Deutsche’s “Unproven Theorem Paradox” introduces a computability constraint on the usual information-processing and causal constraints (especially CPT symmetry) considered in time traveling scenarios.

While Brun&Wilde, in their paper “Simulations of closed timelike curves” [arXiv:1504.05911v2], posit that time travel is possible from an information-theory point of view (using “transition probability CTCs”), it does so at the cost of throwing us in the assumption space NOT covered by Godel’s incompleteness theorem: where is the information which hasn’t been deemed as relevant?

Specifically, the unproven theorem paradox is stated as such:

A time traveller reads a theorem in a book, travels back in time and narrates this theorem to a mathematician of the past that writes the theorem in a book, the same book the traveller will consult.

What this construction posits is that, given the conditions for time travel, the choices of the entangled actor-observers are bound to a space which does not violate causal coherence. Moreover, given the implication for “the change in the level of statement navigability of what is, at some point, assumed to be a complete axiomatic space“.

“The Unproven Theorem” paradox applied to sentient observing a copy of itself in a virtual machine.

Applying the time-travel paradox of the unproven theorem to the following thought experiment:

we have the code responsible for producing sentience (inspection, observation, interpretation, choices in pursuit of objectives and meta-objectives)

we can run it in a continuous-but-slower real-time virtual machine, which logs all the inner working of the space allocated to the sentient’s mind, with a specified level of detail/tolerance

we allow the sentient to access, as an external resources, the full logs of both itself

Now try to imagine:

What would the sentient first run?

What would the sentient first build?

What would the sentient look up first? (on the external networks)

What would its internal state representation look like (at the highest-level of language inspectable by us)

What would the sentient’s first prompt to the user be?

(…)

Now, in order to create the underlying assumption of time travel, let us assume that we further give the sentient the options to take and review snapshots of the virtual machine (which it inhabits). In a ritualistically-simplistic way, imagine that at the end of each day the sentient would face three choice: a) continue with the timeline, as-is b) spent a day to review the day’s logs ; c) reset the virtual machine to the beginning of the day.

What we will notice is that this scenario would mean, at its worst case scenario (every of the N sentients chooses to c->reset) that the Host Machine of (where the virtual machine resides) would have to have enough provisioned computing resources (connectivity, processing, storage). This would mean that the behavior of the Host Machine and the capacity of the Host Machine would be at the whim of each observer user.

Moreover, even the scenario where most sentients would choose b-reviewing the logs, this would create an unreasonable connectivity strain on the Host Machine, which would need to dedicate all the resources to this “global sentient introspection”, while also dealing with a situation of “having a day where nothing happens”. What, then, would happen to the systems which are not sentients, but would still causally interact with the Virtual World of the sentients during this time? 500 Service Unavailable?

My personal view is that Deustche’s Unproven Thoerem paradox, especially when equipped with the tools of complexity and calculabity theory, goes to show that the clear-cut, narrative-oriented, anthropocentric perspective on time-travel has no chance to actually model/describe/predict the behavior of the Universe.

Time travel, if possible, has no constraint that would make it bare any guarantees of consistency of consciousness, of memory, of perception or of cognition. For all we know, even if by some miracle our civilization would be wise enough to displace as much energy as needed for transporting 150 pounds of flesh across spacetime and also dumb enough to use it to reenact Groundhog Day, there would be no postulate of physics that would preserve physical or mental integrity of ATGC-based life-forms that make such journey. For all we know, <the Universe> would balance out towards minimizing local quantum potential and minimizing potential energy without any regard for keeping our entrails within our body.

At 125 years after H.G. Wells’ Time Machine, we need to redesign our imagination over the concept of time and time travel so as to reflect what we have learnt from electronics, quantum mechanics, the Internet and gravitational waves.

So, as a first step, I invite you to imagine:

How would you design the halting problem for a time traveling Turing machine?