#### nomadreid

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- Starting on basics of computational learnability, I am missing the key intuition that allows results about finite processes to reference results from the infinite domain.

**Summary:**Starting on basics of computational learnability, I am missing the key intuition that allows results about finite processes to reference results from the infinite domain.

Snowball:

(a) Hazard brought my attention to a popular article https://towardsdatascience.com/five-machine-learning-paradoxes-that-will-change-the-way-you-think-about-data-e100be5620d7, which had several blatant errors, so

(b) I went to the sources https://www.nature.com/articles/s42256-018-0002-3#ref-CR25 and

(c) https://arxiv.org/abs/1711.05195

which discussed ideas that I had no idea about, so

(e) clicking wildly on Wikipedia.

I did not mean to get in this deeply, and I have no intention of devoting much time to it: I wish to get the main ideas; therefore it is possible that my question is too vague, for which I apologize in advance.

The upshot (most clearly expressed in (c)) is that ZFC is not a sufficient basis for machine learning, which is not surprising; what is more curious is the use of the Continuum Hypothesis (CH). In attempting to see how the authors extend their results to be able to invoke the independence of CH, I find that I am missing a key bit of intuition. I would be grateful for an enlightening explanation of the following:

In the above sources, I find constant references to finite sets, to their maxima, to their cardinality, and to approximations of convergence. Approximations are the order of the day, with some precise results in the form of bounds. The only references to infinity at first are in the limits, and the idea that the functions are over the real numbers (even though, as I understand it, computers only actually use rationals), or over the collection of all finite sets. Then various results are extended to aleph-null, aleph-one, and the cardinality of the reals, bringing it into the realm of the CH.

Knowing that one must be careful when extending results about the finite over to the infinite, and from approximations to exact results, I am trying to find the justification of this jump (if possible, without working through each and every equation in the papers). Could someone enlighten me (knowing that I am presently close to zero in the theory of machine learning, having only begun to read into this field)? I will take into account that any answer on this level is bound to be a simplification, but that is fine with me at this point.

**[Moderator's note: Link removed which breached copyright.]**

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