野菜花 problem revisit

[fzy]

This problem was posted originally by 野菜花 (a very talented lady who frequented WXC about a year and a half ago) at WXC:

空间中是否存在一个无限点集，它在任何平面上都至少有一个点，但都没有无限多个点？

A solution is the curve x=t^5, y=t^3, z=t.

Recently it reappeared there in a different way:

在三维空间中找一条曲线，与任何平面都相交，交点不超过3点。

It was promptly changed to no more than 5 points. Apparently the new poster meant to find the same answer.

Now lets try this version:

空间中是否存在一个无限点集，它与任何平面都相交，但不超过3点？

[fzy]

My solution to this problem uses a standard transfinite argument. This type of solutions would be considered by many as invalid, because of the assumptions behind them (Axiom of Choice). But at least it shows that you cannot disprove the problem.

I still want to see a "real" solution to the problem, a set constructable in the usual sense.

For definition of terminologies, please refer to any text book on set theory.

Recall that c is the cardinal number of the continuum. There are c many planes in a 3d space. Write them as {P_alpha :&nbsp; alpha is an ordinal number < c}. For each ordinal number alpha, we define a set S_alpha using thansfinite induction (an induction on an uncountable set like c) such that S_alpha intersects every plane P_beta for beta <= alpha, but at no more than three points.

Now for given alpha, let S' = U {S_beta: beta < alpha}. If S already intersect P_alpha, let S_alpha = S'. Otherwise we choose a point x in P_alpha such that x is not on a line formed by any two points in S' nor on a plane formed by any three points in S'. Because the cardinal number of S' is less than c, we can find such a point. Let S_alpha = S' u {x}.

Now let S = U {S_alpha : alpha < c}. Then S satisfies the condition:

1) For any plane P_alpha, the subset S_alpha of S intersects P_alpha.
2) Suppose P_alpha intersects S at 4 points, x_1, x_2, x_3, x_4. These 4 points are added to S one by one, and we assume it is in this order. Then when x_4 is added (at S_beta), the other 3 points are already in the set, and x_4 cannot be coplanner with them.

A similar argument can be used to prove that there exists a set in the 3d space that intersects every straight line, but no more than twice.