# Why are Euclid axioms of geometry considered 'not sound'?

The five postulates (axioms) are:

1. "To draw a straight line from any point to any point."

2. "To produce [extend] a finite straight line continuously in a straight line."

3. "To describe a circle with any centre and distance [radius]."

4. "That all right angles are equal to one another."

5. "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, meet on that side on which are the angles less than the two right angles."

What's wrong with them? Which axiomatic system is being used nowadays? Hilbert's or SMSG (School Mathematics Study Group)? I believe in the case of SMSG the list of axioms contains some redundancy. Why do people say Euclid's axioms are 'far from being sound', even if they are all still (I guess) believed to be true? If there's something wrong with them, then maybe our better (Hilbert's or whatever) axiomatic system contains some false statements?

• For example the third axiom is the only one about circles. So strictly speaking, it is hard to prove anything about circles – Hagen von Eitzen Apr 21 '15 at 19:35
• Can you cite a source that finds issue with them? Most criticism I've heard has centered around extra necessary assumptions; things that let us conclude "Yes, these two circles definitely have a non-empty intersection" in proposition 1, for example. – pjs36 Apr 21 '15 at 19:36