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How do Euclid’s postulates imply that there exist more points than provided as assumptions, e.g. in the statement:

Let C1 be the circle centered at A, with radius AB, let C2 be the circle centered at B, with radius BA. let F be a point of intersection of C1 and C2.

How do we know that there is a point of intersection? Indeed how do we know that there’s more than one point on the circle if a circle is defined as “the set of points X which have AX = AB”. Why would there be more than one point? The same question stands when you read things like “extend the line and let M be the intersection…”. Why is there an intersection?

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    $\begingroup$ These are the sorts of things that Hilbert's reconsideration of Euclid's axioms addressed/added... $\endgroup$
    – paul garrett
    Commented Aug 19, 2021 at 20:40
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    $\begingroup$ It turns out that there are a lot of different implicit assumptions in Euclid's axioms which are not stated explicitly. It wasn't until around the end of the 19th century that actually rigorous axiomatic treatments of geometry were presented. See for example Hilbert's axioms en.wikipedia.org/wiki/Hilbert%27s_axioms . $\endgroup$
    – JoshuaZ
    Commented Aug 19, 2021 at 20:41
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    $\begingroup$ My high school geometry book included axioms that a line contained two points; a plane contained three non-collinear points; and space contained four non-coplanar points. Axioms like these are needed for a rigorous description of Euclidean geometry, but after introducing them, the textbook never mentioned them again. $\endgroup$
    – Buzz
    Commented Aug 19, 2021 at 20:52
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    $\begingroup$ There are various good books which trace the mathematical history of filling in various gaps in Euclid's axioms (the one you suggest is not the only gap). Take a look, for example at Geometry: Euclid and Beyond by Hartshorne. $\endgroup$
    – Lee Mosher
    Commented Aug 19, 2021 at 23:33

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