The "Cthulhu numbers" are basically, informally, just "random" numbers. The easiest way to build one is to choose a number at random in the interval [0,1], i.e.

The "Cthulhu numbers" are basically, informally, just "random" numbers. The easiest way to build one is to choose a number at random in the interval [0,1], i.e. by flipping a coin countably many times and choosing each bit in the binary expansion based on the result. With probability one, the number you get will not be the output of any Turing machine, nor described by any particular formula, etc - it will just be a random string of bits whose bits are chosen "for no logical reason." This means it will be one of your Cthulhu numbers. Do we really want to get rid of these and only have non-random structured numbers? Because it's certainly a useful idea to have idealized infinite strings of structureless random bits in plenty of situations. But if you like, you can always try Gödel's constructible universe L.