How set theory implies Euclid's third axiom? I read in an elementary geometry book that:
"... because we are using the language of sets rather than that
of Euclid, it is not really necessary to assume this postulate...(referring to the "circle" postulate, "for every point 0 and every point A not
equal to 0 there exists a circle with center 0 and radius OA").
Can someone explain to me how set theory implies this axiom?
 A: What the author must have had in mind is something like

By definition the circle with center $O$ and one point $A$ consists of the endpoints of all line segments starting at $O$ whose length is $OA$. However, if we believe in set theory, then we can already form the set
  $$ \{ X \mid \overline{OX} = \overline{OA} \} $$
  which is exactly the circle! So we do not need an additional axiom to say that the circle exists.

This is true, so far as it goes, once one decides to base one's geometry on set theory. In fact, if we believe in the usual axiomatic variety of set theory, then we can construct $\mathbb R$ and therefore $\mathbb R^2$ in standard ways and then do plane geometry about subsets of $\mathbb R^2$ without any geometry-specific axioms whatsoever, just adding some definitions of what we take geometrical terms to mean.
However, it looks likely that the book in question is taking points to be primitive objects and then additionally assuming that we can build sets using these elements. Then it would still need some mechanism for talking about distances between its points before it can leave the construction of a circle to set theory. Perhaps your book has provided such a mechanism; perhaps it leaves it to the reader's imagination. But in the latter case it's somewhat disingenuous to pretend that the circle axiom in particular is superfluous -- because in classical geometry circles are the way to know that two distances are the same in the first place.
It is also worth noting that Euclid in fact depends on some properties of circles that he doesn't bother to state as explicit axioms. In particular Euclid appears to consider these continuity properties to be so obvious that they don't even need to be axioms:

If $\overline{OB} < \overline{OA}$, then every line through $B$ has a point in common with the circle through $A$ with center $O$.

and

If $\overline{OB} < \overline{OA} < \overline{OC}$ then every circle that goes through both $B$ and $C$ has a point in common with the circle through $A$ with center $O$.

In the modern view, on the other hand, we do need to state them explicitly as axioms if we want a proper axiomatic theory of geometry that builds on undefined concepts of points. Namely, the set $\mathbb Q^2$ with the usual algebraic definitions of points and distances do not satisfy these properties, but do satisfy the axioms Euclid explicitly states, so the explicit axioms cannot possibly prove the continuity properties.
