Some quetions about set theory.

[MathematicalPhysicist]

so if i define as you suggeted:
(m,n)->RxR for y=mx+n is that enough?
as you said the vertical lines which make the coordinate system aren't considered here, this is why i opted to choose the point (x,y) and therefore considering the set of all lines as a set of sets of points which consist the lines (somehow i feel this will not suffice here too).

p.s
i think that in order to include the vertical lines, if you cannot use your option of (m,n) then why not the pair (x,y) which will include all the points?

 

[matt grime]

Nope, you're still not making sense.

"i opted to choose the point (x,y) and therefore considering the set of all lines as a set of sets of points which consist the lines"

choose the point (x,y) where? You can't pick a point lying on the line to parametrize the line since anyone point in the plane (where we are considering the lines) lies on uncountably many different lines. It is easy to write down a bijection between the set of non-vertical lines and RxR, the line L(m,n) which is the line with equation y=mx+n goes to..., and it can be modified to take account of the vertical lines too; a vertical line can be thought of as a line with slope \infty, so there is a bijection between the lines and (\mathbb{R}u\{\infty\})\times \mathbb{R} and \mathbb{R}u\{\infty\} has the same cardinality as R.

 

[AKG]

but akg, the set of all the straight lines in a plane isn't it the union of the sets?

No. The set of lines is the set of lines. The union of lines is the plane.

if a straight line is set in itself then the set of all the stragiht lines is a set of union of the stragiht lines iteself.

HUH!?

but let's say I'm wrong here, then by your reply the answer should be 2^\aleph

WHAT?!

cause we have a set of sets, which is equal to the power set.

None of what you're saying even begins to make sense.

 

[HallsofIvy]

You are not being precise enough. "A set of sets" is not necessarily a power set. The power set of A is specifically the collection of all subsets of A. Given a set A, it would be possible to select a "set of subsets of A" that is not the power set of any set- just don't include the empty set!

In particular, if A= {1, 2, 3}, with cardinality 3, it's power set is {{}, {1}, {2}, {3},{1,2},{1,3},{2,3},{1,2,3}} which has cardinality 2³= 8. But A= {{1},{2},{3}} is also a "set of sets" and has cardinality 3.