Another question on the Hyperreals - regarding the monad at infinity...

Question

I'm interested in exploring whether there is a monad at infinity. I guess we would define the infinitesimal space surrounding infinity as "A number that is greater than any Real number, but smaller than infinity". I can see some problems with it though - there seems to be no way to differentiate between infinity and near-infinity, since if omega = infinity, then omega - a = omega for all a an element of the reals. At the same time any object that is greater than the reals but smaller than infinity must be infinitesimally close to infinity. But Infinity is an attractor in this sense, in that all numbers that are within any commensurable distance of infinity must be infinite in magnitude. So there cannot be a monad of infinity? Am I even close?

Answer 1

This question seems based on a misconception.

There is not a single thing called "infinity" in the$^1$ hyperreals. Rather, there are many different hyperreals which are infinite and quite distinct from each other. Put another way, there is no "distinguished" infinite hyperreal, any more than there is a "distinguished" infinitesimal hyperreal.

Meanwhile, by definition a hyperreal is infinite iff it is greater than every real number, so trivially there's nothing between finite and infinite in the positive hyperreals.

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$^1$Actually, it's an abuse of terminology to even talk about "the hyperreals" - there is no specific thing called the hyperreals. Rather, there is a notion of "hyperreal field," and we can prove that lots of genuinely different (= non-isomorphic) hyperreal fields exist.

Generally in nonstandard analysis it doesn't matter which we use, so we simply pick one "in the background" and call it the hyperreals, but this is an important subtlety worth mentioning. (And I believe there are more technical situations where we actually do care, to a certain extent at least, which hyperreal field we use - although I can't find an example of this at the moment.)

Answer 2

The problem is that you see infinity on the same axis as normal numbers, leading you to believe the “last” number of normal numbers and the “first” number of that new axis “touch”.

Try seeing those numbers like a new axis. I mean i gets its own axis for much less hefty reasons.

Then in becomes trivial to resolve paradoxes regarding infinity.

And don’t be discouraged by the cultists from the non-scientific branch of mathematics getting triggered by you daring to “offend” their dogma err I mean “axiom”. ^^

A further hint is to think about ”partial” numbers. Think partial functions, but for numbers. 5/0 never needs to be fully resolved to get across the singularity called infinity, if you allow for a new axis to “remember” 5 across it.