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From what I understand Euclid avoided infinity, and so I'm wondering how Euclid might have dealt with the concept of 0 in the Elements.

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  • $\begingroup$ There's a difference between what is implicitly in the Elements and what is explicitly there. Implicitly, it contains the whole first-order theory of the reals. $\endgroup$
    – user466
    Dec 25 '19 at 1:31
  • $\begingroup$ There is very strong evidence that the Elements does not contain the first-order theory of the reals, or even first order logic (on any reasonable interpretation of "implicitly"). There are modern formalizations of Euclid that use much weaker means, like Euclidean fields, and his reasoning is much more closely reconstructed in (non-equivalent) diagrammatic alternatives to first order logic. He did not deal with $0$ or negatives because he had no need for them either. $\endgroup$
    – Conifold
    Dec 25 '19 at 6:35
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The arithmetic books of Elements are books VII - IX.

Euclid begins book VII with his definition of number:

A number is a multitude composed of units.

(This is taken from the Pythagorean notion of number.)

Since "zero" is not a multitude of units it does not satisfy Euclid's definition of number. Euclid did not consider "one" to be a number for the same reason. However, Euclid appears to have been somewhat ambivalent on this point and did treat the unit as a number on those occasions when it helped in stating a general proposition or proving them. For example, Euclid's famous proof (IX.20) that there are infinitely many prime numbers.

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He did not. There is no reference to $0$ in the Elements. As a matter of fact, for Euclid even $1$ was not a number. For him, the numbers are $2,3,4,\ldots$

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