistent logic which isn't uncountable, take the axiom set containing only "The empty set exists". You can conclude nothing else from this, but it is consistent.

(The second incompleteness theorem states that no system capable of basic arithmetic can prove itself consistent without actually being inconsistent.)

So yeah, everything that comes afterwards doesn't follow.

 

[Doomsought]

Let us for a moment consider morality being described through a set of thought experiments.
Let us start with the trolley on a track problem, where twelve people are tied to one track and one person is tied to the other, and you control the switch. There should be at least correct one solution to this problem.
We can change the situation, so that the twelve people are murderous criminals, and the one person is an innocent child. The solution(s) to the problem have changed.
We can do other changed to the problem, such as adding more tracks, or adding conditions of probability, which will all change the solution or solutions.
Thus, by proof by induction we conclude that moral thought experiments may arbitrarily be made more complicated.
Let us imagine the set of moral though experiments that describe all of morality as a list of logical equations, with the a set of details describing a situation on one side of the equation and the set of solutions on the other.
Because we may always arbitrarily alter the situation set by adding or altering details, the length of the list is infinite.
Now let us construct a situation by adding one detail to the situation set of a moral thought experiment by adding one one detail that is different than those in the situation set of each thought experiment on the list, starting from the top of the list and working the our way down.
This new moral thought experiment is not on the list.
Therefore, through diaganalization, we have proven the cardinality of possible moral thought experiments is uncountable.
Because an uncountable set cannot be mapped onto a countable set, the cardinality of an complete description of morality must be uncountable.

 

[Abhorsen]

Therefore, through diaganalization, we have proven the cardinality of possible moral thought experiments is uncountable.

Up until here your are fine.

Because an uncountable set cannot be mapped onto a countable set, the cardinality of an complete description of morality must be uncountable.

This is wrong. An uncountable set cannot be bijectively mapped onto a countable set (i.e. one-to-one and onto, with one-to-one being the thing that cannot be done). But I can definitely map an uncountable set to a countable set: map all of the real numbers to 0, for example. Also, the floor function does the same thing, mapping the real numbers to the integers.

And if I was talking about moral terms, we could use the Catholic morality rule of "don't act in a way that kills people" (roughly stated), as a single axiom that solves uncountably many moral problems of the trolley variety (never pull the lever except to switch to a track with no one on it).

But I do think you are on to something. You might be able to show that any morality system with at least basic arithmetic (any sort of utilitarianism for example, and probably eye for and eye systems also) cannot completely describe all possible situations.