As a mathematician, I don't think I quite agree with your analysis Calrid.
Calrid said:
It's a made up invented term for things that can supposedly exist beyond merely infinite number sets in sets, rather a semantic issue of no real importance, a simple infinite limit will always do in maths.
Hyperfinite numbers were invented by Newton and Leibniz (and before) to give sense to their integral calculus. Sadly enough, the foundations of hyperreal where screwed up, and to fix it, they started to work with limits and epsilon-delta definitions. It is only in the later years, that infinitesimal quantities have found a real foundation. It's not because pure mathematicians wanted to invent something new, it's because they wanted to give a foundation to already existing stuff.
I wouldn't concentrate too much on the details, as this thread highlights mathematicians cannot even conceive of an infinity any more than they can of what would be beyond such a beast and what properties such an invisible unicorn might have, perhaps a shade of pinkness? They just like to play with ghosts of what might be if reality and physical existence was different. It's like fairy stories, imaginary stuff that has no real practical use outside of maths in and of itself.
So infinitesimals have no real practical use? Tell that to the physicists who work with infinitesimals every day. Entire physical theories are built up on the concept of infinitesimals. All mathematics wants to do is to give a foundation to them. This can be done in terms of hyperreals or differential geometry. But I don't think it's fair to call them fairy stories, imaginary stuff and useless.
Hyperfine numbers can be relatively useful (well outside of science or applied maths where infinities are somewhat problematic) however there are some pretty dubious cardinality issues with transfinite numbers.
I'd like to know what you mean with this. You mean the continuum hypothesis? That's not a problem of the transfinite numbers, but of the axioms of mathematics itself. Better axioms could resolve a lot of issues. (Although Godel proved that you cannot choose axioms that resolve all).
Just grasping what an infinity could be is enough for most people to barely comprehend, like the OP. Mathematicians would beg to differ they have visualised something that is by axiom more than just infinite, but meh it's what they do, whether it is philosophically apt to make up ideas of beyond something that can never be reached is a matter of debate only outside of fortress maths. You'll never convince a mathematician that anything he says is epistemologically unjustifiable when all he needs is an axiom. It is true because I say it is, constructive logic and proof is irrelevant as is utility.
Hmmm, you're the first to say that proof is irrelevant to mathematicians...
And transfinite numbers have not been invented because mathematicians thought they were fun. They were invented for a reason. Indeed, Cantor invented transfinite numbers to give sense to Fourier series. And you wouldn't call Fourier series useless do you?
Another big application of transfinite numbers is in probability theory, where the concept of sigma-algebra is fundamental.
As far as transfinite systems go though, even though this has been a firm contention in maths since its inception, amongst the great minds of both maths and philosophy, whom it appears stand in corners according to whether they study philosophy or maths. It is apparently kinda illegal to discuss this subject as a system of "numeration" that will never have any use to anything outside of maths; being called into question is apparently so unsettling that it causes threads to be locked despite the great derth of material on this subject from all sorts of great minds from ancient times to today. It is not appropriate to question what is beyond "God". Perhaps if students of maths were to question their own axioms the whole number system would fall into chaos.
Now you're just making things up. If you would know how real math works, then you would know that the axioms are being questioned every single day. And a student who does not question the axioms of mathematics, is not a good student in my opinion. Calling into question the axioms leads to very fruitful theories, like non-Euclidean geometry and the New Foundations theory. If people propose a new axiomatic system for a mathematical object, then I don't think any mathematician would hesitate to accept it if it were useful.
And as for the threads being locked. I have no qualms in discussing 0.999... and division by zero, if the OP was willing to learn. If somebody with a lot of knowledge about mathematics were to discuss these issues, I would listen and discuss with him/her. But you can't expect us to discuss something like this with somebody who hasn't seen limits and who still thinks that all mathematicians are wrong. If you do not grasp limits, then you have no idea what this question is even about.
In fact, I myself, have once constructed a new system where 1 does not equal 0.999... But the problem was that this system was ugly and not very useful. But don't tell us that we are not willing to change the axioms, because we are. The problem is often that the proposed new axioms do not deliver a nicer theory, on the contrary,...
Perhaps definitions that make any sense are not important. Who knows..?
So, which definitions do you think make no sense?
I think I've said everything I wanted, so I'll stop here. The only things that I want to make clear that mathematicians do not make things up for their amusement. There is often a need to understand something physical/mathematical/philosophical, and this is where the mathematical theories come from.
Are you talking about the space occupied by the numbers if we try to write them all out? What makes you think this has anything to do with mathematics? 
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