It’s just a magnificent toy, this ZF construction. And those others set theories as well. No wonder some people here don’t want it to be broken. With passion, may I add?
Let us take the aleph-zero dimensional R. Countably infinite dimensional Euclidean space, in other words. Then take a point T and all those points which are finitely far away from this point T. By the standard metrics of sqrt(dx1^2+dx2^2+...).
This space is separable into continuum many such subspaces. Where in every such subspace every two points are close. Close means only a finite distance. And every two spaces are far from each other. Where far means an infinite distance between any two points from two different subspaces.
No wonder some people here don’t want it to be broken. With passion, may I add?
Well, whomever is going to show ZF to be contraddictory is sure to receive great glory in the mathematical community, and is going to be probably considered the next Godel. If only ZF wasn’t already been shown coherent with the notion of an inaccessible cardinal. Which has shown to be coherent with the notion of a measurable cardinal. Which has shown to be coherent with the notion of a compact cardinal… and so on. ZF’s coherence is a hairy, hairy subject.
Then take a point T and all those points which are finitely far away from this point T. By the standard metrics of sqrt(dx1^2+dx2^2+...).
Well, you cannot do that: the euclidean norm is not defined for an infinite-dimensional space.
Where in every such subspace every two points are close. Close means only a finite distance. And every two spaces are far from each other. Where far means an infinite distance between any two points from two different subspaces.
I have no idea what this paragraph means. Yes, I can divide a many dimensional space into continuum continuums, that’s a basic property of cardinal arithmetics. And in each subspace two points can be as closed as you what. But the rest of the paragraph I cannot parse. Besides, what is the point of all this? What is it that you’re trying to show?
you cannot do that: the euclidean norm is not defined for an infinite-dimensional space.
Why not? It is the square root of the sum of (dxi)^2, where i goes through all dimensions. Sometimes it is a finite value. Otherwise the distance is infinite.
The points T0(0,0,0,0....) and T1(0,1/sqrt(2),1/sqrt(4),1/sqrt(8)...) are 1 apart.
This is the 3D version of the countable antichain condition (commonly known as c.c.c.).
c.c.c. is implied by a property called separability, which is part of the definition of the real line (the unique linear complete separable order).
So you can fit aleph1 “cubes” only if you operate in a modified notion of space which is not c.c.c.
On the other hand, the real line contains aleph1 points only in some model of set theory. The precise quantity is 2^aleph0.
It’s just a magnificent toy, this ZF construction. And those others set theories as well. No wonder some people here don’t want it to be broken. With passion, may I add?
Let us take the aleph-zero dimensional R. Countably infinite dimensional Euclidean space, in other words. Then take a point T and all those points which are finitely far away from this point T. By the standard metrics of sqrt(dx1^2+dx2^2+...).
This space is separable into continuum many such subspaces. Where in every such subspace every two points are close. Close means only a finite distance. And every two spaces are far from each other. Where far means an infinite distance between any two points from two different subspaces.
You probably are familiar with this also.
Well, whomever is going to show ZF to be contraddictory is sure to receive great glory in the mathematical community, and is going to be probably considered the next Godel.
If only ZF wasn’t already been shown coherent with the notion of an inaccessible cardinal. Which has shown to be coherent with the notion of a measurable cardinal. Which has shown to be coherent with the notion of a compact cardinal… and so on. ZF’s coherence is a hairy, hairy subject.
Well, you cannot do that: the euclidean norm is not defined for an infinite-dimensional space.
I have no idea what this paragraph means. Yes, I can divide a many dimensional space into continuum continuums, that’s a basic property of cardinal arithmetics. And in each subspace two points can be as closed as you what. But the rest of the paragraph I cannot parse.
Besides, what is the point of all this? What is it that you’re trying to show?
Why not? It is the square root of the sum of (dxi)^2, where i goes through all dimensions. Sometimes it is a finite value. Otherwise the distance is infinite.
The points T0(0,0,0,0....) and T1(0,1/sqrt(2),1/sqrt(4),1/sqrt(8)...) are 1 apart.
A metric is supposed to be always finite. Note the round right bracket in https://en.wikipedia.org/wiki/Metric_(mathematics)#Definition.
Fixed link.
This is probably not the intended link.