"which you can not do" is an interesting assertion. In some sense it's trivially true, of course. Generating infinite random numbers would take infinite time. In fact, I'm quite happy to say that approximately 0% of real numbers can exist, for that reason.
But in contrast to that, I'm also happy to grant the existence of constructions with countably infinite length when discussing theories outside of computability. So in that context, is it possible to generate an infinite string of random numbers? Well, it is so long as you believe you can generate any amount random numbers at all. Just repeat the process, whatever it is.
Is that a reasonable thing to believe? That's probably a philosophical question at this point. I'm certainly not equipped to answer it. But assuming it is possible allows for some interesting math, which I support for its own sake.
This is all sort of reminiscent of a mathematician friend's stance on the axiom of choice. If you were to tell them a vehicle is only guaranteed to not explode by the axiom of choice, they wouldn't use it. On the other hand, assuming the axiom of choice is true leads to some useful math, even if it's sort of sketchy ontologically. It's a lot like programming. Sometimes you can't fix the bugs in the underlying system and you just code around them to get your stuff working.
We were talking about somewhat different things, you meant doing it mathematically, sure, no problem, pick an infinite random sequence, I understood the question to mean how to do it in practice.
Actually I had something even stronger in mind, being able to pass the generated number around. For pi I can have a program that I can pass around and that will give everyone access to all the digits of pi, in base 16 we can even have random access to all the digits.
For generating a random Cauchy sequence things are not that easy. Locally I can just use a true random number generator - in case such a thing exists - and generate all the digits I am interested in on demand, storing them in case I have to look at a previously generated digit again. But I can not easily pass that number around, only the digits I have already generated. We would need a shared database where everyone can share all the generated digits.
Or I could try to replace the true random number generator with a pseudo random number generator, then I could just pass the seed around. Everyone would have to either agree how the sequence of random numbers is mapped to the sequence of digits or we would have to use a seekable random number generator. But this raises the question whether using a [specific] pseudo random number generator would still yield a normal number.
But in contrast to that, I'm also happy to grant the existence of constructions with countably infinite length when discussing theories outside of computability. So in that context, is it possible to generate an infinite string of random numbers? Well, it is so long as you believe you can generate any amount random numbers at all. Just repeat the process, whatever it is.
Is that a reasonable thing to believe? That's probably a philosophical question at this point. I'm certainly not equipped to answer it. But assuming it is possible allows for some interesting math, which I support for its own sake.
This is all sort of reminiscent of a mathematician friend's stance on the axiom of choice. If you were to tell them a vehicle is only guaranteed to not explode by the axiom of choice, they wouldn't use it. On the other hand, assuming the axiom of choice is true leads to some useful math, even if it's sort of sketchy ontologically. It's a lot like programming. Sometimes you can't fix the bugs in the underlying system and you just code around them to get your stuff working.