It's not only systems in first order logic that are Goedel complete.
For example Euclidean geometry is Goedel complete even though it
includes universally quantized propositions like "All triangles have
interior angles that sum to a straight angle."
Brent
On 1/14/2025 5:33 AM, John Clark wrote:
On Mon, Jan 13, 2025 at 5:58 PM Jesse Mazer <laserma...@gmail.com> 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?/
*That's Godel's Completeness Theorem which concerns First Order Logic
(FOL), not to be confused with his better-known Incompleteness
Theorems which are concerned with Second Order Logic (SOL). FOL can do
stuff like; Socrates is a man, all men are mortal, therefore Socrates
is mortal. Unfortunately FOL is pretty weak, if something can be
proven in FOL then it can also be proven in SOL, but the reverse is
not necessarily true. In FOL you can _not_ define things like "finite"
or "continuous", you can't use mathematical induction and you can't
even do arithmetic. For those things you need SOL, but it's
incomplete, and it can't prove its own consistency.*
*
*
*John K Clark See what's on my new list at Extropolis
<https://groups.google.com/g/extropolis>*
3e4
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