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i have seen in the "logic to cs" class i take - a theorem that states: "every recursive (computable) function $f$ can be expressed using the arithmetic dictionary {$C_0, C_1, f_+(,), f_x(,), R_\le(,)$} with the structure {$D=\mathbb{N} ,C_0=0,C_1=1, f_+(a,b) =a+b, f_x(a,b)=ab, R_\le(a,b) = a\le b $}"

But we didnt prove this theorem because a part of the students didnt take the "computational models" course (i did take it though)

Where can i find a proof for this theorem? Thanks in advance!

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  • $\begingroup$ Im unsure if this question was asked already in a similar forum. If it was, i will gladly take the question down and look at the answer there :) $\endgroup$
    – nir shahar
    May 28 '20 at 20:56
  • $\begingroup$ Can you define the elements of that "dictionary"? I suspect that the general recursive functions is what you're basically looking for: en.wikipedia.org/wiki/General_recursive_function $\endgroup$
    – Jake
    May 28 '20 at 22:25
  • $\begingroup$ Do you mean "Peano Arithmetic" by arithmetic dictionary? $\endgroup$
    – Beleg
    May 28 '20 at 22:35
  • $\begingroup$ @Beleg yes, using the usual natural numbers structure for this dictionary $\endgroup$
    – nir shahar
    May 28 '20 at 22:39
  • $\begingroup$ To clarify, when i said "can be expressed by the arithmetic dictionary" i meant that there exists a first order logic formula $\phi (z)$ such that $f(x) = y$ if and only if $\phi (y)$ $\endgroup$
    – nir shahar
    May 28 '20 at 22:44
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I'm not sure this is exactly what you are looking for, but you might find what you want in Theorem 3.2.1 of Computability Theory by S. Barry Cooper:

All recursive functions are representable in PA.

that is for any recursive function $f$, there exists a binary predicate $F$ in the language of arithmetic such that for any natural numbers $x$ and $y$ we have $$ f(x) = y ~\Rightarrow~ \vdash_{PA} F(x,y) $$ and $$ f(x) \neq y ~\Rightarrow~ \vdash_{PA} \lnot F(x,y) $$ where $\vdash_{PA}$ means 'PA proves'.

This theorem is central to Gödel's famous incompleteness theorem, so you might also want to take a look at ch. 8 of the mentioned book where it is discussed, and this notion of 'representability' is extended to 'semi-representability', to include the c.e. sets as well.

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  • $\begingroup$ you are right that this is not exactly what i need (and in fact, we have not even started defining provability under PA for first order logic) although, this seems to be very similar to my question so i will take a look at that. Thanks! $\endgroup$
    – nir shahar
    May 28 '20 at 23:02

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