HyperReal numbers

The hyperreals are defined in a particular way. Everything which can be expressed about the real numbers using logical formulae and every last conceivable and inconceivable function upon them, is held to be constant. For example, if it is the case that for every real number there is another number greater than it, then this must also hold for the hyperreals. The difference lies in things which cannot be expressed in logical formulae. In the hyperreal number system, there is a number greater than every real number. Infinite numbers exist as actual numbers which can be pinpointed rather than symbolizing an undefined concept.

Since everything which holds for the reals must hold for the hyperreals, if you add 1 to an infinite number, you must get a different and greater infinite number. If you multiply an infinite number by 2 then you get another infinite number. If you take the reciprocal of an infinite number then you get an infinitesimal. And that's why the hyperreals were invented; the infinitesimals.

Infinitesimals exist between the reals. However, in any finite interval between two reals there must be another real so the infinitesimals seem to exist in no space at all. No real space certainly. This has led some mathematicians to describe the infinitesimals as the ghosts of departed numbers.

When calculus was first invented, infinitesimals were the natural and intuitive way in which it was expressed. It took many decades for mathematicians to invent the epsilon-delta definition of a limit but it took longer for a rigorous definition of infinitesimals. Unfortunately, the intuitive notion of an infinitesimal needed the invention of mathematical logic to be rigorously defined. And now that we have such a definition, calculus students are still taught that infinitesimals do not exist. For all its comparative simplicity, few people learn of non-standard analysis and the hyperreal number system *R.

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I don't think epsilon-delta definitions are really that unintuitive, and they have the advantage of working entirely within the reals, where infinitesimals truly don't exist. But they're still very cumbersome and tend to give miraculous results that should be obvious with a better set up. The only formal construction I've seen of differentials is as members of a cotangent space, which is no help at all. I've heard of hyperreals but never seen a treatment - any chance you could augment the above with a formal construction and/or axiomatization for us less enlightened? Thanks!

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Is there some reason this page is named "HyperReal numbers" rather than "Hyperreal numbers"?