How many "colors" do exist?
Our perception: As far as I know, colors are just different frequencies of light. According to wikipedia, we can see wavelengths from about 380 nm und 740 nm. This means we can see light with a frequency from about $4.051 \cdot 10^{14}$ Hz to about $7.889 \cdot 10^{14} $ Hz. Is this correct? I don't know if time (and frequencies) are discrete or continuous values. If both are continuous, an uncountable number of "colors" would exist. If it is discrete, there might still exist no upper bound.
An upper bound? I found the article Orders of magnitude of frequencies. The Planck angular frequency seems to be by far higher than all other frequencies. Is this the highest frequency which is possible? Do higher frequencies make sense in physics?
Why do I ask this question: I am imagining the vector space $\mathbb{R}^4$ like the $\mathbb{R}^3$, but with colors. I need an infinite amount of colors if this should make sense. In fact the number has to be uncountable.
Answer
A human eye may only distinguish thousands or millions of colors – obviously, one can't give a precise figure because colors that are too close may be mistakenly identified, or the same colors may be mistakenly said to be different, and so on. The RGB colors of the generic modern PC monitors written by 24 bits, like #003322, distinguish $2^{24}\sim 17,000,000$ colors.
If we neglect the imperfections of the human eyes, there are of course continuously many colors. Each frequency $f$ in the visible spectrum gives a different color. However, this counting really underestimates the actual number of colors: colors given by a unique frequency are just "monochromatic" colors or colors of "monochromatic" light.
We may also combine different frequencies – which is something totally different than adding the frequencies or taking the average of frequencies. In this more generous counting, there are $\infty^\infty$ colors of light where both the exponent and the base are "continuous" infinities.
If we forget about the visibility by the human eye, frequencies may be any real positive numbers. Well, if you're strict, there is an "academic" lower limit on the frequency, associated with an electromagnetic wave that is as long as the visible Universe. Lower frequencies really "don't make sense". But this is just an academic issue because no one will ever detect or talk about these extremely low frequencies, anyway.
On the other hand, there is no upper limit on the frequency. This is guaranteed by the principle of relativity: a photon may always be boosted by another ditch if we switch to another reference frame. The Planck frequency is a special value that may be constructed out of universal constants and various "characteristic processes" in quantum gravity (in the rest frame of a material object such as the minimum-size black hole) may depend on this characteristic frequency. But the frequency of a single photon isn't in the rest frame and it may be arbitrarily high.
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