# Gödel, Escher, Bach: b is a power of 10. [closed]

I’d like to verify if my formula correctly expresses that a number is a power of $10$, using the $\sf{TNT}$ language provided by Hofstadter in his famous book Gödel, Escher, Bach: An Eternal Golden Braid. Although Hofstadter uses ‘$b$’ to express the desired number, I’ll use ‘$a$’ just for the sake of clarity. I’ll use common numerals for shortening the formula. Here we go:

$$\exists b: \exists c: \exists d: \exists e: (a = 1) \\$$ $$\lor (((\neg (b = 0) \land (a = 10 \cdot b)) \supset ((b = 10 \cdot c) \lor (b = 1))) \\$$ $$\land (((c = d \cdot e) \land \neg \exists f:(d = 10 \cdot f)) \supset (d = 1)))$$

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## closed as off topic by Andy Putman, Eric Wofsey, Qiaochu Yuan, Bill Johnson, Mark SapirMar 1 '13 at 3:58

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–  Asaf Karagila Mar 1 '13 at 3:12

This is too long for a comment, but it seems that your formula is not correct. In fact, your formula is true in the natural numbers for any value of $a$ simply by taking $b=0$, which fulfills the clause on the second line by denying the antecedent of the implication, and also $c=d=1$ and $e=2$, which fulfills the clause on the last line also by denying the antecedent. Thus, for any value of $a$, even when it is not a power of $10$, we may find values witnessing your existential assertion, and so it does not express the desired property.
To express the property that $a$ is a power of ten in the first-order language of arithmetic (where only the ring operations $+$ and $\cdot$ are allowed), one will have to use some kind of Gödel coding of sequences, in order to encode the recursive definition of the powers of $10$. That is, you want to say something like, $a$ is a power of $10$ if and only if there is a number $r$ coding a sequence of numbers (and this is the difficult part, but it is doable using the Chinese remainder theorem or other tricks, the standard Gödel coding ideas) that starts with $1$ and multiplies by a factor of $10$ in each successive step, such that $a$ appears on the sequence.