To Generalized Extensional Atomic Mereology we add the primitive two place function symbol "distance" symbolized as "d".
Axiomatize:
d(x,x)=d(y,y)
d(x,y)=d(y,x)
x=/=y -> d(x,x)=/=d(x,y)
A point is defined as a mereological atom
A circle is defined as the totality of all points at common distance from a single point.
(i.e. C is circle if and only if there exists points c,x such that for every point y: y is part of C iff d(y,c)=d(x,c))
c is said to be the center point of C.
A line is defined as the mereological totality of all points equidistant from two distinct common points.
(i.e. L is a line if and only if there exists two distinct points a,b such that for every point x: x is a part of L iff d(a,x)=d(b,x))
Axioms:
Two Basic rules
1. From each point x and each distance k there exists a circle drawn from x at distance k.
2. From each points a,b the circles drawn from those points at distance d(a,b) intersect at two distinct points.
Three Dichotomy rules
Define (inside): x inside C <-> C is a circle ^ x is a point ^ for all L: L is a line and x part of L -> L intersect C at two distinct points.
Definition: The interior of circle C is the mereological totality of all points inside C.
Definition: The exterior of circle C is the complement of "C union its interior".
Define (Line piece ab) as the union of a and b with the intersection of the line that passes through a,b, and the interiors of the two circles drawn from a,b each at
distance d(a,b). Here a,b are termed as the 'ends' of the line piece ab.
3. y inside C <-> y is the center point of C or the line piece connecting y to the center point of C doesn't intersect with C.
4. Any line piece connecting a point in the exterior of C with a point in the interior of C must intersect C.
5. Every line L splits its complement into two distinct non overlapping compartments such that every line piece connecting a point from one compartment to a point in the other must intersect L.
Those compartments are said to be the 'sides' of the line L.
Two Intersection rules
6: No two distinct circles can intersect each other at more than two points.
7: No line can intersect a circle at more than two points.
Two Lateralization rules
8: If two circles intersect at distinct points, then there is no more than one intersectional point at each side of the line connecting their center points.
9: If K,L are distinct lines passing through the center point of a circle C, then there can exist only one intersectional point between L and C at each side of K.
Define: the size of a line piece is the distance between its ends.
Define: d(a,b)>d(c,d) if and only if there exists line pieces L,K such that L is a proper part of K and size(K)=d(a,b) and size(L)=d(c,d).
One measure rule
10: The rule of Finiteness of space:
For every line piece ab and line piece ac where ac is bigger than ab, there exists a line piece L such that L is bigger than ac and L is the disjoint union of finitely many line pieces each of which is of the same size as that of ab.
To me all of these rules are conceptually simpler than the fifth postulate of Euclid. However the above system does interpret all of Euclid's system.
Notes:
1) Axiom 9 can be generalized to the following.
If L is a line that passes through the center point of a circle C, then any line K that intersect with L inside C must intersect with C at one point at each side of L.
So line K need not be restricted to the case of passing through the center point of C.
2) The last axiom of finite measure can be replaced with an axiom that do not entail it and yet is enough to interpret the fifth postulate of Euclid and this is:
If G is an angle and K and L are the sides of G then if there are points k,l on K,L respectively such that the line piece kl is perpendicular on L, then for any distance r there do exist points p,q on K,L respectively such that pq is perpendicular on L and d(p,q) > r.
i.e. there is no limit to the size of a perpendicular line dropped from one side of an angle to the other.
This is of course entailed by the finite measure axiom, but it seems to be
weaker than it.
3) Equality between angles is defined here as the existence of equal sized line pieces connecting the sides of the angles at equal distances from the point of the angles per each side. So if we denote an angle by three points with the first point representing the point of the angle, then angles abc and
def are equal if and only if ab=de and ac=df and bc=ef.
4) We can have a version of continuous geometry with overlapping points, where
between each two points there can exist only finitely many non overlapping points. The version bears some similarity to the above one. Here is the exposition of it:
To Mereology add the primitives of "contact" which signify external connection, and distance which is the same one above, and the primitive "point" which informally stands for the area of contact that a compass make with the Euclidean plan (the compass lies outside the Euclidean plan but its ends are in contact with the Euclidean plan) we assume that all compasses have similar ends i.e. the total area of contact of each end of a compass on an end of another compass is exactly that other end. So points here are regions of space. they do have proper parts and even they may have boundary parts with their complements. Now the mereological characteristics of this space is that of endless divisibility, i.e. every part of the space has a proper part, i.e. there are not atoms.
Axioms are of:
[1] Mereological axioms of General extensional atomless Mereology
[2] Topological axioms of external contact relation "C" which are
x C y -> y C x
x C y -> ~ Exist z (z part of x ^ z part of y)
x C y <-> Exist z (z is proper part of x) ^ z C y ^ ~ Exist z (z part of x ^ z part of y)
x C complement(x)
[3] Axioms about points:
Every point is a continuous object (i.e. not the disjoint union of two non contacting parts)
No point have a proper part that is a point.
Every object do overlap with a point.
[4] distance axioms: those are the same ones outlined above in the prior system
[5] Geometric axioms:
That we can draw a circle from each point at each distance between two distinct points.
That a circle is a closed binary partition on space (Divides its complement
into two non overlapping non contacting (i.e. separate) compartments, and have every continuous proper part of it in contact with the complementary part of the circle, which is continuous also, at exactly two points.
That a line is an open (not closed) binary partition on space.
That for each line passing through the center point of a circle there are parts of the circle on each side of that line.
No circles can intersect at more than two points
No line intersect a circle at more than two points
The finiteness measure axiom (see the prior system) or its weaker no limit form (see above).
This kind of space seems much easier to handle than the above one.
Zuhair