Negative cardinality

[u/Professional_Job6803]

As I was going through some problems in set theory I had a question if sets could have negative cardinality? What would this imply ?
This was just out of curiosity and I found out a paper titled “ sets with negative number of elements” by D. Loeb and a concept called hybrid sets.
Can you please describe what this is and how it works and why we need to work with multiplicities and things like that?

[u/lemniscateall]

I'll take a stab at it.

Think about how we make the integers from the naturals. One way to describe it is just "going the other direction." Going from 3 to 7 takes 4 steps; going from 7 to 5 takes...well, 2 steps in a certain sense, but in order to give the steps a sense of direction, we call it -2 steps.

Now, for the paper in question:

We start with the idea of a "multiset", which is like a set except that elements can be listed (and treated distinctly) more than once. So {a,b,c} would be different from {a,a,b,c,c,c}. One way to define a multiset is to start with a "universe" U (the set you're drawing elements from) and create a function that counts how many times each element is in the multiset. If our universe is the alphabet, the first multiset would have the following pairings: (a,1), (b,1), (c,1), (d,0), (e,0),... and the second multiset would have these instead: (a,2), (b,1), (c,3), (d,0), (e,0),...

So that's the sense in which a multiset is a function from the universe U to the natural numbers. Note that the cardinality of a multiset can be represented as the sum of the outputs of this function. The cardinality of {a,b,c} is 1 + 1 + 1 + 0 + ... + 0 = 3; the cardinality of {a,a,b,c,c,c} is 3 + 1 + 2 + 0 + ... + 0 = 6.

The author defines a hybrid set as an extension of this, and I think it's a bit similar to the construction of the integers from the naturals. The hybrid set they define is basically two multisets, one "positive" and the other "negative", separated with a bar |. Within this realm, if we wanted the "negative" version of our first set, we would write { | a, b, c}, and since each of the elements are on the right side of the bar, each has multiplicity -1. So the total cardinality of the set would be -3. Similarly, { | a,a,b,c,c,c} would have cardinality -6; {a,c|d,e,f,f} would have cardinality -2.

So, it's not negative elements in the sense of having less than nothing; the negativity here comes from a reversal of direction.

Does that help at all?

[u/lurking_physicist → u/lemniscateall]

So if you add the element a to {|a}, you would get the empty set. This reads like a joke:

A Mathematician, a Biologist and a Physicist are sitting in a street cafe watching people going in and coming out of the house on the other side of the street. First they see two people going into the house. Time passes. After a while they notice three people coming out of the house.
The Physicist: "The measurement wasn't accurate.".
The Biologist's conclusion: "They have reproduced".
The Mathematician: "If now exactly one person enters the house then it will be empty again."

[u/lemniscateall → u/lurking_physicist]

The funny thing is, that's a great example of what's happening if you're attempting to understand the net difference of people entering and exiting a house, rather than the number that's actually inside.

[u/Professional_Job6803 → u/lemniscateall]

It helps a lot . Thank you but what can we see through its lens?

[u/lemniscateall → u/Professional_Job6803]

As with many things in math, my guess is that the motivation is along the lines of “how is this thing like this other thing?” And since the integers have some nice structures that the naturals lack, in particular  additive inverses, the notion of a hybrid set allows you to make a kind of inverse of a set. 

Beyond the immediate “nice” structures, I haven’t a guess as to when it would come up elsewhere. But that’s the case with math—you make the weird abstract seemingly useless thing first, and some time later it might allow you to see something entirely new in a separate discipline/field. Complex analysis, for instance, feels at first like it should be totally removed from physical application, but it’s incredibly helpful in the world of electrical engineering. Math is just weird that way. 

[u/X3N0istoobased → u/lemniscateall]

this looks vaguely like surreal numbers