Conceptually: A set whose elements can only be probabilistically characterized?

Question

Sorry for the informality here, but was musing over the basic concepts around describing a set in real world usage: A finite set of explicitly named elements, this apple and that apple, nothing more straightforward. Then implicitly defined sets, like 'all pink rabbits' -- might be a finite set of rabbits, or maybe an infinite set including imaginary rabbits, but essentially you can rely on the fact you know it when you see it.

However, I was considering if you can't have sets where this 'know it when you see it' sort of property isn't true. For instance, how to think about a set of all 'good' people? I don't mean to get at the notion of people having different subjective definitions of 'good' -- Instead assume there is single definition of 'good' that can partition any group of people into good and not good provided total knowledge of how each person behaves in every given possible situation.

Based on the above understanding of inclusion, I'd think it's valid to define a set this way. But, it bothers me that it's impossible to determine whether a given element belongs to the set or not. We can only see how people will react in some finite subset of possible situations, but every situation or combination of situations is certainly some large infinity.

Instead there's an intuitive probabilistic notion based on the observable information -- Well, he walked that old lady across the street, looks normal, has friends, spoke about other people in a fair way, he is very probably a 'good' person. You still can't be certain he isn't a 'serial killer next door', because you never observed the relevant circumstances.

At this point should I just chuck out any notion of a set since it's a probabilistic inclusion operator instead of a concrete one? Would it be better to think of this in statistical terms with infinite dimensionality? Anyway, apologies if this comes off as the unmathematical rambling of some crank.

Answer 1

Your problem has nothing to do with "the notion of a set". It's just that you don't have an algorithm to determine whether a particular individual is a member of a certain set. This is a not uncommon situation, whether in the real world or in the world of mathematics. In mathematics, the situation can be worse: most sets of integers have no description at all.

Answer 2

This is a good candidate for Bayesian Inference. I wouldn't say you are dealing with a probabilistic inclusion operator...a person is either good or not good, according to your post. Instead, think of this as a "prediction" problem:

Let our sample space be $\Omega:=$Set of all people who will ever be born and $G\subset \Omega$ be the subset set of all good people and $I_G(\omega \in \Omega)$ be a random variable where $I_G(\omega)=1$ if $\omega \in G$ and $0$ otherwise. In this case, what you will need to do is define the joint distribution $F(I_G,C_1,C_2,C_3...)$ where the $C_i$ represent data/observations that are thought to be related to being good or not good. $F$ can be a subjectively defined distribution based on your opinions plus any general data you have. Then, once you observe someone, you can condition on the observed $C_i$, integrate out the $C_i$ you didn't see, and apply Bayes rule to get the "posterior" probability that $I_G=1$ or $0$.

Answer 3

Just because we can't figure out if something is in a set does not mean we should disregard the set. For example the set of all irrational numbers. This set definitely exists and is well defined. There are numbers, however, for which we do not know if they are irrational. $\pi + e$ is one example of such a number.

We can take it a step further and consider the set of all Turing machines which halt after a finite number of steps. Once again this set definitely exist. Not only do we not know any way to test if a Turing machine halts. We in fact know for sure that there is no such algorithm. A sketch of the proof and some more info is here.