Give an example of a statement which is independant of the incidence axioms which is shown to be independent by the model of the usual Euclidean plane and the integer lattice plane.
Thanks
Give an example of a statement which is independant of the incidence axioms which is shown to be independent by the model of the usual Euclidean plane and the integer lattice plane.
Thanks
You can borrow it from me if you give me thy soul.Yes, that's the textbook.
Let us start with #3. The integer lattice $\displaystyle \mathbb{Z}\times \mathbb{Z}$ has countable many points. While Euclidean geometry is an example of incidence geometry (in $\displaystyle \mathbb{R}^2$) and it has uncountably many points. Thus, the notion of "being countable" is an independent statement in incidence geometry.can you elaborate on one of your answers above.
Look at #1. Consider the statement "for any line $\displaystyle l$ and for any point $\displaystyle P$ not on the line there exists a unique line $\displaystyle m$ such that $\displaystyle l$ and $\displaystyle m$ are parallel". In the integer lattice there are more than one unique line (give an example!) but in standard Euclidean geometry there is percisely one such parallel. Thus, this statement is independent in incidence geometry.