1

In the Hilbert's Foundation there is a $I.2$ axiom:

Any two distinct points of a straight line completely determine that line; that is, if $AB = a$ and $AC = a$, where $B \not= C$, then is also $BC = a$.

But Hilbert don't defines a relation "distinct". (He defines only theese: "are situated", "between", "parallel", "congruent", "continuous").

So, it looks like that in this formulation, this cannot be an axiom.

Mikhail
  • 11
  • 2
  • 1
    Find a different text. That one has a bunch of horrible translation errors. (See https://math.stackexchange.com/questions/2778914/a-confusion-about-the-second-connection-axiom-of-euclidean-geometry and https://math.stackexchange.com/questions/2961169/redunduncy-of-paschs-axiom-of-hilberts-foundations-of-geometry.) – Eric Wofsey Nov 11 '23 at 00:51
  • I have read an original text. So it also contains this relation without defenition – Mikhail Nov 11 '23 at 09:25
  • 2
    "Equality" is usually taken as primitive when you're defining an axiomatic theory. In the same way for instance the theory of groups has axioms like "$g \cdot e = g$" without saying what $=$ means. Semantically, it really just means that the objects in question are the same object. If you're proving something, you can assume $=$ is an equivalence relation with the substitution property. – Izaak van Dongen Nov 11 '23 at 12:25
  • But the theory of groups use sets, so it means that it’s a part of the set theory and it (theory of groups) must use ZFC, where we define “equality” – Mikhail Nov 11 '23 at 15:11

1 Answers1

0

I have come to the conclusion that in fact, Hilbert's axioms inherently refer to the logic in which equality is defined

Mikhail
  • 11
  • 2