Connection between Set Theory and Navier-Stokes equations?

[maxulu]

Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at some point the fluid is switching the dimension, which are independent yet interconnected. just a feeling

 

[martinbn]

No.

 

[maxulu]

alright, thank you lol

 

[maxulu]

I found this article: https://www.researchgate.net/publication/239805005_Systems_of_Navier-Stokes_Equations_on_Cantor_Sets

this doesn't at all relate?

 

[suremarc]

I’m not aware of a specific solution to Zeno’s paradox proposed by Cantor.

Turbulent flows are well-known to have fractal-like properties. I’d say that fluid dynamics are “related” to set theory insofar as fractals are. It’s worth noting that fractals existed before set theory — they came up in the early days of analysis — and set theory is much broader than topology, calculus, or geometry, all of which are central to the study of fractals.

 

[Jarvis323]

Hi, I saw this video by numberphile, and near the end they mention how at the point of a right angle the equation shows infinite velocity for fluids. I'm wondering if this isn't perhaps related to Cantor's solution to Zeno's Paradox of distance (there's always a midpoint). Because I feel like at some point the fluid is switching the dimension, which are independent yet interconnected. just a feeling

Isn't calculus the solution to Zeno's paradox? This is relevant to not only Navier-Stokes, but pretty much all systems of differential equations I guess.

In many cases, with our continuous models, we do have points where the model breaks down due to it being a continuous model that assumes the underlying state-space of the phenomena can be infinitely divisible. In reality, at some deeper level than the system of equations you're using, there will be something different, and probably more discrete like. In flows, there are actual discrete molecules moving around. If you zoom in on a corner into the infinitesimal, there will be no molocules there, only vacuum.

 

[maxulu]

thank you for the responses

I will say that the highest math I ever took was Calculus II and I'll be if I remember anything from it or Calc I other than finding derivatives/integrals and why that's important.

My thought was that the issue with the corner was perhaps related to the fact that you are switching dimensions, and a more than infinite number of layers are needed to make a 2D object into 3D. Hence my idea about Cantor.

Yes, Cantor developed set theory, if I understand it correctly, to solve Zeno's (most famous) paradox. Zeno was quite an interesting philosopher and my favorite is his Paradox of Place (what is space "contained" in - if nothing, how can it exist, if something, what is that something contained in and so on)

 

[suremarc]

Yes, Cantor developed set theory, if I understand it correctly, to solve Zeno's (most famous) paradox.

If so, then he failed to follow up properly, because nowhere has he directly mentioned Zeno’s paradox in any form as far as I know. I think Cantor’s set theory helped advance set theory and topology by dealing with infinities, but the concept of limits and other constructions in analysis existed long before set theory.

My thought was that the issue with the corner was perhaps related to the fact that you are switching dimensions, and a more than infinite number of layers are needed to make a 2D object into 3D. Hence my idea about Cantor.

“More than infinite” doesn’t mean anything. Something is either finite, or it is not.

 

[maxulu]

If so, then he failed to follow up properly, because nowhere has he directly mentioned Zeno’s paradox in any form as far as I know. I think Cantor’s set theory helped advance set theory and topology by dealing with infinities, but the concept of limits and other constructions in analysis existed long before set theory.“More than infinite” doesn’t mean anything. Something is either finite, or it is not.

I'm sorry, but I don't agree with that at all. Cantor's set theory specifically explains Zeno's Paradox: the fact that a finite distance has an infinite number of midpoints. And the fact that there's different cardinalities, I hope you know, that the irrationals outnumber the rational numbers by a higher set of infinity.

this isn't a discussion of Cantor and our knowledge of it. I'm curious about a connection to Navier-Stokes. please stick to the topic and don't start arguments

 

[suremarc]

I'm sorry, but I don't agree with that at all. Cantor's set theory specifically explains Zeno's Paradox: the fact that a finite distance has an infinite number of midpoints.

This is exemplified in the geometric series, which was known to mathematicians centuries before Cantor.

And the fact that there's different cardinalities, I hope you know, that the irrationals outnumber the rational numbers by a higher set of infinity.

A “higher set of infinity” is still infinite nonetheless. There may be multiple infinities, but they are all infinite. It is meaningless to say something is “bigger than infinite” because “infinite” does not imply a specific cardinality; it is a property that sets may or may not have.

this isn't a discussion of Cantor and our knowledge of it. I'm curious about a connection to Navier-Stokes. please stick to the topic and don't start arguments

You say I am starting arguments, but I’m trying to have coherent discussion. Perhaps there is something interesting to be said about Navier-Stokes and set theory, but nothing will come of a discussion based on premises that are vague, incorrect, or not even wrong.