I deliberately leave vague what is in the theory of the mind, but simply assume a small number of things about consciousness:
1) That there is a linear dimension called (psycholgical) time, in which the conscious mind find itself embedded 2) The observations are a form of a projection from the set of subsets of possibilities onto the same set. We identify a QM "state" with a subset of possibilities. 3) The Kolmogorov probability axioms 4) The anthropic principle 5) Sets of observers are measurable
Also I assume the existance of the set of all descriptions (which I call the Schmidhuber ensemble, but perhaps more accurately should be called the Schmidhuber I ensemble to distance it from later work of his). This is roughly equivalent to your Arithmetic Realism, but probably not identical. It is the form I prefer philosophically.
(I think this is the exhaustive set of assumptions - but I'm willing to have other identified)
I only treat continuous time in Occams razor (hence the differential equation) however I do reference the theory of timescales which would provide a way of extending this to other types of time (discrete, rationals etc). In any case, contact with standard QM is only achieved for continuous time.
The justification for assuming time is that one needs time in order to appreciate differences - and differences are the foundation of information - so in order to know anything at all, one needs to appreciate differences hence the need for a time dimension.
Note - computationalism requires time in order to compute mind - therefore the assumption of time is actually a weaker assumption than computationalism.
comp assumes only that the sequence 0, 1, 2, 3, 4, 5, 6, ... "lives" in Platonia. 3-person time apparantly does not appear. 1-person time appears through the S4Grz logic.
In terms of the above assumptions, 1) is a consequence of computationalism, which I take is a basis of your theory (although I've never understood how computationalism follows from COMP).
? Wait a bit. COMP refers to computationalism. I don't understand.
2) corresponds to your 1-3 distinction. Indeed I refer to your work as justification for assuming the projection postulate.
That is not clear for me.
3) Causes some people problems - however I notes that some others start from the Kolmogorov probability axioms also.
No problem at all with Kolmogorov proba axioms.
4) I know the Anthropic principle causes you problems - indeed I can only remark that it is an empirical fact of our world, and leave it as a mystery to be solved later on.
No problem with the so called Weak Anthropic Principle. Although obviously I prefer a Turing-Universal-Machine--thropic principle ...
5) Measurability of observers. This is the part that was buried in the derivation of linearity of QM, that caused you (and me too) some difficulty in understanding what is going on. I spoke to Stephen King on the phone yesterday, and this was one point he stumbled on also. Perhaps this is another "mystery" like the AP, but appears necessary to get the right answer (ie QM !)
Of course a more detailed theory of the mind should give a more detailed description of physics. For example - we still don't know where 3+1 spacetime comes from, or why everything appears to be close to Newtonian dynamics.
Stephen King is cooking up some more ideas in this line which seems interesting...
Thanks for your clarification,