# Why is it empty and why is there only one empty set?

## Main Question or Discussion Point

Hi, I joined recently and I'd like to start a do-discussion about the empty set. Why is it empty and why is there only one empty set?

It's "full of nothing", right?

Apparently the ontology of a singular but empty set is still a hurdle for the philosophers, yet mathematicians use this logical, infinitely 'variable' thing with 'nothing' in it blithely, or regularly. Is it the only regular thing in existence, since it contains the future or all future imaginary events? Does it, in fact, represent imagination?

from wikipedia: (my underlines)

Axiomatic set theory

In Zermelo set theory, the existence of the empty set is assured by the axiom of empty set, and its uniqueness follows from the axiom of extensionality. However, the axiom of empty set can be shown redundant in either of two ways:

* A logic such that provability and truth hold for both empty as well as nonempty domains is called a free logic. Set theory is almost never formulated with free logic as its background logic; hence many theorems of set theory are valid only if the domain of discourse is nonempty. Canonical axiomatic set theory assumes that everything in the (nonempty) domain is a set. Therefore at least one set exists; call it A. By the axiom schema of separation (a theorem in some theories), the set B = {x | x∈A ∧ x ≠ x} exists and, having no members, is the empty set;

* The axiom of infinity, included in all mathematically interesting axiomatic set theories, not only asserts the existence of an infinite set I (from which B in the preceding paragraph may be constructed), but typically requires that the empty set be a member of I.
Assertion is therefore the 'realization' that the imagined (empty thing) in {}, is in fact real...?
Do we then imagine time itself, since all epistemological 'experience' initially is in {}?

And of course, we epistemologically know that the universe was in {}, and now there is exactly one (empty) universe, because it's full of "things" from the empty future that the full past asserts.

What do you think - do we require that there be exactly one {} (which is provable mathematically), so that it's always empty, since otherwise the universe would not have been asserted which, obviously it has been?

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russ_watters
Mentor

There is a difference between "empty" and "full of nothing". "Full of nothing" implies that "nothing" is a "thing". It isn't. It is the absence of "things".

apeiron
Gold Member

There is a difference between "empty" and "full of nothing". "Full of nothing" implies that "nothing" is a "thing". It isn't. It is the absence of "things".
Yet to have an absence of things is an argument that there is a context against which that absence can be measured - and so there is something.

That was the point of the empty set approach. Even a void has dimensionality.

As to Deferro's comment: "Apparently the ontology of a singular but empty set is still a hurdle for the philosophers, yet mathematicians use this logical, infinitely 'variable' thing with 'nothing' in it blithely, or regularly."

The empty set was the basis of Russell's attempts to establish a philosophical foundation to mathematics as an enterprise. It did not really pan out. Set theory has since been subsumed under category theory - the new putative foundations for math.

And on the general idea that the purpose of the universe is to bring about a state of nothingness, a cold expanding void, I think there is indeed mileage in that.

russ_watters said:
There is a difference between "empty" and "full of nothing". "Full of nothing" implies that "nothing" is a "thing". It isn't. It is the absence of "things".
The empty set doesn't exist?
But, it does; otherwise there "would be nowhere to put" future events that don't exist yet; there would be: an empty set which is always empty, therefore it exists because "there is/is not a place to put events that haven't happened".

Therefore, logically: "the empty set is infinitely full of all infinitely extended events"; and, there is exactly one empty set.

1) the empty set is the absence of things
2) the empty set isn't a thing
3) there is a difference between empty and full
4) the empty set is full, but empty = always full of emptiness
5) the empty set is infinitely full of the difference between full, and empty

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Math Is Hard
Staff Emeritus
Gold Member

The empty set doesn't exist?
But, it does; otherwise there "would be nowhere to put" future events that don't exist yet.

Therefore, logically: "the empty set is infinitely full of all infinitely extended events"; and, there is exactly one empty set.

