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• Chris S
in this sense:

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What do these symbols represent?

I remember I heard my teacher in the course 'real variable function' said in the start of the course: Zeno's paradox show that the Completeness Axoim for R is necessary. But I can't understand it. I think the crux is understanding the model that time is R, and if we want to use the completeness axoim, I guess we should understand time contains N, Z, Q step by step (but I cannot understand why time is pass by) That is what I am asking in this post, literally, what is time and why time is R?
I agree that ultimately the question is why time passes by and this I think is implicit in the paradox of the arrow. The intuition of time is inherent in consciousness and seems continuous. The intuition of continuity is part of the way our minds model the world outside of us. That is probably why one thinks of modeling time as R.

I think that the big advance in thinking during the modern scientific revolution was the idea that Nature obeys laws of motion. These laws do not tell you why there is time or motion but they do tell you what motion looks like and how to predict it. And in order to do this, they tell you that there must be certain instantaneous properties of things called initial conditions - at least in classical physics. This seems like a type of solution to Zeno's Paradox. There are laws of motion that objects obey and in order to obey them they must have intrinsic properties that exist in each instant of time. They can be seen to confirm Zeno's point that in order for motion to exist there must be instantaneous properties of motion. Interestingly, initial conditions exist prior to the motion and are not mere mathematical limits derived from the motion.

We may never understand the why of things. As someone once said - maybe Feynman - but I couldn't find the reference, ' Physics tells you nothing. Do you really think Nature does all of those calculations?'

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Delta2
I thought it would be interesting to look at examples of continuous paths that illustrate problems with the idea of continuous motion.

Here is a start:

There is a continuous path called the Devil's staircase that is defined on the unit interval and which starts at a value of zero at 0 and rises monotonically to a value of one at 1. Except on the Cantor set this function is constant in the sense that it is constant on each middle third. The Cantor set has measure zero so if Achilles is following this path he spends all of his time lounging around without moving.

All of the motion takes place on the Cantor set but the amount of time spent on it is zero. The motion is instantaneous.

One could have a sequel paradox which says that motion can not happen because in this case, Achilles reaches the tortoise without spending any time moving.

Since Cantor set is uncountable, Achilles traverses an uncountable number of points instantly. He also is converging on an uncountable number of Cauchy sequences. This is because the Cantor set though totally disconnected is complete and has no isolated points. Every point in it is the limit of Cauchy sequence. At the same time it contains no intervals no matter how small.

The Devil's staircase is an example of a continuous path that can not be explicitly written down but is known to exist because it is the uniform limit of a sequence of piecewise linear continuous functions. The key theorem is that the uniform limit of continuous functions is continuous. This theorem can be used to construct many unintuitive examples of continuous functions.

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I thought it would be interesting to look at examples of continuous paths that illustrate problems with the idea of continuous motion.

Here is a start:

There is a continuous path called the Devil's staircase that is defined on the unit interval and which starts at a value of zero at 0 and rises monotonically to a value of one at 1. Except on the Cantor set this function is constant in the sense that it is constant on each middle third. The Cantor set has measure zero so if Achilles is following this path he spends all of his time lounging around without moving.

All of the motion takes place on the Cantor set but the amount of time spent on it is zero. The motion is instantaneous.

One could have a sequel paradox which says that motion can not happen because in this case, Achilles reaches the tortoise without spending any time moving.

Since Cantor set is uncountable, Achilles traverses an uncountable number of points instantly. He also is converging on an uncountable number of Cauchy sequences. This is because the Cantor set though totally disconnected is complete and has no isolated points. Every point in it is the limit of Cauchy sequence. At the same time it contains no intervals no matter how small.

The Devil's staircase is an example of a continuous path that can not be explicitly written down but is known to exist because it is the uniform limit of a sequence of piecewise linear continuous functions. The key theorem is that the uniform limit of continuous functions is continuous. This theorem can be used to construct many unintuitive examples of continuous functions.
Phyisics normally assumes everything is k times differentiable, k at least 3; often, 'smooth' is assumed. That also addresses that motion is defined in an instant - a derivative must exist. Why this must be so is typically considered 'outside of physics'.

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Delta2
Phyisics normally assumes everything is k times differentiable, k at least 3; often, 'smooth' is assumed. That also addresses that motion is defined in an instant - a derivative must exist. Why this must be so is typically considered 'outside of physics'.
What about a cold and a hot body put into contact? At that instant the temperature will be discontinuous.

*now* and Delta2
What about a cold and a hot body put into contact? At that instant the temperature will be discontinuous.
I guess that‘s a good counterexample. However I did say normally, not always.

What about a cold and a hot body put into contact? At that instant the temperature will be discontinuous.
Temperature is not well enough defined to be either continuous or discontinuous. Like pretty much every other measurable property.

*now*, hutchphd and PeroK
Temperature is not well enough defined to be either continuous or discontinuous. Like pretty much every other measurable property.
It is not about the measurable property, it's about its mathematical model. Is temperature modeled by a smooth function? Not always.

Delta2
It is not about the measurable property, it's about its mathematical model. Is temperature modeled by a smooth function? Not always.
We all have different views of the elephant.

Your point seems to have been that our models are not always smooth and that, in particular, we feel free to switch from one smooth model to another smooth model and not quibble about a lack of smoothness at what we choose to model as a discrete transition. Agreed.

My point was that the smoothness in the two models (and the discreteness in the transition) was the result of an idealization, rather than a true and measurable consequence of the physical reality.

We all have different views of the elephant.

Your point seems to have been that our models are not always smooth and that, in particular, we feel free to switch from one smooth model to another smooth model and not quibble about a lack of smoothness at what we choose to model as a discrete transition. Agreed.

My point was that the smoothness in the two models (and the discreteness in the transition) was the result of an idealization, rather than a true and measurable consequence of the physical reality.
Well, my point is that you can never know if it was just an idealization or a faithful representation of reality. You can only make a finite number of measurements with finite accuracy, so you can never know if something is discrete or continuous/smooth. But the smooth models are very good and successful. You assume at any stage as much regularity as possible. My example was that sometimes that means non-continuous is the best possible.