Difference between revisions of "Public vs Private Networks"

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== Private Networks Can Always Be Made Strict Subsets Of Public Networks ==
== Private Networks Can Always Be Made Strict Subsets Of Public Networks ==


This is a proof demonstrating that there are no private (restricted) Grassland type networks that can't be made a strict topological subset of a public/decentralized one. Such that there are no private networks, S<sub>p</sub>, or Public Networks, S<sub>d</sub>, in which S<sub>p</sub> ⊂ S<sub>d</sub> is impossible.  
This is a proof demonstrating that there are no private (restricted) networks that can't be made a strict topological subset of a public/decentralized one. Such that there are no private networks, S<sub>p</sub>, or Public Networks, S<sub>d</sub>, in which S<sub>p</sub> ⊂ S<sub>d</sub> is impossible.  


This isn't to say that private networks are bad, necessarily. But they should be used with caution and on a limited scale since the act of creating a private "stack" on the earth's "heap", would represent a loss of information for all other parties and remove the mutual knowledge required to maintain peaceful [https://en.wikipedia.org/wiki/Nash_equilibrium Nash Equilibrium].  
This isn't to say that private Grassland networks are bad, necessarily. But they should be used with caution and on a limited scale since the act of creating a private "stack" on the earth's "heap", would represent a loss of information for all other parties and remove the mutual knowledge required to maintain peaceful [https://en.wikipedia.org/wiki/Nash_equilibrium Nash Equilibrium]. And could lead to assured destruction scenarios.





Revision as of 13:45, 3 June 2020

Private Networks Can Always Be Made Strict Subsets Of Public Networks

This is a proof demonstrating that there are no private (restricted) networks that can't be made a strict topological subset of a public/decentralized one. Such that there are no private networks, Sp, or Public Networks, Sd, in which Sp ⊂ Sd is impossible.

This isn't to say that private Grassland networks are bad, necessarily. But they should be used with caution and on a limited scale since the act of creating a private "stack" on the earth's "heap", would represent a loss of information for all other parties and remove the mutual knowledge required to maintain peaceful Nash Equilibrium. And could lead to assured destruction scenarios.


As Topological Spaces:

It is sensible to define Grassland networks as topological spaces (Topological space). That is to say, a set of points along with a set of neighbourhoods (Topological Neighbourhood) for each point.

1) We'll define each network, as a 3D, topological set, S, upon and the surface of an earth shaped spheroid

2) All the points of S are made up of each camera's raycasted principal point and the neighbourhood of points around it within the frame. The raycasted principal point being the initial point at which an imaginary line drawn outwards from the center of the camera frame intersects an external, real-world object (an unprojection from the camera to the world).

3) We'll denote any private network as Sp, and any decentralized (public) network as Sd.

4) And we'll define the "domination" of one set, S, by another set, S', to be whenever the points (raycasted pixel points) in S can be approximated (substituted) to a higher degree by points (raycasted pixel points) in S' than the other way around. To put it another way, the probability that any given x ∈ S is arbitrarily close to some y ∈ S' is higher than the probability that any given y ∈ S' is arbitrarily close to some x ∈ S. Therefore, the number of points in S' is greater than the number of points in S (even if it's just one more camera/node)


Since both Sp and Sd are on the surface of the earth, a spheroid having no edges, no set, S, can contain ALL its limit points (Limit Point) to the effect of keeping points of another set, S', from approximating points of S. And it goes without saying, that this is true along the vertical axis. So even continent's aren't topologically compact. They don't contain their limit points in regards to camera placement since along vertical axis the "sky's the limit". Which means there's always some way for points in an unrestricted Grassland network, Sd, to approximate more points in Sp. Topologically speaking, a greater number of points in Sp can be shown to be merely a point in a neighbourhood of some point in Sd.

As Infinite Sets:

We can also demonstrate this by finding an embedding of S into the infinite sets. And showing any differences in cardinality between them. In the sense that some infinities are larger than others. The points of a private Grassland network would have to be embedded in the set of all Integers. It can theoretically stretch infinitely in both directions, [-infinity, ..., -3, -2, -1, 0, 1, 2, 3, ..., infinity], with an infinite amount of raycasted pixel points from a countably infinite amount of cameras. But it's still bound to certain values; Sp has more restrictions than Sd by definition. Therefore, a decentralized (public) Grassland network, because it doesn't have those restrictions, finds a complementary embedding into the Reals, [-infinity, ..., -Pi, ..., -1.001, ..., -0.001, ..., 0, ..., 0.001, ..., 1.001, ..., Pi, ..., infinity]. It's an infinity too but it's a much larger, uncountably infinite, infinity because it can theoretically stretch to infinity even between the Integers.

And it therefore approximates the Integers. Sp's point at "1" can be approximated by an infinite number of points in Sd. Moreover, "1" in Sp can be approximated to an arbitrary degree of accuracy by points in Sd from both sides, "0.9999...9" and from "1.0000...1". This is because y ∈ Sd has the freedom to approach any x ∈ Sp along every dimension of the space they're embedded into.

Conclusion:

So since the more points of Sp can be approximated to an arbitrary degree of accuracy by points in Sd than the other way around, and since Sd therefore contains the limit points of Sp, we can say that the set, Spn, comprised of the neighbourhoods of all points in Sp is merely a smaller subset of the set, Sdn, a set comprised of the neighbourhoods of all points in Sd. And therefore Sp is dominated by Sd according to point 4 above.