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I'm back to basics as I'm preparing a science test as an adult student and I'm struggling to understand the significance of some of Hilbert's axioms.

Specifically, the third Incidence axiom tells us that:

There exist at least two points on a line. There exist at least three points that do not lie on the same line.

My gripe is with the second sentence. I understand you can identify a line with two points but I'm brought to think here that, since there are infinite points on a plane, well of course there exist at least three points that do not lie on the same line. You can actually find any number of points that do not lie on the same line.

Likewise, the eight Incidence axiom says:

There exist at least four points not lying in a plane.

Again, I realise that a plane can be identified with three points, but there are any number of points outside of a plane.

What am I not undestanding? Why make these statements?

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    $\begingroup$ 1) Who said that there are infinitely many points in a plane? 2) Even if there were, why shouldn't they all be aligned? $\endgroup$ – user228113 Mar 26 '18 at 10:14
  • $\begingroup$ Isn't the definition of a plane a "flat, two-dimensional surface that extends infinitely far"? You can identify any number of points such a surface, in any directions, as long as they are, well, on the same plane. $\endgroup$ – Alessandro Macilenti Mar 26 '18 at 10:20
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    $\begingroup$ No. It's the definition of two-dimensional surface which is "something that looks locally like a piece of the plane". In the context you are studying, you do not define what a plane, a point or a straight line are. You just state that they are objects which stay in mutual dependence with one another. That's what it means to axiomatise (a theory). $\endgroup$ – user228113 Mar 26 '18 at 10:27
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This is what you have to understand:

Ad 1. Your intuition tells you that in the plane there are points not sitting on a given straight line. You are right. But logically the axiom at stake nails down what the intuition tells. Another example for congruent segments "If $AB=CD$ and $CD=FE$ then $AB=FE$." Isn't this obvious? Yet an axiom has to ensure that purely logical thought processes may use this property of congruency.

If you want to stay on a straight line and you want to prove very simple statements about intermediacy then you may use another axiom: There is only one straight line. There are no points outside of it.

Ad 3. If you want to deal with the 3D space then you have to formally nail down that there are points outside of a plane. But if you want to stay on the plane then you may say that there are no points outside of your plane.

Also If you want to stay in the 3D space then you say that there are no points outside of it. However, if you want higher dimensional geometries then you may say that there are points outside of the space. This may sound strange. But formally, this is not different form the first axiom mentioned.

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  • $\begingroup$ Thanks. This makes the intention of the axioms clear. $\endgroup$ – Alessandro Macilenti Mar 26 '18 at 10:57
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I think I understand now. Maybe I'd rephrase as:

Given three points, there is no guarantee they lie on the same line.

Given four points, there is no guarantee they will lie on the same plane.

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