Can Space-time be Continuous?

[sayak83]

In another thread, which I cannot find now, the argument was made that from Zeno's paradox one sees that space and time cannot be Continuous and has to be discrete. Quite apart from what science shows about space time eventually, it has been showed that Zeno's paradox can be resolved without much of a problem in modern Mathematics and set theory. Thus finite line segments can indeed be composed of an infinite number of zero dimensional points and such a line can indeed be composed out of an infinite number of ever decreasing intervals... both for intervals of time and intervals of space. The proof for this was requested. I believe the best one is the visual proof of Balzano-Weirstrass theorem (see below) which shows the idea of line segments made of infinite number of points or an infinite number of decreasing intervals is quite consistent.

Bolzano–Weierstrass theorem - Wikipedia

 

[Brickjectivity]

It was explained to me that the problem could be resolved by using a Cartesian Coordinate System, something Zeno did not have. In Mathematics there is an anologous problem to the question you are proposing that is called the Continuum Hypothesis.

 

[Revoltingest]

It was explained to me that the problem could be resolved by using a Cartesian Coordinate System, something Zeno did not have.

I'd explain it to Zeno by pointing out that it's all fine if he takes into account
that covering ever shorter distances requires ever shorter times.

 

[Brickjectivity]

I'd explain it to Zeno by pointing out that it's all fine if he takes into account
that covering ever shorter distances requires ever shorter times.

Zeno was arguing for continuous time I think, and the tortoise was his proof that time must be continuous in contrast to what Parmenides insisted.

 

[sayak83]

Zeno was arguing for continuous time I think, and the tortoise was his proof that time must be continuous in contrast to what Parmenides insisted.

A good commentary on his various paradoxes and solutions
Zeno’s Paradoxes | Internet Encyclopedia of Philosophy

 

[Kemosloby]

I don't know.

 

[bobhikes]

I'd explain it to Zeno by pointing out that it's all fine if he takes into account
that covering ever shorter distances requires ever shorter times.

Interesting so once the distance covered requires less than the standard measure of time, it is then no distance. Basically there is a unit of distance that is finite. Space time has a minimum size.

 

[Revoltingest]

Zeno was arguing for continuous time I think, and the tortoise was his proof that time must be continuous in contrast to what Parmenides insisted.

Continuity certainly exists on the macro level.
But that's the only level I know.

 

[Revoltingest]

Interesting so once the distance covered requires less than the standard measure of time, it is then no distance. Basically there is a unit of distance that is finite. Space time has a minimum size.

I don't see evidence that space & time exist as quanta.

 

[bobhikes]

I don't see evidence that space & time exist as quanta.

That's OK, I'm still on the fence that space time actually exists.

 

[Revoltingest]

That's OK, I'm still on the fence that space time actually exists.

For all practical purposes, it exists.
We use it to do things like finding a sammich place using my GPS.

 

[Aupmanyav]

@Revoltingest
That is because you have dismissed the other level.

 

[Revoltingest]

@Revoltingest
That is because you have dismissed the other level.

I don't dismiss it.
I recognize that things can exist even though I don't notice them.
Example....
Quantum mechanics is the realm of particle physicists.
But the only thing about it which affects me is the emergent
property of gears, shafts, bearings, levers, pulleys, nuts & bolts.

 

[Aupmanyav]

But the only thing about it which affects me is the emergent property of gears, shafts, bearings, levers, pulleys, nuts & bolts.

Is that true? What we think of the other level effects our actions and our interaction the other people.

 

[Revoltingest]

Is that true? What we think of the other level effects our actions and our interaction the other people.

Note that I'm not talking about reality....just what I perceive, aware of some severe limitations.

 

[Polymath257]

It was explained to me that the problem could be resolved by using a Cartesian Coordinate System, something Zeno did not have. In Mathematics there is an anologous problem to the question you are proposing that is called the Continuum Hypothesis.

The Continuum Hypothesis doesn't directly deal with Zeno's paradoxes. It has to do with different sizes of infinite sets and whether there is a 'size' between that of the counting numbers (1,2,3,..) and the set of real numbers (all possible decimal numbers).

The remarkable fact that the the CH is independent of the other axioms of set theory. If set theory with the current assumptions is consistent, then you also get a consistent system by adding CH. BUT, you will also get a consistent system by adding the negation of CH!

That leads to two, different, set theories!

 

[Polymath257]

Interesting so once the distance covered requires less than the standard measure of time, it is then no distance. Basically there is a unit of distance that is finite. Space time has a minimum size.

Rather, the possible *measurable* distances may be quantized. Even in theories with quantized space and time, the underlying structures are continuous (usually Hilbert spaces).