Why non-standard analysis is not used?

[Alpharup]

In analysis, we do encounter tougher epsilon-delta proofs instead of more intuitive algebraic methods( those involving infintesimals). I have read that there is branch where infintesimals are rigorized like epsilon-deltas. My question is why people don't use that?
Also, is it logically sound? How long does it take to understand axioms?
Sorry, if I appear to be less informed.

 

[WWGD]

Yes, it is sound, it is elementary equivalent to the standard Reals, meaning every 1st order statement from the standard Reals is also true for the non-standards. Only difference is in the Archimedean principle. And I think it is a matter of force of habit to a good degree.

 

[Alpharup]

Can least upper bound axiom of a set of real numbers, be extended to hyper-real numbers?

 

[micromass]

Can least upper bound axiom of a set of real numbers, be extended to hyper-real numbers?

Yes. Any statement about the reals can be extended to hold true for the hyperreals. This is the transfer principle. This does not mean that you should just take the same statement. Obviously, the upper bound axiom for the hyperreals does not hold if it is stated like "any set that is bounded above has an upper bound". This will be false. You'll need to do some more modifications. I highly recommend Goldblatt's "Lectures on the hyperreals" for a nice introduction.

Then, why is it not taught? This is a good question since the infinitesimal approach is useful in math and outside math. I think it's a shame it's not taught. But there are reasons for not teaching it. Whether they are good reasons is something you must decide:

1) They have been teaching calculus the modern way fine for close to 100 years.
2) Hyperreal numbers are logically sound and useful, but are a fringe topic. Over 99.99% of research papers and books nowadays will not use the language of hyperreal numbers. This means that if you don't teach the standard approach, your students will not be able to read books and papers later on. This is obviously a very bad thing.
3) In an analysis course, you make calculus rigorous. Making the real numbers and standard calculus rigorous is already not so easy. But making the hyperreal numbers rigorous would be impossible for an undergrad course. It requires a great deal of advanced logic. This means that mathematics students need to take the hyperreal numbers on faith until grad school and possibly forever. This is against the spirit of mathematics.

As you see, not teaching the standard approach is a very bad thing. I am personally a proponent of teaching both approaches. But then you might not have enough time to accomplish this.