Re: More complex numbers than reals?

On 7/9/2024 12:08 PM, Chris M. Thomasson wrote:

On 7/9/2024 3:02 AM, FromTheRafters wrote:

FromTheRafters pretended :

Chris M. Thomasson explained on 7/9/2024 :

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?

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Corrected.
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In a sense there are 'more' since the reals are all on the x axis line whereas the 2D R x R space is filled with complex numbers. R is contained in C. In another sense they are the same size set, C being basically R by R in the same sense as Q being Z by Z).
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Are there any other sizes of sets between countable Q and uncountable R? How about between uncountable R and uncountable C?

 Seems to boil down to:
 Is uncountable infinity the same "size", as any other uncountable infinity? Say reals vs. complex numbers...

This is where I like to ponder on so-called, density of infinity. The reals are "denser" than the naturals... Agreed? The complex numbers seem denser than the reals. Oh well, that is in my mind for some damn reason.
:^)