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What does it mean for one geometrical axiom to be considered equivalent to another geometrical axiom?

For example consider Playfair`s axiom:

In a plane, given a line and a point not on it, at most one line parallel to the given line can be drawn through the point.

This is always described as being equivalent to Euclid's parallel postulate (the 5th postulate) which states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

However in Euclidean geometry two points are required to define a line. Therefore I would argue Playfair's axiom is not logically equivalent to Euclid's fifth postulate since it posits the existence of lines through a single point. Playfair's axiom is often characterized as a more streamlined version of Euclid's postulate but I suspect this is because it tacitly uses ideas that are not present in Euclidean geometry.

Often when mathematicians talk about Euclidean geometry I have found they are actually talking about another geometry which was devised to serve as a rationale for non-Euclidean geometry. The claim is basically this: Euclidean geometry = Neutral geometry plus the parallel postulate. However the right side of this "equation" incorporates ideas that are not present in Euclidean geometry.

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  • That, form the with others Euclid's axioms plus Ax-1 we can prove Ax-2 and vice versa. Commented Jun 15, 2021 at 11:34
  • Two points uniquely identify a single line. Through a point, we can draw an infinity of lines. Commented Jun 15, 2021 at 11:36
  • See Parallel postulate: Equivalent properties Commented Jun 15, 2021 at 11:59
  • In Euclidean geometry a straight line is constructed by first specifying two points. We can draw an infinity of lines through one point but that way of making lines is inconsistent with the rules of Euclidean geometry. Commented Jun 15, 2021 at 12:03
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    @Methadont, well said. Commented Jun 15, 2021 at 12:28

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I would say your error lies in this assertion:

However in Euclidean geometry two points are required to define a line.

What the axiom actually says is a 'line' (modern interpretation: a line segment) CAN be drawn between two points. It does NOT say that you HAVE TO HAVE two points before you can draw a line. In point of fact, you can draw a line entirely without regards to any points, if you so wish. It is merely useless to do so, as then there is no logical connection between the drawn line and the rest of the figure, so no logical deductions can be applied to it (unless this is the first part of the figure drawn, and other parts are subsequently drawn in relation to it).

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  • Yes one is free to draw such lines. The problem is mathematicians have given themselves the license to make logical connections between such lines and the rest of the figure. Commented Jun 16, 2021 at 12:46
  • In other words if logical connections are to be made a line first needs to be specified by two points. Commented Jun 16, 2021 at 12:52
  • If I committed an "error", it is one of tone rather than of logic. Commented Jun 16, 2021 at 13:01
  • You need at least two points to draw a line. In fact you need infinitely many. There is no error. So two points are required.
    – user52804
    Commented Jun 17, 2021 at 14:26
  • @Methadont I think what Euclid meant by a straight line is more like a string drawn taut between two points rather than a line drawn on a plane surface. Points can be assigned to such a line like beads on a necklace but the line itself is not made of the points. Circles on the other hand are described on a surface with a string of fixed length. Instructional video: "How to use a string line" youtube.com/watch?v=1Eto9MJa67A Commented Jun 21, 2021 at 11:20

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