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Given the two equivalent terms $A \vee B$ and $B \vee A$, Gödel numbering returns two various codes $2^{4}.3^{\overline{A}+1}.5^{\overline{B}+1}$ and $2^{4}.3^{\overline{B}+1}.5^{\overline{A}+1}$, respectively. These two codes do not generally equal to each other, which is an unfortunate if one would like to observe the equivalence after the mapping. Thus, is there any alternative logic-to-numeral mapping which takes such symmetries into account, for example, by yielding one code corresponding to both $A \vee B$ and $B \vee A$?

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  • $\begingroup$ $A\vee B$ and $B\vee A$ are two different words. The equivalence that you mention is an interpretation that you are giving to the existence of another word, which constitutes a proof of that equivalence. To produce what you want you would need to be more precise on what words you want to consider 'equivalent'. $\endgroup$ – plop Jul 10 '20 at 13:05
  • $\begingroup$ @plop: By equivalent, I indeed mean those words whose elements can be symmetrically replaced by each other. In a more rigorous term, those whose truth tables read each other. $\endgroup$ – Roboticist Jul 10 '20 at 13:11
  • $\begingroup$ @plop: By the way, would you please explain why the two assumed words are "different"? $\endgroup$ – Roboticist Jul 10 '20 at 13:27
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    $\begingroup$ The first character of $A\vee B$ is $A$, while the first character of $B\vee A$ is $B$. The reason why you are seeing them as equivalent is because probably in your language there is a third word, which you consider the proof of that equivalence. For example, the word $A\vee B \equiv B\vee A$, or $A\vee B\iff B\vee A$. I don't know. It depends on your language. $\endgroup$ – plop Jul 10 '20 at 13:28
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    $\begingroup$ Since the truth table for $A\vee B$ and for $B\vee A$ (ordering the inputs in ascending order) is $\begin{pmatrix}0&0&0\\0&1&1\\1&0&1\\1&1&1\\\end{pmatrix}$, its rows are the numbers $0,3,5,7$. Then we encode it as $2^0\cdot 3^3\cdot 5^5\cdot 7^7$. $\endgroup$ – plop Jul 10 '20 at 13:39

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