*C*2 = 01000110110000010100111001011101110000 . .

.

which consists in the concatenation of all binary strings enumerated in quasi-lexicographical order [76].<sup>19</sup> A a Chaitin world number is given by a Chaitin Ω*U* number (or halting probability), that is the probability that the universal self-delimiting Turing machine *U* halts [77]. Chaitin world numbers "hold proofs" for almost all mathematical known results; such as as Fermat Last Theorem (in the 400 initial bits), Goldbach's conjecture, or important conjectures like Riemann Hypothesis (in the 2745 bits initial bits) and P vs. NP (in the 6,495 initial bits; cf. [78]. Both world numbers are Borel normal in the sense that every binary string *x* appears in these sequences infinitely many times with the same frequency, namely <sup>2</sup>−|*x*|, where |*x*| is the length of *x*. In such a world every text—codified in binary—which was written and will be ever written appears infinitely many times and with the same frequency, which depends only on the length of the text. In particular, any correlation appears in such a world infinitely many times. However, these worlds are also very different: A Champernowne world number is computable, but a Chaitin world number is highly incomputable because it is Martin-Löf random. As a consequence, while both number worlds have all possible correlations repeated infinitely many times, the status of those correlations are different: in a Chaitin world number these correlations are spurious (because of its randomness), but in a Champernowne world number they are not (because its computability, hence highly non-randomness).

How an embedded observer would "feel" to live in such a world? This is a deep question which needs more study. Here we will make only a few simple remarks (see also [36]).

First, no observer or rational agen<sup>t</sup> could decide in a finite time whether they live in a Champernowne or Chaitin world. Second, any observer or rational agen<sup>t</sup> surviving, or at least recording experimental outcomes, a sufficiently long time will see many of the previously discovered accepted "laws" being refuted.

Third, suppose intrinsic observers embedded into a mathematical universe experience and "surf" these number worlds by their interactions with them; that is, they perceive long successions of initial

 .

<sup>17</sup> Again, one should not think that this means that there are no computable world numbers, see Section 6. The result follows from the fact that the computable sequences form a countable set.

<sup>18</sup> A sequence is bi-immune if its corresponding set of natural numbers nor its complement contain an infinite computably enumerable subset.

<sup>19</sup> In base 10, *C*10 = 12345678910111213141516 . . .

bits of their defining infinite sequences. Assume now that these sequences are Champernowne or Chaitin sequences. Because of the Borel normality of these sequences, the strings surfed by observers are Borel normal as finite objects, that is, they are distributed uniformly up to finite corrections [79]. How would intrinsic observers experience such variations? In one scenario one may speculate that intrinsically such "interim" periods of monotony may not count at all; that is, these will not be operationally recognised as such: for an embedded observer [6,13], the world number will remain "dormant" while the number world remains monotonous.

Another option, maybe even more speculative, is to assume that, as long as the world number allows for a sufficiently wide variety of substrings, the intrinsic phenomenology will, through emerging character of (self-) perception, "pick" its own segmen<sup>t</sup> or pieces (of numbers) from all the available ones. Indeed, it might not matter at all for intrinsic perception whether, for instance, the cycle time is altered (reduced, increased), or whether the lapse of cycles is arbitrarily exchanged or even inverted: as long as there are still "sufficient" patterns and number states emerging could "process" and "use," lawfulness and consciousness will always ensue [80].
