What can 2-adic arithmetic tell us about the Continuum Hypothesis?
More precisely, what happens to the continuum hypothesis if we define an alternative metric on the size of sets saying that two sets are close in size if their sizes differ by a large power of 2 and ask the same question, i.e. "is there some intermediate size between the infinite set and its power set?", based on this metric?
If we think of induction of the superset relation as analogous to counting in integers, then we might consider the infinite set as the limit point of this process in the same way infinity is the limit of the integers.
What set theory appears to lack currently, if we wish to continue the analogy, is some metric by which the difference in size between some set and the infinite set can be measured and thereby some set-theoretic equivalent of Cauchy sequences defined.
If we measure any finite set $X$ and measure the difference between the cardinality of its power set and itself we have $2^{\lvert X\rvert}-\lvert X\rvert$
Obviously $\lim_{n\to\infty}2^n-n$ diverges but what about $\lim_{n\to\infty}\lvert2^n-n\rvert_2$? if this tended to zero then we would have an alternative metric of set size which has many desirable properties, and in which the continuum hypothesis is true.
According to my limited capacity to complete the analogy, should we use this analogous metric for sizes of sets, it would appear to value the size of the power set of the integers as $0$ and therefore in this theory the space between $2^{\aleph_0}$ and ${\aleph_0}$ can accommodate a set the exact size of $\aleph_0$.
It tells us absolutely nothing. CH is about cardinal arithmetic. No field arithmetic is compatible with cardinal arithmetic, so there is nothing it can say about it.
Might as well endow the natural numbers with another operation denoted by $2^n$ defined as $2^n=n$ for all $n$, and argue that Cantor's theorem is false because in this structure $2^n=n$.
The point is that the analogy about "metric on sets" is wrong, so there is no point in continuing it any further, which would make it "even more wrong". There is no "metric on sets", and certainly not one which is defined by size.
Set theory, as a theory (from a holistic point of view), might be missing some things, but it most certainly does not miss a metric based on size differences and powers of $2$. That is a myopic view that misses the entire big picture, and most of the small details.
So by hamfisting this analogy in the first place, you get a weird notion that there should be a notion of Cauchy sequences on sets in this metric. Let me postulate, then, the obvious question. Why should this be a metric, and not a valuation which considers even longer sequences? Why not sequences of cofinality $\omega_1$? Why not all type of sequences?
Metric talks about "distance", and so what does it even mean that two sets are close or distant? Are sets of cardinality $500$ and $510$ close? What is the distance between them? Not to mention that since cardinality subtraction is very much undefined, this leads to ultimately odd questions. What is the distance between $\aleph_1$ and $\aleph_0$? What is the distance between $\aleph_{42}$ and $42$? Why don't you just end up with a discrete metric?
The analogy falls apart once you try to ask for details which are not cherry picked for this $2$-adic approach.