In article <ITLUh.37486$4S5.email@example.com>, Aatu Koskensilta <firstname.lastname@example.org> writes: > email@example.com wrote: >> Godel's argument depends on the ability (roughly speaking) of the >> system to contain am embedded model of itself. This means that >> the system needs to be both powerful and describable. A system can >> escape Godelization by being either too simple to contain itself or by >> being too complex to fit into itself. > > I'm afraid your comments offer but a marginal improvement over those by > Tony012 and John Bailey.
OK. I'm sure you're right. You've demonstrated significant expertise in the area. Unfortunately, your remarks leave me guessing on where I went wrong.
I suppose one answer is that it doesn't matter much whether the theory in question can directly encode a model of itself. What matters is whether it can encode a model of a particular subset of arithmetic.
Then, as best as I can fill in the details from various popularizations, what matters is whether the encoded model of arithmetic can be used to encode a model of the original theory in such a way that provable statements in the original theory become provable statements in the doubly-encoded version of the theory and vice versa.
With those two pieces in hand, one can demonstrate the existence of a well formed formula G which can be interpreted as asserting its own unprovability.
[I'm not clear on whether the existence proof for G is constructive. The popularization I'm looking at right now does not spell that part out]
Am I back on track now? Or still heading into the weeds?
Assuming I'm on the right track...
It doesn't matter whether the theory is "too complex" by reason of not being able to fit into itself. It does matter whether the theory is too complex to permit a finite encoding of the theory in the language of arithmetic. In particular, the theory must not contain an infinite number of axioms unless all but a finite subset of those are redundant and can be eliminated.
It doesn't matter whether the theory is "too simple" by reason of not being able to contain itself. It does matter whether the theory is "too simple" in the sense of not being able to encode a model of an appropriate subset of arithmetic.