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It would be very convenient if we had, at least non-constructively, a correspondent formal system that could reproduce any causal event within the universe.

The strength would be that naturalism would hold just from the mere fact that axioms exist and that deduction is a sequent which can be done cut-free, thus every proven statement would be a subformula of some of the premises.

Thus, naturalism would be a crystal clear conclusion.

Perhaps it is a bit much to ask how to solve Hilberts 6th problem, so to ask differently:

Under what circumstances could reality not be reduced to a formal system?

Could it be that spontaneous indeterminacies that are observed in quantum physics violate the demanded rigidity of such a formal system?

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    "reduced"? Maybe described.
    – Mauro ALLEGRANZA
    Commented Feb 9 at 9:53
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    You cannot take reality and just write it (you will never end formalizing even a grain of sand). First, you need to model reality with some GOAL (that is the core matter), and only then you can express such model as a formal system. Example of modelization for a goal: a map is a model of reality which is useful to travel, but not to enjoy the landscape. Example of formalization: if your goal is to be the owner of the universe, you can model such reality (create a logically consistent concept) and then express it as a formal system: Axiom 1: all things are mine [∀x: Mine(x)]. Done.
    – RodolfoAP
    Commented Feb 9 at 10:21
  • Sounds like the often discussed Tegmark's MUH is not music for your ears, then your concerned antecedent should be the existence of some multifacted thing at one of its ends to encompass the all-inclusive aspects of those which could be possibly experienced of your said and intended totality of the necessary reality...
    – Double Knot
    Commented Feb 9 at 23:45
  • How can reality be reduced if even algebra can't be reduced. That is if we assume that algebra is part of the reality.
    – spacemonkey
    Commented Feb 10 at 7:39

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None. Uncertainty arises whenever we attempt to model outcomes in general as the product of a specified deterministic process.

Chaos theory takes a deterministic system which is characterized as sensitive to initial conditions. The system is unstable. Run the same deterministic model in a computer simulation, starting with the same initial conditions, and the simulation produces distinct outcomes, which are recognized as non-deterministic or unstable outcomes of the theoretically deterministic process.

Complex systems are interactions between the system and its surroundings which do not reduce to deterministic models. Physical models of reality show complexity as well as human history, politics, and economics cannot be separated from agent-based simulations.

ScienceDirect.com:

Chaos theory describes the qualities of the point at which stability moves to instability or order moves to disorder. For example, unlike the behavior of a pendulum, which adheres to a predictable pattern a chaotic system does not settle into a predictable pattern due to its nonlinear processes.

Under complexity theory, complex systems are viewed as open systems that interact with their environments, implying a need to understand the systems' environments before understanding the systems.

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  • The this logic answer -- that formal systems are intrinsically incomplete, is an excellent one. One could supplement it with the pragmatic observation that our physics as observed is also incomplete, hence no formal system will match it all. Here is an answer with links pointing this out: philosophy.stackexchange.com/questions/68224/…
    – Dcleve
    Commented Feb 11 at 18:12

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