(i) A system is partially closed by definition (otherwise it is indistinguishable from the environment, not observable

(ii) No systems are completely closed (or there would be, or have never been no interaction and therefore not observable).

(iii) From (i) and (ii), all systems are partially open by definition.

(iv) A partially open system interacts with its environment or other systems (if information can flow it will flow)

Stuff happens: axiom 1.

(v) The Universe is a closed system, therefore total number of bits in universe remains constant

Universe is a closed system: axiom 2.

(vi) A universe with an observer must contain a system. All universes where anything is observed to happen consist of at least 2 systems (observer and system A) and the environment (rest of universe). A single system and its environment has no observer. Therefore systems exist and there are at least 2 of them.

(vii) Rephrasing the logic of a Shannon style (encoder, channel, receiver) system as a whole, Information exchange is not transfer of a piece of information from one place to the next, but the syncing of two systems. (Sender and Receiver now know the same thing, so the difference between them is less)

(viii) From (vii), the total number of bits required to describe system A plus B system tends to decrease over time. Therefore given (v) total number of bits required to describe the environment increases over time (increase in entropy of A + environment (if B is the observer, B cannot measure her own entropy)) NB this is number of bits in Kolmogorov sense, i.e. compressed form.

(ix) Logical and Thermodynamic entropy are the same.This followed from Szilard and Landauer, and was solution to Maxwell’s Demon paradox i.e. over time, number of bits needed to describe systems interacting with their environment or other systems tends to increase.

(x) If the universe is a closed system (v) and logical and thermodynamic entropy are the same (ix), then the 2nd Law applies relative to an observer, i.e. observer may see perceived increase in entropy of system as a whole but entropy of the universe (number of bits needed to describe the universe) must remain constant. This follows because any description of a finite universe must be contained within it.

Conclusion: 2nd Law as perceived by an observer is necessary consequence given finite universe where anything happens. However, entropy increase is relative to observer since by definition if the universe is a closed system, absolute entropy does not increase, the 2nd Law by definition is relative to an observer.

(xii) Partially open systems interacting will have their interactions affected by the environment (there is no channel that is a closed system). i.e. there is noise in all communication.

(xiii) If there is noise in all communication then any action (state change in B governed by a rule) based on a message that is communicated from A to B will eventually become a different action. Any state change rule will mutate over time.

(iv) A 1 dimensional (2 state, nearest neighbor) cellular automata is the simplest model of a rules based system. [proof?]

(vi) Variable inheritance of rules applied to a system in a finite environment is equivalent to the parameters which explain natural selection: Biological natural selection = structure (information store – DNA, phenotype), inheritance (breeding), mutation, finite environment (competition for survival), where rules are the inheritance and noise is the mutation. However, there is no information store – A 1 dimensional cellular automaton is not necessarily a computer.

(xiv) If there are a finite number of rules and an infinite amount of time, all rules will be cycled through in an information exchange.

(xv) Rule 110 in a 1 dimensional, 2 state CA is proven to be Turing complete (Mathew Cook). i.e. there are 1 dimensional cellular automata that are computers.

(xvi) In the simplest model of the universe as bits with systems that are also strings of bits, where there a physical laws that govern the state of those bits, a rule that is Turing complete will eventually be applied to a system which when Turing complete will have all of the ingredients to operate under natural selection.

(xvii) A system which is Turing complete can store information, it can learn.

(xvii) A system which can learn will self configure to extract information from other systems, and when observed will be seen to maximize the rate of entropy production relative to others.

(xvix) Turing complete systems will dominate the universe, over time, via natural selection which is a product of the 2nd law which in turn is a product of a finite universe where stuff happens.

BTW – If the universe is infinite and stuff happens, for the second law to apply you don’t have the observer problem but you need the concept of bit splitting as systems interact and learn.