MHB Can We Enumerate All Primitive Recursive Functions?

AI Thread Summary
Primitive recursive functions can be enumerated by listing all possible derivations and definitions. This enumeration is relevant in demonstrating that Ackermann's function is not primitive recursive. The proof of Ackermann's non-primitive recursive status involves showing properties applicable to all primitive recursive functions through this enumeration. The discussion clarifies that enumeration includes all basic functions and their definitions through compositions or primitive recursions. Understanding this enumeration is crucial for exploring the boundaries of primitive recursive functions.
mathmari
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Can we enumerate the primitive recursive functions?
 
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Of course, just enumerate all possible derivations (definitions).
 
Is the enumeration of primitve recursive functions an other way to show that Ackermann's function is not primitive recursive? Or isn't it possible?
 
In the proof that Ackermann's function is not p.r. you prove something for all p.r. functions, and you do this by enumerating all possible derivations.
 
Evgeny.Makarov said:
In the proof that Ackermann's function is not p.r. you prove something for all p.r. functions, and you do this by enumerating all possible derivations.

By "enumerating all possible derivations" do you mean that we enumerate all possible cases how the p.r. function is defined, if it is one of the basic functions (constant, successor, projection), or is defined by compositions or primitive recursions?
 
Yes.
 

Thread 'Deductive proof in logic formal systems'

I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...
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