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[parent] example of Gödel numbering (Example)

We can define by recursion a function $ e$ from formulas of arithmetic to numbers, and the corresponding Gödel numbering as the inverse.

The symbols of the language of arithmetic are $ =$, $ \forall$, $ \neg$, $ \rightarrow$, 0, $ S$, $ <$, $ +$, $ \cdot$, the variables $ v_i$ for any integer $ i$, and $ ($ and $ )$. $ ($ and $ )$ are only used to define the order of operations, and should be inferred where appropriate in the definition below.

We can define a function $ e$ by recursion as follows:

  • $ e(v_i)=\langle 0,i\rangle$
  • $ e(\phi=\psi)=\langle 1,e(\phi),e(\psi)\rangle$
  • $ e(\forall v_i \phi)=\langle 2,e(v_i),e(\phi)\rangle$
  • $ e(\neg\phi)=\langle 3,e(\phi)\rangle$
  • $ e(\phi\rightarrow\psi)=\langle 4,e(\phi),e(\psi)\rangle$
  • $ e(0)=\langle 5\rangle$
  • $ e(S\phi)=\langle 6,e(\phi)\rangle$
  • $ e(\phi<\psi)=\langle 7,e(\phi),e(\psi)\rangle$
  • $ e(\phi+\psi)=\langle 8,e(\phi),e(\psi)\rangle$
  • $ e(\phi\cdot\psi)=\langle 9,e(\phi),e(\psi)\rangle$

Clearly $ e^{-1}$ is a Gödel numbering, with $ \ulcorner\phi\urcorner=e(\phi)$.



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deductions are $\Delta_1$ (Theorem) by mathcam
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Cross-references: order of operations, integer, variables, language, inverse, Gödel numbering, numbers, arithmetic, formulas, function
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This is version 4 of example of Gödel numbering, born on 2002-08-24, modified 2004-04-11.
Object id is 3345, canonical name is ExampleOfGodelNumbering.
Accessed 2554 times total.

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AMS MSC: 03B10 (Mathematical logic and foundations :: General logic :: Classical first-order logic)
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