It would also interesting to note that the program can't run and optimise itself simultaneously. Probably it need to copy its source code, edit it, than terminate itself and start the new code. Or edit only subagent which is not in use in current moment.

To be fair, it seems that recently almost everyone can speak before a some kind of UN panel.

Wildberger's complaints are well known, and frankly not taking very seriously. The most positive thing one can say about it is that some of the ideas in his rational trignometry do have some interesting math behind them, but that's it. Pretty much no mathematican who has listened to what he has to say have taken any of it seriously.

This question presupposes that the task will ever be done Sure. It's called super-tasks.

From mathematics we know that not all sequences converge. So the sequence of distributions that you gave, or my example of the sequence 0,1,2,3,4,... both don't converge. Calling them a supertask doesn't change that fact.

What mathematicians often do in such cases is to define a new object to denote the hypothetical value at the end of sequence. This is how you end up with real numbers, distributions (generalized functions), etc. To be fully formal you would have to keep track of the sequence itself, which for real numbers gives you Cauchy sequences for instance. In most cases these objects behave a lot like the elements of the sequence, so real numbers are a lot like rational numbers. But not always, and sometimes there is some weirdness.

From the wikipedia link:

In philosophy, a supertask is a countably infinite sequence of operations that occur sequentially within a finite interval of time.

This refers to something called "time". Most of mathematics, ZFC included, has no notion of time. Now, you could take a variable, and call it time. And you can say that a given countably infinite sequences "takes place" in finite "time". But that is just you putting semantics on this sequence and this variable.

The limit of your distributions is not a distribution so there's no problem.

If there's any sort of inconsistency in ZF or PA or any other major system currently in use, it will be much harder to find than this. At a meta level, if there were this basic a problem, don't you think it would have already been noticed?

Why do you believe that? And do you also believe that ZF is inconsistent?

At least two major classes of existential risk, AI and physics experiments, are areas where a lot of math can come into play. In the case of AI, this is understanding whether hard take-offs are possible or likely and whether an AI can be provably Friendly. In the case of physics experiments, the issues connected to are analysis that the experiments are safe.

In both these cases, little attention is made to the precise axiomatic system being used for the results. Should this be concerning? If for example some sort of result about Friendliness is proven rigorously, but the proof lives in ZFC set theory, then there's the risk that ZFC may turn out to be inconsistent. Similar remarks apply to analysis that various physics experiments are unlikely to cause serious problems like a false vacuum collapse.

In this context, should more resources be spent on making sure that proofs occur in their absolute minimum axiomatic systems, such as conservative extensions of Peano Arithmetic or near conservative extension?