The problems with judging a civilisation by its energy consumption have been laid out well by ThisWriter. The main issue is that Universes, and probably higher -verses, are probably infinite; a flat or hyperbolic spacetime (as well as some of the other five Thurston spacetimes that I haven’t brought up) would be infinitely large, and therefore contain an infinite amount of energy.
This isn’t directly an issue, but it leads to two unfortunate consequences:
- There are no friendly qualitative descriptors of Kardashev levels. 1 is Planet, 2 is Star, 3 is Galaxy, but 4 and above don’t mean anything special.
- The qualitative descriptors explode. Universe is Type Infinity. Multiverse is also Type Infinity, but maybe a bigger infinity.
This may be recoverable by switching to a different number system. Infinity is not scary.
Surreal numbers are usually very handy for this kind of thing; they allow you to do fun things with infinities and infinitesimals. Let’s try those first. A civilisation with a power output of ω would have a Kardashev level of 1⁄10 (log ω - 6), which is totally allowed in the surreals.
But, of course, both of those have the same cardinality. Is there are difference between them? You could just number a countably infinite number of stars, and depending on how you renumber them you’d get a vastly different amount of energy consumption. The actual ordinal number one uses is completely irrelevant; you could just pick a single representative ordinal.
Single representative ordinals that have a single defined size are a very well-studied concept, and we call them cardinal numbers. These have the inconvenient property that you can’t generally take logarithms of them, because cardinal exponentiation is defined differently to ordinal exponentiation in a way that matches perfectly with finite numbers but doesn’t for infinite numbers. Briefly, ordinal exponentiation takes limits of finite numbers of mappings between two sets, cardinal exponentiation looks at the number of mappings directly between the sets.
You could probably define a fake ℵ-1 for which |2ℵ-1| = ℵ0 and which doesn’t exist because it would break a lot of stuff (aleph null is no longer a limit ordinal in this system!). This opens the can of worms of ℵ-2 and maybe also aleph numbers for real numbers and other surreals but it’s slightly better than just allowing surreals.
The other option is just to declare, by fiat, that ℵ0 corresponds to a countably infinite amount of energy consumption, and ℵ1 is the smallest uncountably infinite amount of energy consumption, and this doesn’t match up with the finite Kardashev scale because all life if suffering.
On the whole, I think the ℵ-1 solution is slightly more elegant, if only because being unable to take the logarithms of cardinal numbers is mostly a contrivance so that all cardinal numbers describe different numbers of things. It’s a very, very natural contrivance, and is really important if you care about the “cardinal” part more than the “number” part, so try not to tell your mathematician friends if you break it.
Using cardinal numbers, however, leads to another problem, somewhat similar to the surreal problem. A universe is a countable infinity of particles, and a multiverse is a countable infinity of universes, which means that a multiverse is also a countable infinity of particles. A Universal civilisation would be able to beat a Multiversal civilisation just by renaming all its stars, and vice versa.
The resolution to this could be to just ignore it. Infinite civilisations are weird like that, and it’s only actually a problem to we finite folk who are used to only dealing with small numbers like Rayo(10100) and stuff like that. The only truly more powerful civilisation is one that has an uncountable infinity of particles.
Maybe a multiverse could contain an uncountable infinity of universes, and a megaverse contains ℵ2 multiverses, which would give you a lovely progression.
This also gives the megaverse a nice property of being actually qualitatively bigger than a multiverse, which might actually be a useful distinguishing characteristic between a megaverse and just a really big multiverse. So, my proposed resolution looks something like this: