On 1/14/2025 1:56 AM, PGC wrote:
On Monday, January 13, 2025 at 11:58:38 PM UTC+1 Jesse Mazer wrote:
Doesn't Godel's theorem only apply to systems whose output can be
mapped to judgments about the truth-value of propositions in
first-order arithmetic? A cellular automaton would seem to have
"evolving quantities and/or qualities through numerical or some
other equivalent formalism's means", but Godel's theorem places no
limitations on our ability to compute the behavior of the cellular
automaton for N time-increments, for any finite value of N, so I
would think Godel's theorem would likewise place no limitations on
our ability to compute the physical evolution of the universe's
state for any finite time interval. For some cellular automata it
may be possible to set up the initial state so that the question
of whether some theorem is ever proved true or false by the Peano
axioms (or other axioms for arithmetic) is equivalent to a
question about whether the automaton ever arrives at a certain
configuration of cells, so Godel's theorem may imply limits on our
ability to answer such questions, but this is a question about
whether something happens in an infinite time period. I assume
there are similar limitations on our ability to determine whether
certain physical states will ever occur in an infinite future
(straightforwardly if we build a physical machine that derives
theorems from the Peano axioms, or a machine that derives
conclusions about whether various Turing programs halt), but most
of what physicists do is concerned with predictions over finite
time intervals, I don't see how Godel's theorem would pose any
fundamental obstacles to doing that.
Jesse
Gödel’s first incompleteness theorem states that any sufficiently
strong formal system (capable of arithmetic) contains statements that
are undecidable—neither provable nor disprovable within that system.
The second theorem says such a system cannot prove its own consistency
from its own axioms. These are statements about provability in formal
theories. They are not directly about whether you can compute a finite
number of steps in a system like a cellular automaton.
You can absolutely compute, step by step, the evolving states of a
(finite) cellular automaton for N time steps, and Gödel’s theorems do
not say you can’t. They say something deeper: if your formal axioms
are strong enough to represent integer arithmetic (like Peano
Arithmetic or any Turing-complete formulation), there will be
statements expressible within that framework which it cannot resolve.
That’s a statement about what can or cannot be proven within the
system, not about whether a machine can run a simulation for some
finite time.
But if it is some proposition that can be settled by finite computation
it can't be one that's that unprovable, even if the required computation
is impracticallly long.
You also assume that Gödel’s incompleteness only restricts what can
happen in an “infinite time” scenario. For example: “Well, sure, there
might be some question about whether a certain configuration arises
eventually, but for finite intervals we have no Gödel-limited
obstacles.” This misreads Gödel: Gödel’s first theorem does not hinge
on infinite time steps; it is about the intrinsic logical structure of
the formal system. Even for trivial seeming statements involving
finite objects (e.g., “this specific integer has property P”), the
theorem shows there can be statements that the system cannot prove or
disprove. It’s not that you can’t “run the simulation long enough,”
but rather that the theory itself cannot settle certain propositions
at all.
Yes, questions about infinite evolution (like “does the automaton ever
reach configuration C at some unbounded time?”) can become
unanswerable in principle if they encode the halting problem or an
arithmetic statement. But Gödel’s point is more general: the existence
of some undecidable statements is guaranteed, quite apart from whether
they manifest in an infinite-time scenario or not.
If a cellular automaton is known to be Turing-complete (i.e., it can
replicate the behavior of a universal Turing machine), then in
principle it inherits exactly the logical limitations that come from
being universal. That means there will be statements about the
automaton’s global behavior (for instance, whether it will ever reach
a certain state from a certain initial condition) that are
undecidable. This “undecidability” is not about lacking the ability to
compute it up to step N—rather, it’s about no possible proof or
disproof existing within a certain formal system’s axioms regarding
some carefully constructed properties of that automaton.
Put another way, if the theory describing the automaton and its states
is as powerful as something like Peano Arithmetic, then Gödel’s result
shows there exist statements about that automaton’s behavior that the
theory can’t settle.
This doesn’t contradict our practical ability to simulate the
automaton for finite steps. It simply highlights that some high-level
questions (like a halting-like problem for the automaton) could be
undecidable in the strong sense Gödel identified.
A final misconception is that the existence of straightforward
computable truths (e.g., “2 + 2 = 4” or “state s follows from state
s_0 after N steps”) somehow negates Gödel’s theorems. It does not.
Gödel never claimed “no statement is decidable,” but rather that
“there exist some statements that are undecidable.” The presence of
plenty of decidable truths or effectively computable processes is not
an escape hatch from incompleteness. A formal system can easily prove
2 + 2 = 4, yet still fail to prove or disprove a Gödel-type sentence.
Therefore, you seem to mix up “we can compute local steps easily” with
“therefore there’s no deeper limit on provability.” A huge jump.
Incompleteness is about the existence of certain statements the theory
cannot decide at all—not about whether we can check how a system
evolves over N steps.
It can help to see Gödel’s message as a reminder that once you allow
arithmetic (or an equivalent notion of proof) into your theoretical
framework, you automatically inherit logical “gaps” you cannot fill
from within that framework. Consequently, every time a new observation
or phenomenon appears to contradicts some current theory of the
universe in the news feeds—be it quantum mechanical or
cosmological—standard practice is to say, “We’ll just extend or refine
the system,” hoping that, in principle, you can patch all holes. From
a Gödelian perspective, however, such hopes are naive, even if made by
top researchers constantly; you cannot achieve a neat, airtight
completeness once you’re in the realm of arithmetic-level
expressiveness. Each new theory you devise may bear fruit—often very
real, practical fruit—and it may rectify certain anomalies, but it
cannot be the final word, because no single formal system can be both
consistent and complete about everything you can say in it.
That’s why it’s no surprise to someone with a minimum of familiarity
in Gödel’s work that the mind spawns endless re-interpretations and
frameworks—especially in quantum mechanics—that run counter to some
“standard” stance. They’re not necessarily wrong or right in a final
sense; rather, they’re new attempts at formalizing a larger or more
nuanced slice of reality. As Gödel implies, there is an unbounded
supply of possible formal systems you can keep proposing, and many do
yield fresh insights or engineering progress. But none will eliminate
the fundamental incompleteness that arises once you assume enough
structure to reason arithmetically. So the cycle of “discover
contradiction → revise theory → see new contradictions” is essentially
built into the logical fabric of how we formalize our knowledge—Gödel
merely gave us a precise way to see why that cycle never fully ends.
I don't think how we formalize our knowledge, and certainly not of
physics, involves discovering logical contradictions. Mathematicians
sometimes add axioms to extend a theory, but not in face of a
contradiction. If you reach a contradiction in mathematics, then
everything is provable.
Brent
This is also why a ToE, when reliant on evolving quantities/qualities
and including physics, has to be “informal” with Gödel in mind. And
that’s what this list got so wrong with so many years of splitting
hairs with Bruno: informal reasoning can be rigorous. And regarding
ToE post Gödel, it has to be that way! All the attacks on Bruno of the
“You can’t be serious!” or “that is not a rigorous argument/proof”, or
“how can informal thought experiments be taken seriously” - sort just
miss the point/lack familiarity with Gödel.
I hope this provides some clarity. I’ll have to ask forgiveness for
not being able to participate in discussions and reply as much as I’d
like but I have too many time restraints atm and not enough time to
even read. Hence, my call for either some moderation, more restraint,
or whatever is needed to have a more organized list focused more on
the ToE topic, instead of spoon feeding, bad faith arguments, or troll
engagement/permissiveness. Yes, of course we can talk Trump politics,
but not in some trivial "caveman truth" asserting way. I hope you
understand. In closing, I suggest looking at the work (Gödel) and
making up your own mind to decide for yourself whether Gödel’s
theorems apply or not, so you don’t have to take my word for it, as
you shouldn't.
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