Dear Arved Hübler,
good work, although some people may consider it as purely metaphysical.
In the past, i have considered similar lines of reasoning and came also to the conclusion that mathematics must have been created. My current answer to this question is, indeed it was, but i have to make some restrictions on what you exemplified.
Firstly, one has to recognize that even before the creation of maths, there had to be some kind of logic, because otherwise the creation process you describe would not be necessarily appear at all. If the latter would be true, mathematics would be the result of an absurd change from an empty set to a - relatively - meaningful existence. For differenciating impossible existence from potential existence - as you have done stringently - one (and i think not only a thinker, but also 'nature' at is fundamental level, means at the level of an empty set) needs some kind of logics which allows nature to differentiate between impossible and potential existence. Otherwise even impossible existence would be possible! And what could the latter mean other than that your creation process is just a lucky fluke, an appearance of mathematics without any (and i emphasize *any*) reason whatsoever, a kind of nice and consistent nightmare (for some people) out of the very blue. There could well be other such emergences, not so nice and totally devoid of any meaning and order - but with observers in it! It your creation process does not presuppose some kind of logics, the latter scenario cannot be ruled out.
My answer to the question of what initiated the creation process of maths is that it can only be a higher consciousness who has set the stage for this. I therefore take it for highly probable that there exists an intelligent entity, usually called God, from whom the needed logic originated.
Let me say that this would be consistent with what you wrote. An induction process needs, in my opinion, something more than an empty set, it needs logics. Otherwise, as already stated above, the appearance of your creation process would be some kind of magic, or if one likes, a totally irrational event from out of a nothingness which is logically not limited to further produce all kinds of irrational events - all the time.
Induction traditionally is understood as collecting some data which does form a certain pattern and then induce something from it. In the case of an empty set, one has just a half bit of information: there is an empty set (or stated from the view we humans have now: there is no pattern). For your creation process to start, you need 1 bit that can make a logical difference. But an empty set is not 1, but 0. Surely, in retrospective view, an empty set can be equated with 1, because we can now contrast it with existing things, but if we talk about 'nothing', this traditionally means that there is nothing, not even an empty set (yes, nothing in the traditional sense is very hard to imagine; i assume because it simply is impossible!). So what you have done is to induce from the fact that there is existence to the conclusion that there has always been some existence, even if it is only an empty set. So, from this perspective, you have the 1 Bit, the empty set and the consequences of it - namely, the non-empty set. But you have induced it with what couldn't be present before your creation process, namely logics. You circumvent this by saying that the empty set is a mathematical set, an homogenous background and thereby tacitly introducting the needed logics for your creation process to unfold. But if logics is already there, then also maths is potentially there. Moreover, one then has to ask where the logics came from, what i do here.
What to you appears as an empty set in your description, could well be the full gamut of God. In this empty set there would be contained all eternity. Hard to swallow and to understand, but some near-death experiences do report impressions like this. Surely this empty set would then also include the needed logic. I think if there is this entity called God, he/she very well knows what happens if his domain would be contrasted with something that is not this empty set. Therefore, in some sense, it seems to be very natural that if something (or someone) does try to transcend/leave the realm of God's existence (due to free will), it results in a very real illusion of existence that is relatively devoid of all the good properties which are ascribed to God (i think if we wouldn't have been created as eternal souls and in the picture of God's own existence, the world would be not only relatively devoid of all the good properties which are ascribed to God, but absolutely devoid of it) Moreover, this realm of existence would seem to successfully prove its own absurdity and in some sense it does indeed prove it. Only take for example what i have wrote in my essay about extrapolating mathematics to be the only fundamental level of reality. This would lead to a world which is exclusively only ruled by mathematical relationships and therefore is strictly deterministic. No free will whatsoever, but only absurdity at the core level. I don't buy such a conclusion but further stick to logics and it tells me that there must be more than a mathematical empty set at the roots of existence. Because an empty set as a stanalone feature of some existence - the existence of the set itself! and not more - does not make much sense.
Nonetheless your essay is a brave piece of work to tackle the big questions. Thanks for an enjoyable reading!
Best wishes,
Stefan Weckbach