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3 Joel's question

Such a set of lines can be constructed by transfinite induction of length $\mathfrak c = 2^{\aleph_0}$.

Enumerate all possible directions and all possible points in order type $\mathfrak c$. (This uses a well-order of the reals.)

In the i-th step, do the following two steps:

Step (1,i): If the i-th point is covered by the lines selected so far, do nothing. Otherwise, choose a line through this point which has a direction that was not used by previously selected lines, and that does not meet any intersection point of any two previous lines.

Step (2,i): If a line with the i-th direction has already been selected, do nothing. Otherwise, choose a line in the i-th direction which does not contain meet any intersection point of lines chosen so far.

Each of the choices above uses a well-order of the reals. Each such choice is possible because there are $\mathfrak c$ many possibilities a priori, and fewer than $\mathfrak c$ are disqualified by the requirement to avoid certain points.

Unless I have made a mistake, this construction by transfinite induction is quite standard. If I recall correctly, Ciesielski's book "Set theory for the working mathematician" contains more examples of such constructions. ADDED: Also, Joel's question mentions several similar (plus a few dissimilar) constructions, both on mathoverflow and elsewhere.

2 fewer, not less

Such a set of lines can be constructed by transfinite induction of length $\mathfrak c = 2^{\aleph_0}$.

Enumerate all possible directions and all possible points in order type $\mathfrak c$. (This uses a well-order of the reals.)

In the i-th step, do the following two steps:

Step (1,i): If the i-th point is covered by the lines selected so far, do nothing. Otherwise, choose a line through this point which has a direction that was not used by previously selected lines, and that does not meet any intersection point of any two previous lines.

Step (2,i): If a line with the i-th direction has already been selected, do nothing. Otherwise, choose a line in the i-th direction which does not contain meet any intersection point of lines chosen so far.

Each of the choices above uses a well-order of the reals. Each such choice is possible because there are $\mathfrak c$ many possibilities a priori, and less fewer than $\mathfrak c$ are disqualified by the requirement to avoid certain points.

Unless I have made a mistake, this construction by transfinite induction is quite standard. If I recall correctly, Ciesielski's book "Set theory for the working mathematician" contains more examples of such constructions.

1

Such a set of lines can be constructed by transfinite induction of length $\mathfrak c = 2^{\aleph_0}$.

Enumerate all possible directions and all possible points in order type $\mathfrak c$. (This uses a well-order of the reals.)

In the i-th step, do the following two steps:

Step (1,i): If the i-th point is covered by the lines selected so far, do nothing. Otherwise, choose a line through this point which has a direction that was not used by previously selected lines, and that does not meet any intersection point of any two previous lines.

Step (2,i): If a line with the i-th direction has already been selected, do nothing. Otherwise, choose a line in the i-th direction which does not contain meet any intersection point of lines chosen so far.

Each of the choices above uses a well-order of the reals. Each such choice is possible because there are $\mathfrak c$ many possibilities a priori, and less than $\mathfrak c$ are disqualified by the requirement to avoid certain points.

Unless I have made a mistake, this construction by transfinite induction is quite standard. If I recall correctly, Ciesielski's book "Set theory for the working mathematician" contains more examples of such constructions.