1) the empty set is the absence of things
2) the empty set isn't a thing
3) there is a difference between empty and full
4) the empty set is full, but empty
5) the empty set is infinitely full of the difference between full, and empty
Can't we just say the empty set exists as a mental construct, and that's all?

Can't we just say the empty set exists as a mental construct, and that's all?
Sure, but half the fun of philosophy 101 is watching students contort through misapplied abstraction.

Another thing in the empty set is potential energy.

Or, topologically speaking, potential is nullspace.

Any logical space has to have room to extend into (see quote above), and there has to be potential for that extension - nullspace must exist.
You can't construct a regular solid if it doesn't. Or divide it - it has to be divided through its nullspace or sectional 'function'.
Or, in ordinary language you have to have a way to cut or section a solid in order for it to exist.

This is fundamental to a certain paradox, that says you can section or divide a solid and re-construct it with twice its volume.
A sphere is twice its volume if you section it a certain way.

Another one is, the explanation of the origin of universal mass; and the location of a photon emitted by a particle with charge and spin.

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It is problematic to define things negatively. Could we not say that an empty set is a set that contains precisely zero elements and then that zero is the identity for addition? Does this avoid a negative definition?

Can we say that addition isn't production?
Since you can't actually increase the volume of a Platonic solid, or a sphere by sectioning it as the paradox implies (the Banach-Tarski). Nonetheless you can imagine doing this paradoxically, so the addition of volume is the increase or expansion in the nullspace, but where is it?

This has a connection to switching theory and networks, a Benes net is a mesh that is a map of spatially sectioned "signals" where the sectioning function is the crossbar w/2 dimensions or inputs, as they get called. This is the time function, so that the sectioning creates a "timed mesh" representation in a regular, Platonic sense; that is, of a regular solid sectioned with time, so the potential or nullspace is the logic itself.

Assertion is the "we know how to" proposition of algorithmic design, in that if you assert you know "how to write a program" in a Turing sense, then you also know how to induce a predecessor for such a program, which "is not" the Turing machine, it's the Turing question.

Has anyone heard of a /the data-barrier model?

OK, since we're in the lounge, let's try a little recursive descent.

Since I was on the Banach-Tarski which is mere mathematics, and since there is no practical way to construct a sphere with twice its volume by sectioning it.

If we assume a cut can be made between points p,p' on the sphere -> +p,-p; and we can measure the area projected on the section, it's an outer wedge product of the triangle produced by the two points and the origin. This is a circular measure, of singular points.

Try to map these to some regular solid in a data-compressive fashion so the outer product has two Cartesian 'transfer functions' that map the area to an open disc as two points z(0,1) where $$\phi(z_{0,1} ,w) = \frac { ( z^4 + w^3 ) } {|z^2 + w^3|}$$
and $$w$$ is "just a formula". We first compress it, so that the numbers 2,3,4 are represented as a phase-space of numbers of some kind; note that any encoding for three things is available - these are the familiar base 10 numerals we understand, but there are only three of them in the formula, so we can abstract, or Godelize them as much as we feel free to.

The formula appears to map z(0) and z(1) to a single z polynomial, we might want to keep a map of both initial values for z; however, two things can be represented by 0,1 since this has a direct transformation, in base2 to -1,1, which induces the Hadamard recursion, but we need a rotation so that we can have 1,1 (as either a row,column), this is only available using a direct product, so we need to extend the transfer (from a single volume to two copies of it) to a Boole sum and product formalism, or Clifford's algebraic decomposition, we need a set of binary functions;
luckily we can polarize electron spin, and things like photons of light when we look through filters (sunglasses), s.t. we are operating on the transfer function(s) of an involutional transform (represented by a filter map of sunglasses) on polarization of spatial extent in a set of "packets of energy"; we call this visible light 'colored light', mnemonically.The other mnemonic is that, since we untie or unwrap the knot, we have to slice space in a Cartesian way.
If we try to slice the time knot, encoded spatially by photons, the physical momentum product from the vacuum, it isn't a knot anymore.

