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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.

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No.

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alright, thank you lol

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this doesn't at all relate?

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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.

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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.maxulu said:

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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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

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)

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

“More than infinite” doesn’t mean anything. Something is either finite, or it is not.maxulu said:My thought was that the issue with the corner was perhaps related to the fact that you are switching dimensions, and amorethan infinite number of layers are needed to make a 2D object into 3D. Hence my idea about Cantor.

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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

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

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: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.

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.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

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.

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.

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.

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.

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.

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