9 has 4 digits as "1001" in binary and 1 in decimal, so no function from integers to their size. There is no such thing as the size of a integer independent of any digit system used (well you could refer to some set constructions but then the size would be the integer itself).
As surreals we could have ω pieces of equal probability ɛ that sum to 1 exactly (althought ordinal numbers are only applicaple to orders which can be different than cardinal numbers. While for finites there is no big distinciton from ordinal and cardinal, "infinitely many discrete things" might refer to a cardinal concept. However for hypotheses that are listable (such as formed as arbitrary lenght strings of letters from a (finite) alphabeth) the ωth index should be well founded).
It is not about arbitrary complexity but probability over infinite options. We could for example order the hypotheses by the amounts of negation used first and the number of symbols used second. This would not be any less natural and would result in a different probability distribution. Or arguing that the complexity ordereing is the one that produces the "true" probailities is reframing of the question whether the simplicity formulation is truth-indicative.
If I use a complexity-ambivalent method I might need to do fewer eliminations before encountering a working one. There is no need to choose from accurate hypotheses if we know that any of them are true. If I encounter a working hypthesis there is no need to search for a more simpler form of it. Or if I encounter a theory of gravitation using ellipses should I countinue the search to find one that uses simpler concepts like circles only?
I think this is relevant: https://en.wikipedia.org/wiki/Bertrand_paradox_(probability)
The approach of the final authors mentioned on the page seems especially interesting to me. I also am interested to note that their result agrees with Jaynes'. Universability seems to be important to all the most productive approaches there.
Or arguing that the complexity ordereing is the one that produces the "true" probailities is reframing of the question whether the simplicity formulation is truth-indicative.
If the approach that says simplicity is truth-i...
This essay claims to refute a popularized understanding of Occam's Razor that I myself adhere to. It is confusing me, since I hold this belief at a very deep level that it's difficult for me to examine. Does anyone see any problems in its argument, or does it seem compelling? I specifically feel as though it might be summarizing the relevant Machine Learning research badly, but I'm not very familiar with the field. It also might be failing to give any credit to simplicity as a general heuristic when simplicity succeeds in a specific field, and it's unclear whether such credit would be justified. Finally, my intuition is that situations in nature where there is a steady bias towards growing complexity are more common than the author claims, and that such tendencies are stronger for longer. However, for all of this, I have no clear evidence to back up the ideas in my head, just vague notions that are difficult to examine. I'd appreciate someone else's perspective on this, as mine seems to be distorted.
Essay: http://bruce.edmonds.name/sinti/