We have a constrained quantum algebra for 5 dimensions of the domain of mass,color,spin,charge,force which is the Georgi-Glashow model in the rotation group SU(5). Then propositionally:

if SU(5) then SU(3),SU(2)
if SU(3)xSU(2) then SU(2),SU(1)
if SU(2)xSU(1) then U(1)
if SU(1) then gamma,e

where e is a fermion in the vacuum and gamma is a distribution function for the "energy scale" of the symmetries, over m which is total mass algebra (as sums,products). A triple quark algebra acts on fermions at distance scales that expand su(2) algebras that act at electroweak scales to expand the circle group. This is encoded in an su(5), the GG model. It requires a successor algebra so then GG is presumably the trivial case for it.

We note that su(5),su(3),su(2),U(1) is not 4,3,2 in our imaginary formula, the sliced sphere at points p,p' = z(0,1). If we find a cross-product su(2)xsu(2) -> su(1)xU(1) s.t. 2x2 = 2 + 2; and then 2 + 2 -> 1 + 1 in imaginary units of space,time -> energy, we might unlock the algebra (so it can be extended); since, the origin or vertex of "mass generation" is unknown, but conjectured. We are still constructing the [machine that answers the Turing questions in] spatial/temporal extent that encodes the vacuum geometry at near the SU(5) full symmetry.

Then if GG is a subgroup g' of a 'higher symmetry' s.t. g(v,e) -> g' we will have a map, which is possibly the trace of the convex involution H, represented in the transition (o|0,1) -> 1,-1; for the H' pullback where o is n observers on a sphere, or spheroid, a lattice L(V,E). Faces over the lattice are a convex hull in graph g'.

Here, L represents a density of vertices, or the convex product is a sum of observers o'(n) in the convex section of a sphere who can triangulate (the time-lattice) on Sx2. Or a density $$\rho$$; then $$(\rho|vT)$$ is Boltzmann-Faraday complete as the map M|(g,g'); this density operator is the 'coloring function' of spacetime's causal structure. Mere math, however, an algebra of mixed/pure states.

ed: I would like to write a textbook for those young students who "show promise" in academic pursuit, particularly those with good creative or artistic (coloring, music, mathematics) skills, which as it happens my neice and nephew are, so I want to discuss, and develop ideas for young minds that aren't a lot of confusing symbology - since the kids would rather play than study, you have to turn it into a kind of search for something mysterious and hidden - a color that is never there but extends itself constantly (nullspace contains nullcurves and nulltime line elements) and coloring Platonic solids reveals the symmetry in the causal substructure = space,time they observe it in. (and reapply for my postgrad ad eundum, so far my BE,BSc isn't research-level, so this is all homework for Dr Lloyd's assignment: uncover the symmetries of spacetime that reveal su(2) as Pauli's algebra)

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And for those stuck with how to step simply, or drive from algebra and sectioned surfaces/volumes to density matrix mechanics without going hard through Heisenberg-Shcroedinger and turning into Dirac-Pauli, you can derive your very own personal su(2) algebra, sigma-finite and evrything, and also su(2)xSO(3,1), and called a slice group, also called a Rubik's group (of colored and sectioned puzzles), or handy mnemonics.

You can scramble and unscramble them over a surface too, as an exercise, which is then a way to flatten something, a color-algebra. Other derived products that are volume sections (or products in a quotiented space) relate to the icosahedron and dodecahedron which induce the cube in R3.

Which can be handy. Or otherwise illustrates why the puzzles are buildable in that their predecessor was a pile of wooden blocks and some elastic rubber sectioned into loops, wadaya know? He could have colored these bands - constraints on blocks b(1,2,...,n) -and realized the same construction, or perhaps derived another kind, after all we use similar devices to keep crystals of semiconductor together, after we 'condense' them on a surface.

And of course, we epistemologically know that the universe was in {}, and now there is exactly one (empty) universe, because it's full of "things" from the empty future that the full past asserts.

What do you think - do we require that there be exactly one {} (which is provable mathematically), so that it's always empty, since otherwise the universe would not have been asserted which, obviously it has been?