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]

0oohttps://0oo.li/proj...nality-sets-project Quantified Cardinality Sets Project -- let it be the initiative to bring about a specific axiomatization of negative and generally quantifiable cardinality into broader use. [Inyuki, Jun 03 2021]

Rubix Cube and Group Theoryhttp://people.math....0Rubik's%20Cube.pdf Here traditional sets underly group theory, which in tern is applied to solve some problem (it's a nice read too as it happens) can the same be extended for i-sets? [zen_tom, Jun 04 2021]

[+] 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.

[pertinax], the ideas had spotted me. Thanks for the bun. Just thinking,
how would the foundations of mathematics look like from this
perspective. Will share any more significant updates here.

These look like sets, but they deviate from the definition of what a set is, by way of this new
negation feature. So, what we have are collections of objects that are largely set-like, but which
allow for pairwise negation of associated elements. You could call them N-Sets or Inuki-Sets or
something, but since they're defined differently to established sets, then they are different beasts.

You could still build an edifice of mathematics on top of these things however, but I think you lose
the link between sets and cardina1l/natural numbers - perhaps you could build a definition on top of
Inyuki-sets that underpins the integers instead - but cardinal numbers really don't have negatives,
it's not what cardinality *does*.

Integers do have negatives, sure - we're not saying negatives don't exist, but if you smear cardinals
and integers together, you destroy what it means to be cardinal. The |size| of a set is its size, while
the contents of a set normally carry whatever signature they carry. A set of {1, 1', 2, 2'} *does* have
4 elements, I can see them, and I can count them. There are definitely 4. Perhaps you could define an
I-Count property where the i-count of x and x' is zero, and the i-count, or i-cardinality of the above
set is zero, but counting remains counting and there are 4 elements in that set.

If you change what cardinality means, then any number of sets, {}, {1,1'}, {1,1,1',1'} and {0,0',...,n,n'} all have an i-cardinality of zero,
while having traditional cardinalities from truely being zero to a size of |infinity|. In effect, i-cardinality becomes a 2-value parameter,
(x,y) where x is positive and y is negative and x-y is the i-cardinality. Actually, that depends - it's 2-values if you can match *any*
positive element with any other negative element. If negation has to be based element-wise - i.e. 1 is only negated by 1', and 999 is only
negated by 999', then things become tricky - e.g. what's the i-cardinality of { 1, 2',3, 4', 5 }, is (+3,-2) a valid i-cardinal value, or would this have
a value of |5| ?

[zen-tom] the idea states //subtracting an element that is not in a set// so I don't think you can just subtract numeric values. Its more like antiparticles maybe? //Just like antimatter in Physics// A proton plus an antiproton cancel one another out, but a proton plus an anti-neutron don't.

Or like pegs and holes, a round peg and a round hole combine to give an empty set with no pegs and no holes, but a round peg and a square hole don't combine and so their set has two elements. Take another set containing a square peg and a round hole, you now have two sets each with two elements in. Combine the two sets and the combined set has no elements since the pegs fit in the holes.

Though fitting pegs into holes doesn't usually result in humungous amounts of energy being released.

Your set { 1, 2',3, 4', 5 } I suppose has 5 elements; but you could combine it with a set of three elements {1', 2, 3} to end up with a set [3, 3, 4', 5} with four elements.

Am I reading this right?

In which case there need not be only two different "flavours" of elements? It could all go a bit rock-paper-scissors if we are not careful...

[pocmloc] yes, I think so - I just wanted to be sure what's being proposed and what that might lead to - messing
with cardinality is pretty fundamental.

How about a different application, in group theory, there's the idea of a traditional set whose elements are such
that two elements in the set, combined with an operation leads
you to another element in the set. So using clock-integers, {0,1,2,3,4,5} the operation +, and knowledge that for
+, 0 acts as the identity (x+0=x), you can build a transition
table where each element relates to another element by means of some + operation. So the table:

tells us that 2+3=5, and 1+1=2, 3+3=0 and 4+4=2. Setting up this structure lets you do interesting things like
crack codes, solve rubix cubes and other tricky problems.

The axioms to make group theory work are:
1) Closure - All the elements of the set are reachable from a combination of some pair of other elements of the set
via the chosen operation. any a o b is an element of G
2) Associativity - ( a o b ) o c == a o ( b o c )
3) Identity - there must be some element i for which i o a = a
4) Inverse element - for every a, there must exist some b such that a o b = the identity element i

You can extend this to non-clock-like, infinite sets and a whole bunch of really applicable situations and all that
together makes the starting place for group theory - to make
i-sets interesting, it would be good to come up with an operation that allows you to get from the x world into the
x' universe, or which might explain what x o y' might yield -
which might be interesting in of itself, but isn't yet clear. It might also be interesting to come up with some
simple axioms that would let you recreate group-theory on sets
that included negating inverses - similar things have been thought about, for examples regards the group theory
covering rotations and mirrorings - where mirroring might be
analogous to x vs x' - i.e. one undoes the other via axiom 4, except that the translation allows both things to
exist, only in "flipped" or mirrored form rather than
annihilations - the annihilations part kind of smashes up lots of otherwise useful maths.

Put another way, sets are useful because they serve as the basis for lots of other things (like group theory), to
make i-sets equally interesting, you'd have to come up with
some working examples of analogous structures that could be built on these alternative foundations. But I guess the
idea is suggesting someone does just that, so fair-play.