The invention of negative numbers proved to be useful in mathematics. Set operations, however, do not have the idea of having less than 0 elements. We reach the empty set {}, and then stop, but why should we? Imagine a property: subtracting an element that is not in a set creates a potentiality to annihilate
such element. Such potentiality could be marked as elements with an apostrophe. I.e., {1,2',2} = {1}.

This idea was inspired by "World’s Most Exclusive Club," when thinking about the super-exclusiveness.

Fuzzy sethttp://en.wikipedia.org/wiki/Fuzzy_set Close idea, however, membership is in the range [0, 1]. (Membership function is non-negative.) [Inyuki, Jun 01 2012]

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[+] As soon as I get my membership card from the World’s Most Exclusive Club, I am going to publish a paper in their journal on proving the existence of Sets with Negative Cardinality.

// What would you do with a set containing negative elements? //

Answer: set operations.

Btw., calling them 'negative elements' would be a misnomer: {-1, -2} U {2} = {-1, -2, 2}, whereas {1', 2'} U {2} = {1'}. Calling them something like "anti-elements" would allow to avoid such an ambiguity.

[Inyuki], I agree with {1', 2'} U {2} = {1'}, upon which an entire algebra could probably be formulated. The set {1', 2'} owes someone a 2, so it has to pay it back under the Union operation. Sweet.

What you have just invented, Inyuki, is something called 'accountancy' and what you are describing is a balance sheet made up of assets & liabilities. It's a splendid idea but it already exists.

But the novelty is in the idea of a mathematical set that is missing some elements. And further, that addition by the Union operation can be used on such nonstandard sets to achieve accountancy. Those don't exist in the accountancy literature I would venture.

[scad] - As said above, "upon which an entire algebra could probably be formulated." Your proof, which is correct, shows that under this new algebra, the associative law does not apply to the U operator. That's why its called a new algebra, and is no less valid than the usual one. Useless maybe, but valid.

[UB] everyone knows that you have made XZ an "anti-member" of your set. Does this mean that XZ can still be a nuisance by increasing the equal and opposite quantity of anti-XZ ideas and annotations that you have to read?

// Isn't this like saying you have a non-empty set of anti elements? //

Suppose you have a non-empty set of elements and anti-elements: {1', 2', 1, 2}, the classical set would still have 4 elements, but this non-classical set has 0 elements. So, it is not just the idea of anti-elements. It's a property of the set, in context of anti-elements.

//array of negative length to see what happens// Yesterday I was messing around with some python code, writing a class to handle some geolocation data. It was built to always reject the first line of data (column headers), so it would, kinda, be like starting with a negative size.

This reminds me of a joke: A mathematician is looking at a house, and one person enters the house and then two people come out. His remark: "If one more person goes into the house, it will be empty!"

this reminds me of the importance of understanding the difference between math and science and how it's important to understand exactly what the math is meant to represent.

I think half the lunacy that occurs in quantum physics can be explained by this misunderstanding, since apparently they actually come up with theories almost purely from math (from what I read). And it is indeed lunacy; there's no way there could be more than three spatial dimensions, and if there arem they will always be meaningless to us with our limited perception. Any math trying to model such a world will always be meaningless on the human scale, and can't be truly groked by humans.

[EdwinBakery], I agree with what you say, except for 4 statements, starting with // there's no way there could be more than three spatial dimensions //.

//Any math trying to model such a world will
always be meaningless on the human scale//

If by "meaningless" you mean "useless" or
"academic" then, with all due respect, bollocks.
Quantum mechanics may be only an
approximation, but it's essentially abstract and
you wouldn't be reading this on a computer if it
weren't directly applicable.

As for being understandable, I suspect that
understanding comes partly with experience. If
some way were found to make some of the
stranger predictions of physics experiable, I think
there's a good chance humans would be able to
understand them.

[+] I didn't particularly enjoy sets education in grammar school, but found it was much more interesting when I as able to experience it with members of the opposite sets.

So a union of a set with it's anti-set (the set of
anti-elements for each element of the initial set)
would be the empty set.

In this regime, unions of elements and elements,
or anti-elements and anti-elements would all work
as usual. As would intersections.

But unions of sets and antisets would become
their symmetric difference, and as scad scientist
points out, it screws up the order invariance of
set operations.

However, there is a generalisation; instead of just
having elements and anti-elements, have
countable numbers of elements which can be
negative. So traditional sets are just sets with a
count of 1, and a union is a max(nA,nB) of the
numbers of the element in A and B.

Thus if
A = {1(-1), 2(-1)}
B = {2(+1)}
C = {2(+1), 3(+1)}

Then a count-keeping version of union Uc:

A Uc B Uc C:
A Uc (B Uc C) = {1(-1), 2(-1)} Uc {2(+2), 3(+1)} =
{1(-1), 2(+1), 3(+1)}
(A Uc B) Uc C = {1(-1)} Uc {2(+1),3(+1)} = {1(-1),
2(+1), 3(+1)}.

In this algebra, every set (including the empty set)
is in fact a set of all possible elements, where any
elements not in the sets are equivalent to
members with a count of zero.

//Those don't exist in the accountancy literature I would venture.//

Accruals. For example, I finish Year 1 with £25 income owing to me, so I enter a journal crediting my I&E account with an extra £25 income and debit £25 to debtors on the balance sheet. The journal is reversed into the next accounting period (Year 2) giving me a starting position of £25 debit on I&E and net total movements of zero on my balance sheet debtors. When the payment eventually comes in, you post a debit against cash on the balance sheet and the other side of the entry cancels out the accrual in I&E so that my net income for the period shows as zero. Accountancy.