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This is a question arising from the discussion in another thread.

It's my understanding that among systems of natural units, the Planck units system sets not only "all three of those to 1", but also does the same with the Coulomb and Boltzmann constants, leaving the elementary charge not similarly normalized, because trying to normalize that too, would introduce an inconsistency.

That's part of my impression from the Wikipedia Natural units article; however, @PeterDonis said you can't set ##c = 1, \hbar = 1, G = 1## -- that no more than 2 of those can be set to 1 without inconsistency.

So far, it's been my experience that when he flatly disagrees with me about something, he's right and I'm wrong, but in this instance I'm still in doubt:

The cited article says:

and later,

That seems to me to show that in the Plank units system of natural units, not only are ##c, \hbar,## and ##G## being set to 1, but so are ##k_e## and ##k_B##, while ##e## is not.

But @PeterDonis posted while I was editing, so I'll read his reply now, which I'm sure will be insightful.

It's my understanding that among systems of natural units, the Planck units system sets not only "all three of those to 1", but also does the same with the Coulomb and Boltzmann constants, leaving the elementary charge not similarly normalized, because trying to normalize that too, would introduce an inconsistency.

That's part of my impression from the Wikipedia Natural units article; however, @PeterDonis said you can't set ##c = 1, \hbar = 1, G = 1## -- that no more than 2 of those can be set to 1 without inconsistency.

So far, it's been my experience that when he flatly disagrees with me about something, he's right and I'm wrong, but in this instance I'm still in doubt:

The cited article says:

Thus, we cannot set all of ##k_e, e, ħ,## and ##c## to 1, we can normalize at most three of this set to 1.

[In that statement, ##e## refers to the elementary charge.]and later,

Planck units are defined by

##c = ħ = G = k_e = k_B = 1##,

where ##c## is the speed of light, ##ħ## is the reduced Planck constant, ##G## is the gravitational constant, ##k_e## is the Coulomb constant, and ##k_B## is the Boltzmann constant.

That seems to me to show that in the Plank units system of natural units, not only are ##c, \hbar,## and ##G## being set to 1, but so are ##k_e## and ##k_B##, while ##e## is not.

But @PeterDonis posted while I was editing, so I'll read his reply now, which I'm sure will be insightful.

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