Rakih_Elyan
He/Him- 4,408
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- #321
Why did transcending Infinite Dimensions in the tier page only equal to Aleph-1 then?Oh yeah, you have a point. I wasn't quite careful with the exact formulation there. In fact, I was double not careful regarding something else as well, so I have to revise some stuff I said earlier, as I came up with a good argument for why it's wrong.
The case that I literally wrote there equates to "can destroy an n-dimensional space for every natural number n." In this you're correct. That would be arbitrarily high into 1-B, just below High 1-B, and we would not give it a +1 ('cause this statement implies to qualitative superiority over the spaces). But if we added transcendence then it would be High 1-B.
If it's instead "can destroy all the nested n-dimensional spaces at once" then it's something infinite dimensional, because the unification of all finite-dimensional spaces is infinite-dimensional. Here lies the second part I was not careful with. Let's say we, somewhat intuitively, identify the union of all finite-dimensional spaces with the space of infinite sequences R^N. (I say intuitively, as one could consider the 1D space as the subspace build by infinite sequences in which only the first entry may change, the 2D space the one where the first two may change etc. and with no specific limit regarding distance. i.e. only equipped with the product topology) Now, despite what it might look like at first, the cardinality of the dimension of R^N is not aleph_0, but aleph_1. (Here's a proper explanation on that.) But I kinda didn't think that far when I made my initial statement.
Hence, in the case of destroying all finite-dimensional spaces at once, the tier would be Low 1-A. Honestly, that's a much better reason to start the (Low) 1-A realm there instead of at aleph_0, than that separation argument or whatever it was...
With that in mind I then have to correct earlier statements I made as well. For this, and none of the other reasons that were given before, I agree that it makes sense to tier transcendence over arbitrarily many (finite) dimensions as 1-A (aleph_2), as being 1 infinity above the unification of finite-dimensional spaces is aleph_2, as the unification itself is aleph_1.
So, while for different reasons, I suppose I'm actually fine with your initial 1-A proposal.
So what will the tier of these statements be?