Re: More complex numbers than reals?

Op 09/07/2024 om 22:05 schreef Chris M. Thomasson:

On 7/9/2024 12:38 PM, sobriquet wrote:

Op 09/07/2024 om 07:24 schreef Chris M. Thomasson:

Are there "more" complex numbers than reals? It seems so, every real has its y, or imaginary, component set to zero. Therefore for each real there is an infinity of infinite embedding's for it wrt any real with a non-zero y axis? Fair enough, or really dumb? A little stupid? What do you think?

>
How can you compare them if they are not even in the same dimension?
It seems that instead of comparing the real number line to the complex plane (or the Cartesian plane for that matter), you might as well compare a unit of length to a square unit of area.
Numerically they might be the same, but they are not identical, because
a line has no area, so a unit of length 1 has an area of 0 units squared.

 Are the complex numbers just as infinitely dense as the reals are? Or are there somehow "more" of them wrt the density of the reals vs, say, the rationals and/or the naturals? Is the "density" of an uncountable infinity the same for every uncountable infinity? The density of the complex numbers and the reals is the same?

How do you define sets exactly?
Is there a specific set that corresponds to sqrt(2)?
Does this set have an infinite number of elements analogous to the sqrt(2) having an infinite decimal expansion?
It seems that the existence of something like sqrt(2) is already rather dubious.
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).
So if the concept of irrational numbers like sqrt(2) turns out to be nonsense, that means the concept of the number line is nonsense likewise and we don't even need to consider higher dimensional nonsense.