# Thread: Incidence Geometry and Integer lattice

1. ## Incidence Geometry and Integer lattice

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

2. Originally Posted by hercules
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
Are you by any chance in CCNY? (I know you are from New York because of your ISP).

1)Parallel postulate
2)Transitivity of parallels
3)Uncountably many points.

3. Originally Posted by ThePerfectHacker
Are you by any chance in CCNY? (I know you are from New York because of your ISP).

1)Parallel postulate
2)Transitivity of parallels
3)Uncountably many points.
How do u know my ISP?
Thanks for the ideas.
I still need to get the class books to start studying geometry.

4. Originally Posted by hercules
How do u know my ISP?
I know everything.
Thanks for the ideas.
I still need to get the class books to start studying geometry.
Is it Poincare' Half-Plane? I have it.

5. Originally Posted by ThePerfectHacker
I know everything.

Is it Poincare' Half-Plane? I have it.
can you elaborate on one of your answers above.

Yes, that's the textbook.
And I know who you are....whohahahah

6. Yes, that's the textbook.
You can borrow it from me if you give me thy soul.

can you elaborate on one of your answers above.
Let us start with #3. The integer lattice $\mathbb{Z}\times \mathbb{Z}$ has countable many points. While Euclidean geometry is an example of incidence geometry (in $\mathbb{R}^2$) and it has uncountably many points. Thus, the notion of "being countable" is an independent statement in incidence geometry.

Look at #1. Consider the statement "for any line $l$ and for any point $P$ not on the line there exists a unique line $m$ such that $l$ and $m$ 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.

7. Originally Posted by ThePerfectHacker
You can borrow it from me if you give me thy soul.
Well I agreed to buy it from a old classmate yesterday...and i can't go back on the deal...and i will get it tomorrow. Thanks for the offer (W).

What classes are you taking this semester?

And Double thanks...see you around at school.

8. Originally Posted by hercules
What classes are you taking this semester?
The only math class that I have to go to is Set Theory.