As a mathematician, I don't think I quite agree with your analysis Calrid.
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.
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.
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).
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.
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,...
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.
I never said limits weren't useful an hence infinitesimals are useful if we accept infinity cannot ever be equalled only approached we must also accept that nothing cannot be represented physically but can only be approached. It isn't limits that are the problem or even hyper reals, it's transfinities, what it means to have infinite infinite sets where the problem becomes epistemologically inexplicable. Which rather makes the rest of your arguments redundant at least if you mean anything that is bound to a limit like calculus etc. For example is pi closer to pi at aleph 0, infinity in natural numbers, or is it closer to infinity at aleph 1 or aleph 2, or aleph omega? What does it mean to set up limits that are more than infinite or less? is it conceptually viable, will what is beyond reality ever have utility unless imagination is of course just a part of the set that exists.
Does the photon have 0 mass, or have we only measured it to a lower bound to which the difference is practically inconsequential?
I acknowledged hyperreals have utility in pure calculus issues. What I don't acknowledge is that beyond infinity ever could make any sense to anyone. What is beyond that which we cannot even imagine except sophistry and religious fervour or fairy tales?
You have no idea what value infinity has, and like wise you have no idea how to cardinalise an actualy infinite value, because you could never reach its limit. This may allow us to say that infinity ^ infinity is akin to aleph omega, but this actually means nothing, nor ever could. It is eternally philosophical arm waving. It does not actually mean infinity, unless we make the destinction between something like the size of the universe, a countable infinity and infinity a number in which no matter how long one spent trying to approach it, one would never reach it. It is beyond definition. To define it is as many philosophers have said is to define God: that which cannot be comprehended or defined. So what is beyond that which is beyond all that exists exactly, and why should we care?
I agree that .999... = 1 at infinity but that is only the case if we do not use transfinities, otherwise it is more or less equal depending on what set you are using. Can you see why such mental masturbation is useless? We only need one limit for any proof in any field of maths you care to name, we can derive all the rules of maths from simply having infinity as 1 asymptotic non defined value. To be honest we can probably get away without limits in most of maths with the exception of course of calculus and set/ number theory which itself underpins mathematical axioms. Science it doesn't even get a mention as its physically impossible. It is a very useful and purposeful limit when it is undefined.
The threads that were locked were in philosophy and general. I don't have an issue with that although an explanation would of been nice, particularly when I requested one. You know like thread locked pending moderation is not really an explanation..? But meh whatever.
I was being sarcastic about mathematicians questioning axioms you'll note also hence the smilie.
The only real axiom that makes sense I think is how can I apply this to reality, how might I use this: beyond all that exists? I guess that is where applied mathematicians and mathematicians differ. Cantors continuum is not even a non constructive proof, it cannot even define its terms as they are indefinite by every axiom outside of that one. Not that I argue with: if that given axiom is accepted without question then it must be true, but axioms don't need to be deductive or require proofs they just need to be accepted. I don't think we should accept that axiom because it has no utility or function and it cannot be iteratively proven only alluded to. As Kant said existence is not a predicate, by which he meant nothing exists just because it has a property we can imagine. It certainly doesn't exist logically, except as a limit to reality, if we cannot even comprehend it, much less what lies beyond it has or could have any utility to anything except circular self referential a priori assumptions.
I don't think mathematicians actually gained any real further understanding from imagining what infinite infinities might be simply because they cannot even comprehend an infinity in the first place without making it something it is not. The universal set on its own would be enough to define all that is to which all sets are part of, it and all mathematical branches from topology to algebra likewise can be contained in a set of definitive values, not illusory ones or not much better allusory (is that a word) ones. Sure it's a semantic issue, but aren't semantic issues sometimes very important?