[...] In the narrative, "eternal recurrence" is just a hypothesis put forward by the author. None of the exhaustive arguments, axioms, theorems, syllogisms, etc., required to prove or support a "philosophical" theory, are ever given either here, or any other works of Nietzsche. Eternal Recurrence is just an idea, a concept, thrown at us much like a ghost story. [...]

I wonder whether even such axioms as the the ones that follow would be accepted by this person as "proper foundations" for a system of thought

Axiom 1: Things which are equal to the same thing are also equal to one another.

Axiom 2: If equals be added to equals, the wholes are equal.

Axiom 3: If equals be subtracted from equals, the remainders are equal.

Axiom 4: Things which coincide with one another are equal to one another.

Axiom 5: The whole is greater than the part.

Postulate 1: It is possible to draw a straight line from any point to any other point.

Postulate 2: It is possible to produce a finite straight line continuously in a straight line.

Postulate 3: It is possible to describe a circle with any center and any radius.

Postulate 4: It is true that all right angles are equal to one another.

Postulate 5: It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.

It is often and erroneously asserted that Euclid's parallel postulate is equivalent to Playfair's axiom, named after the Scottish mathematician John Playfair, stating that "Exactly one line can be drawn through any point not on a given line parallel to the given line." This axiom is actually more powerful than Euclid's parallel postulate, as it assumes that a single parallel line exists. This does not follow from Euclid's postulate. In fact, it is possible to develop spherical geometry without contradicting the parallel postulate, as it does not assert that the lines will not meet again on the side of the obtuse interior angles. Euclid himself believed he had shown in his Proposition 1.27 that parallel lines exist independently of the parallel postulate, which would have ruled out spherical geometry. However this proof depends on an implicit assumption made in Proposition 1.16 which Euclid does not appear to have recognized. This assumption along with the parallel postulate are together equivalent to Playfair's axiom.

Some of the other statements that are equivalent to the parallel postulate or to Playfair's axiom appear at first to be unrelated to parallelism. Some even seem so self-evident that they were unconsciously assumed by people who claimed to have proven the parallel postulate from Euclid's other postulates. Here are some of these results:

- The sum of the angles in every triangle is 180°.

- There exists a triangle whose angles add up to 180°.

- The sum of the angles is the same for every triangle.

- There exists a pair of similar, but not congruent, triangles.

- Every triangle can be circumscribed.

- If three angles of a quadrilateral are right angles, then the fourth angle is also a right angle.

- There exists a quadrilateral of which all angles are right angles.

- There exists a pair of straight lines that are at constant distance from each other.

- Two lines that are parallel to the same line are also parallel to each other.

- Given two parallel lines, any line that intersects one of them also intersects the other.

- In a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the other two sides (Pythagoras' Theorem).

- There is no upper limit to the area of a triangle.

However, the alternatives which employ the word "parallel" cease appearing so simple when one is obliged to explain which of the three common definitions of "parallel" is meant - constant separation, never meeting or same angles where crossed by a third line -- since the equivalence of these three is itself one of the unconsciously obvious assumptions equivalent to Euclid's fifth postulate.

Two alternatives to the parallel postulate are possible in non-Euclidean geometries: either an infinite number of parallel lines can be drawn through a point not on a straight line in a hyperbolic geometry (also called Lobachevskian geometry), or none can in an elliptic geometry (also called Riemannian geometry). That other geometries could be logically consistent was one of the most important discoveries in mathematics, with vast implications for science and philosophy. Indeed, Albert Einstein's theory of general relativity shows that the real space in which we live is non-Euclidean.

The parallel postulate in Euclidean geometry states, for two dimensions, that given a line l and a point P not on l, there is exactly one line through P that does not intersect l, i.e., that is parallel to l. In hyperbolic geometry there are at least 2 distinct lines through P which do not intersect l, so the parallel postulate is false. Models have been constructed within Euclidean geometry that obey the axioms of hyperbolic geometry, thus proving that the parallel postulate is independent of the other postulates of Euclid.A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.