Axiomatic foundations of geometry

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Problem II.4.17 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter II.4 "The Language of the Predicate Calculus"

Let MM be the set of points, lines, and planes of 3-dimensional Euclidean space, with the following predicates:

T(x)=T⇔x is a point; T(x) = \text{T} \Leftrightarrow x \text{ is a point}; L(x)=T⇔x is a line; L(x) = \text{T} \Leftrightarrow x \text{ is a line}; Pl(x)=T⇔x is a plane; Pl(x) = \text{T} \Leftrightarrow x \text{ is a plane}; On(x,y)=T⇔x lies on y. On(x, y) = \text{T} \Leftrightarrow x \text{ lies on } y.

Write the following formulas:

?
(a)

through every two points a line can be drawn; if these points are distinct, such a line is unique;

(b)

through every three points not lying on one line a unique plane can be drawn;

(c)

the definition of parallel lines;

(d)

the definition of parallel planes.

Problem II.4.18 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter II.4 "The Language of the Predicate Calculus"

In the model from Problem II.4.17, write:

?
(a)

Euclid's axiom on parallel lines;

(b)

Lobachevsky's axiom on parallel lines.

Problem II.4.19 from Lavrov I.A., Maksimova L.L. "Задачи по теории множеств, математической логике и теории алгоритмов" [LaMa02]More problems in Chapter II.4 "The Language of the Predicate Calculus"

Choose predicates and write:

?
(a)

Hilbert's axioms for Euclidean geometry;

(b)

the axioms for Lobachevsky geometry;

(c)

the axioms for Riemannian geometry.

References

[LaMa02] Lavrov I.A., Maksimova L.L. Задачи по теории множеств, математической логике и теории алгоритмов. 5-е изд., Физматлит, 2002.