How big is a point? (Can be that the Natural Numbers are Finite?)

[ydoaPs]

On 10/1/2021 at 11:01 AM, Conscious Energy said:

Which law says a point has absolutely no space and a minimal length? How is it reasoned that a point of space has no physical extent?

When you're talking about volumes and hypervolumes, your set needs to be equipped with a measure function. The technicalities are too much to go into in a single post, but see the wiki link if you're interested.

The standard measure used on the Reals is the Lebesgue measure where the length (in one dimension) is just the difference of the endpoints.

So, if we're looking at the length of a point p, we need to just take p-p, which is, of course, 0.

[Markus Hanke]

A point in 3-space can be considered as the limit of the volume of a sphere with r-> 0. You can formally set up the volume as an integral function, and then take the limit. 

[MigL]

I thought it was enough for a 'point' to be defined as dimensionless.
If Conscious Energy wants to define something else, with physical extent, he should call it something else.

[studiot]

18 hours ago, ydoaPs said:

When you're talking about volumes and hypervolumes, your set needs to be equipped with a measure function. The technicalities are too much to go into in a single post, but see the wiki link if you're interested.

The standard measure used on the Reals is the Lebesgue measure where the length (in one dimension) is just the difference of the endpoints.

So, if we're looking at the length of a point p, we need to just take p-p, which is, of course, 0.

You do not need a measure in standard set theory to define a point, any more than you need a coordinate system or a dimension theory.

A point, eg in the Reals, is a subset (partition) with exactly one entry.

But yes using using the apparatus of epsilon delta, or of measure theory leads to your definition or Markus' limit.

 

6 hours ago, MigL said:

I thought it was enough for a 'point' to be defined as dimensionless.
If Conscious Energy wants to define something else, with physical extent, he should call it something else.

Conscious Energy has left the room, courtesy Swansont.

But it was never clear whether they were referring to a mathematical or physical point.

[Country Boy]

It's NOT a law, it is a definition.   A geometric point is not a physical object.  It is an a mental concept that identifies a position and has  no other properties.

[joigus]

1 hour ago, Country Boy said:

It's NOT a law, it is a definition.   A geometric point is not a physical object.  It is an a mental concept that identifies a position and has  no other properties.

I couldn't agree more. That doesn't mean definitions are free from responsibility to be useful. 👍

[Country Boy]

_{I think that ConciousEnergy, saying "in space" is confusing mathematics and physics.  In mathematics a point is defined by its position.  It has NO "area" or "volume".   Talking about a "point in space" you are talking about physics and when you apply mathematics to physics it only matches approximately,}  

[SuperSlim]

I heard a prof say in a lecture on gravity, on youtube of course but it was a university in Germany, that a point is like dust. The Euclidean plane is dust, the only thing keeping all the points together is a relation. There isn't any "glue" between two points no matter how close they are.

Edited February 13, 2022 by SuperSlim

[studiot]

54 minutes ago, SuperSlim said:

I heard a prof say in a lecture on gravity, on youtube of course but it was a university in Germany, that a point is like dust. The Euclidean plane is dust, the only thing keeping all the points together is a relation. There isn't any "glue" between two points no matter how close they are.

Hello and welcome.

I expect your 'prof' was talking about the Cantor set or Cantor dust.

This reference may be a bit advanced but has some good diagrams

https://en.wikipedia.org/wiki/Cantor_set

 

Ask if you want something simpler.