To clarify what I am doing there: I am starting with an intuitive depiction of what a (strict, total) order should be, convert it to a definition which only needs set-theoretical concepts: (strict total) order is a subset O of P(S x S) such that for every x != y from S, either (x,y) in O, or (y,x) in O, and there is no finite sequence x1,x2,x3,...,xn such that (x_i,x_(i+1)) in O, and also (xn,x1) in O (I said it less formally by not containing any directed cycles). By the way, usual definition uses the properties of "irreflexivity", "asymmetry", "transitivity", and "total", which is equivalent. Using such definition of an ordering, the elements on the left of x are exactly such y such that (y,x) in O. (analogously, the elements on the right) However, this is a definition not yet present at 10:15. There, I just rely on an intuitive idea what an "ordering" should look like, and what "on the left / right" should mean in order to figure out a formal definition that would match such intuitive notion.
Great video Mirek!!
excellent!
10:15 How do we define what is *left/right* of the point?
To clarify what I am doing there: I am starting with an intuitive depiction of what a (strict, total) order should be, convert it to a definition which only needs set-theoretical concepts: (strict total) order is a subset O of P(S x S) such that for every x != y from S, either (x,y) in O, or (y,x) in O, and there is no finite sequence x1,x2,x3,...,xn such that (x_i,x_(i+1)) in O, and also (xn,x1) in O (I said it less formally by not containing any directed cycles). By the way, usual definition uses the properties of "irreflexivity", "asymmetry", "transitivity", and "total", which is equivalent.
Using such definition of an ordering, the elements on the left of x are exactly such y such that (y,x) in O. (analogously, the elements on the right)
However, this is a definition not yet present at 10:15. There, I just rely on an intuitive idea what an "ordering" should look like, and what "on the left / right" should mean in order to figure out a formal definition that would match such intuitive notion.
I don't know how most of the calculus was developed without any set theory.
first view