Can anyone please explain me what is Hilbert's Axioms?
I am going through "Foundations of Geometry" by David Hilbert. What I feel the concept of point, line and plane can be substituted by tables, chair etc. It unified both plane geometry and solid geometry. What I mean to ask you is that the axioms of Hilbert introduced, did it try to evade Euclid's axioms or it has been substituted?
There are certain things which I would like to know on Hilbert's axioms:
(1) Axioms of Connection
(2) Axioms of Order
(3) Axioms of parallel
(4) Axioms of congruence
(5) Axioms of continuity.
Apart from 3 and 5 which are Euclid and Archimedes, the other 3 are they new axioms invented by Hilbert? Or are they the same 5 axioms of Euclid, which has been proven in a different manner?
Did Hilbert tried to prove something like: Geometry can be proven WITHOUT AXIOM?
Can anyone please explain the following lines:
"Axioms are not taken as self-evident truths. Geometry may treat things, about which we have powerful intuitions, but it is not necessary to assign any explicit meaning to the undefined concepts. The elements, such as point, line, plane, and others, could be substituted, as Hilbert says, by tables, chairs, glasses of beer and other such objects. It is their defined relationships that are discussed." (Source David Hilbert - Wikipedia, the free encyclopedia)