# Connection between Set Theory and Navier-Stokes equations?

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• maxulu
In summary, the video discusses how at the point of a right angle, the equation for fluid velocity shows infinite velocity. It's possible that this is related to Cantor's solution to Zeno's Paradox of distance. However, I don't think that the article actually provides a solution to the paradox.

#### 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

nuuskur
No.

pbuk and fresh_42
alright, thank you lol

this doesn't at all relate?

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.

maxulu said:
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.

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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)

maxulu said:
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.

maxulu said:
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.

suremarc said:
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

maxulu said:
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.

maxulu said:
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.

maxulu said:
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.

## 1. What is the connection between set theory and Navier-Stokes equations?

Set theory is used to mathematically describe the concept of a set, which is a collection of objects. In the context of Navier-Stokes equations, sets are used to represent the spatial domain in which the equations are solved. This allows for a more rigorous and precise mathematical treatment of the physical phenomena described by the equations.

## 2. How does set theory help in understanding Navier-Stokes equations?

Set theory provides a foundation for the mathematical framework of Navier-Stokes equations. It allows for the precise definition of concepts such as boundaries, domains, and subsets, which are essential for solving and analyzing the equations. Additionally, set theory enables the use of advanced mathematical techniques, such as topology and measure theory, to study the properties of solutions to the equations.

## 3. Can set theory be used to simplify Navier-Stokes equations?

While set theory is a powerful tool for understanding and solving Navier-Stokes equations, it does not necessarily simplify the equations themselves. The equations are inherently complex and cannot be reduced to a simpler form using set theory. However, set theory can aid in organizing and analyzing the equations, making them more manageable for study.

## 4. Are there any limitations to using set theory in relation to Navier-Stokes equations?

Set theory is a fundamental mathematical concept and is applicable to many different fields, including fluid dynamics and Navier-Stokes equations. However, like any mathematical tool, it has its limitations. Set theory may not be able to fully capture the intricacies of physical phenomena, and it may not be suitable for certain types of problems. In these cases, other mathematical methods may be more appropriate.

## 5. How does the use of set theory impact the solutions to Navier-Stokes equations?

The use of set theory does not directly impact the solutions to Navier-Stokes equations. The equations themselves remain the same regardless of the mathematical framework used to analyze them. However, set theory can provide a more rigorous and systematic approach to studying the solutions, leading to a deeper understanding of the physical processes described by the equations.