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asdf60
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So I'm a bit confused about these metatheorems about first order logic, partly because I haven't read any of the real proofs, but I just want to know the results for right now. Here is what I understand:
Soundness means that any derivation from the axioms and inference rules is still valid. As I understand, first order logic is sound.
Completeness means that any formula that is valid can be proven from the axioms. Once again, FOL is complete.
Now comes the confusing part. Godel's Incompleteness and Undecidability.
So godel's incompleteness theorem says that in a first order language with a model/interpretation defined sufficient to talk about mathematics, there exists truths that are not provable through the axioms. How does this not contradict the Completeness of FOL discussed above? Is it because the the completeness of FOL talks about truths that are valid for ANY model, while Godel's Incompleteness talks about truths that are valid in a more specific class of models?
Also, undecidability. What exactly does this mean? And how does it not contradict the completeness of FOL. If a formula can not be proven, then completeness assures us that the formula must be false. So what is the undecidability.
Thanks,
ASDF
Soundness means that any derivation from the axioms and inference rules is still valid. As I understand, first order logic is sound.
Completeness means that any formula that is valid can be proven from the axioms. Once again, FOL is complete.
Now comes the confusing part. Godel's Incompleteness and Undecidability.
So godel's incompleteness theorem says that in a first order language with a model/interpretation defined sufficient to talk about mathematics, there exists truths that are not provable through the axioms. How does this not contradict the Completeness of FOL discussed above? Is it because the the completeness of FOL talks about truths that are valid for ANY model, while Godel's Incompleteness talks about truths that are valid in a more specific class of models?
Also, undecidability. What exactly does this mean? And how does it not contradict the completeness of FOL. If a formula can not be proven, then completeness assures us that the formula must be false. So what is the undecidability.
Thanks,
ASDF