User:GFauxPas/Sandbox

Welcome to my sandbox, you are free to play here as long as you don't track sand onto the main wiki. --GFauxPas 09:28, 7 November 2011 (CST)

Tarski's Axioms of Geometry
Yay I found a source:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.27.9012

Primitive terms:

Points

Betweenness

Congruence.

Equidistance.

Let $B\left({abc}\right)$ mean b is between a and c.

Let $ab \equiv cd$ mean that the relationship of equidistance holds between ab and cd.

In the following axioms, the quantifiers range over points.

I'm gonna nail up the easier axioms and then work on the more sophisticated axioms later.

A4: $\exists x : B\left({qax}\right) \land ax \equiv bc$.

That is, given two points $q, x$, there is some $a$ in between them. The relationship of equidistance holds between $ax$ and two other points $bc$.

A5: $\forall a, b, c, d, a', b', c', d': \left[{a \ne b \land B\left({abc}\right) \land B\left({a'b'c'}\right) \land ab \equiv a'b' \land bc \equiv b'c' \land ad \equiv a'd' \land bd \equiv b'd' }\right] \implies cd \equiv c'd'$

Whew! A diagram is helpful, I e-mailed the author for permission to use his, otherwise I guess I'll create my own.

For a non-degenerate case, this means:

Construct two triangles $\triangle{acd}$ and $\triangle{a'c'd'}$

On side $ac$ of $\triangle{acd}$ pick a point $b$ between $a$ and $c$.

On side $a'c'$ of $\triangle{a'c'd'}$ pick a point $b'$ between $a'$ and $c'$.

If 4 of the 5 line segments of each triangle are equidistant, the fifth sides of the triangles are equidistant as well.

Presumably, the author's degenerate case is that the triangles are not distinct.

A7: $\forall a, b, c, p, q, : B\left({apc}\right) \land B\left)({bqc}\right) \implies \exists x: B\left({pxb}\right) \land B\left({qxa}\right)$

This means that if a line intersects a triangle on one of its sides, and does not intersect a vertex, it intersects another side of the triangle.

A8: $\exists a, b: a \ne b$

There exist two distinct points.

The author has several variants of all of these axioms but I'm still trying to understand his intent in having more than one formulation of them, as they seem trivially equivalent? I'm almost certainly missing something. I will of course read it much more thoroughly before this leaves my sandbox, but I'm showing the development of my understanding here.

A9: Don't understand this one yet.

A10: $\forall a,b,c,d,t: \left({B\left({adt}\right) \land B\left({bdc}\right) \land a \ne d }\right] \implies \exists x,y: B\left({abx}\right) \land B\left({acy}\right) \land B\left({xty}\right)$

This means:

Draw an angle $\angle{bac}$. Let $t$ be between the lines $ab$ and $bc$.

There exists some line passing through $t$ that intersects both sides of the angle.

A11: don't understand this one yet.