/sci/ - Science & Math

>>12687867
the surreal number line has recursive depth, meaning 0 to ϵ (where 1/∞ ~= ϵ) has infinite subdivisions such as 1ϵ^2, 2ϵ^2, 3ϵ^2 etc.

Still interested in developing this more if other people are also. It's an interesting topic and comes up often in one form or another.

The real question eventually becomes, even if there is some incomplete symmetry in the current number line design, and even if some symbols could fix it, similar to how imaginary numbers created the field of complex math; the real question becomes what useful math can now be done with theses new symbols.

How can symbols such as symbols for the infinitesimals, the knot, a smallest number, and a largest number, be implemented to solve problems and make calculations?

I think sci should keep developing this topic. I'd even support the idea of "Hyper-reals General thread." Why not have sci develop it since the general larger math community seems uninterested or unwilling. I'll bet after a while it would become a useful system or something good would come from it.

>>11548212
[math]
\aleph_1
[/math]

>>10563127
>he doesn't realize that some infinities are bigger than other infinities
>he doesn't realize that if the space of all possibilities is a bigger infinity than the total number of chances for things to happen, all possibilities won't necessarily happen

So apparently the reason why 0.999...=1 because we use real number system. If we don't consider infinitesimal as 0, It is possible to be 0.999... is not equal to 1.
There is a number system that holds those ideas, such as hyperreal number system, or surreal number system.

This question should come naturally. How "real" are real numbers? Why do we accept real number by axiom? Why don't we put hyperreal and surreal ahead?

I know It is more of a philosophical subject that whether number really exists or it's just creation of human mind. What I only want to know is what mathematical theorems put real number as more "real" than other systems.
Is there a better answer, except for the "nah that is more useful" kind of example?

>>9728741
Here's a helpful diagram for you.
These are the hyperreals.
Only big brain phenotypes can understand it.

Obviously, as we all know, it does in the reals. This is because the real number system has the Archimedean property of having no largest or smallest number. But what about the hyperreals? There is a smallest number there, the infinitesimal.

What does /sci/ think of hyperreals?