Re: More complex numbers than reals?

On 7/21/2024 3:56 PM, Chris M. Thomasson wrote:

On 7/10/2024 2:30 PM, Chris M. Thomasson wrote:

On 7/9/2024 3:16 PM, Moebius wrote:

Am 09.07.2024 um 22:27 schrieb sobriquet:
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How do you define sets exactly?

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Actually, we don't _define_ the concept of /set/ by a "proper definition".
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Is there a specific set that corresponds to sqrt(2)?

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Well, rather a sequence (which is a certain kind of set in the context of set theory):
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(1, 1.4, 1.41, 1.414, ...)
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Does this set have an infinite number of elements analogous to the sqrt(2) having an infinite decimal expansion?

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Yes. See above. This sequence (called an /infinite sequence/) has infinitely many terms.
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It seems that the existence of something like sqrt(2) is already rather dubious.

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Oh, really?
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If you say so.
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So in your "math" there is no /number/ x such that x^2 = 2.
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Ok, if you can live with(out) that, fine.
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In reality, things are finite and space and time might also be finite (composed of atoms of space and time that can't be subdivided with
the parts retaining their original spatial and temporal properties).

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Yes, they could.
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So if the concept of irrational numbers like sqrt(2) [etc.]

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Hint (1): You won't find numbers like sqrt(2) IN (PHYSICAL) REALITY.

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If we draw a unit square, sqrt 2 is in there by default, right? From the unit square all other squares can be constructed.
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Hint (2): You won't find numbers like 1, 2, 3 there neither/either (?).
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 Also, from the unit circle, all circles and ellipses can be created. The ellipse aspect is that of a circle rotated in 3d is an ellipse when viewed on the projected plane. Check this out in VR:
 https://skfb.ly/6RozT

Think of a circle in 3d. Looking at it dead on as a circle. Then, change the camera view. It becomes an ellipse.