Before you can understand the physics of mathematical infinity properly you will need to understand the role of local sovereignty in the act of counting. The fundamental structure of infinity is not defined by layers and layers of division-of-divisions which grow larger in quantity but smaller in substance (for example, counting the number of atoms in the universe leads to unfathomably huge numbers, while the size of each unit is equally unfathomably small, and counting the quarks makes the gap even larger, etc.). Instead, it is defined at a primordial level by the requirements needed to sustain Local Sovereignty at all stages and levels of the development of the physical universe.
"Local sovereignty" is at first a curious term, which may seem to those new to the idea to be an old-fashioned political, economic, or legal concept, roughly translating to something akin to "every man is a king" known early in the American experiment when the idea flourished as a popular political innovation.
However, it has a meaning within physics which goes to the finest structures representing the tiniest infinitismal divisions at the deepest sub-quantum layer of the physical universe -- and is equally present in the opposite direction, outward to the largest structure, the whole universe itself. It is an idea about a structural division of intelligence which is necessarily entangled with the idea of division itself. It is an idea which ontologically precedes the nature of physical matter, and epistomologically precedes counting.
Materialists would prefer that the idea of intelligence, or consciousness, be utterly separated from the tiniest structures within physics, but that takes Occam's razor one step too far -- dividing a subject of inquiry past the point of meaning, and thereby leaving an enormous question mark about the relationship between consciousness and matter that they cannot answer and must therefore avoid, deny, and circumlocute around.
This approach establishes at the foundation of positivist inquiry an intellectual dishonesty that then permeates the self-centered presumption that theirs is the only sensible interpretation. This is a zero-sum approach which falls apart when compared with the more tenable presumption that their approach is only one of many, and like all similar presumptions, has limits, about which it ought to be honest, in order to allow for a greater syncretic understanding which is able to hold all such presumptions in a single, blended, insight greater than the sum of its pieces.
Local sovereignty is why the physically-smallest structure of the universe has within it the ability to rule its own domain -- whatever it be, however it will -- in its place within the larger picture. And so does every other structure. In other words, "freedom is embedded in the fabric of physics." We tend to think of something as sophisticated as sovereignty to begin operating only at a high level of sentience in intelligent beings, so to consider that this is embedded in the finest structure of the physical universe implies big things immediately: First, that sovereignty operates in the range at or below the simplest Planck scale all the way out to the largest and most complex structure possible, permeating everything everywhere. Second is that what we know about all this from here where we sit, roughly in the middle of that enormous spectrum, is only a very little so far.
The sea of eyes is the ultimate infinity
To say it more poetically: The sea of eyes is the ultimate infinity in terms of sheer numeric countable quantity, but its size is uncountable, both because this sea ontologically precedes counting, and because, physically, the "eyes" move; coming and going faster than we can count. The motion within every eye is happening independently of each other eye (locally sovereign) -- but also responding intimately to each other eye (globally united). How this can happen is similar, conceptually, to the scale-free correlation which happens when a flock of birds swirls and sweeps through the sky at sunset, or schools of fish, or the leaves of trees in a forest, and other large movements or structures in nature comprised of smaller components working in intimate harmony with each other on a scale discernible at least one layer of abstraction higher.
Each infinitismal "eye" has within it a direct, unbroken, unseparated, undivided, contiguous connection to the entire singularity form of the universe -- both inward toward a single unity which holds all, and outward, toward ever more division until the art of abstraction begins to unite everything again. To divide the global connection from the individual cell is "to divide too far."
Both directions, inward and outward, terminate with the single, unified, truth that the universe is, was, and ever will be undivisible, and therefore uncountable except in a rough approximation sort of way. The existence of the universe is in all respects more fundamental than the ability to count -- which itself is an artifact of local division, not of global unarity. Because a good portion of our intelligence operates from within division, we think the two forms of unity (the universe as a whole, and the underlying singularity at sub-quantum level) are separated, but look at the simple linear diagram below understanding that both unities have the same referent:
unity -- division -- unity
This is how we see things, from within division. But from within unity, the matter is more spherical, less linear, and the essential two-ness of division, which is embedded in our way of seeing, is seen from both ends simultaneously, and a linear way of representing it just won't capture the idea, but here is one attempt:
Motion is outside the domain of mathematics
Mathematics measures and studies and models upon things which may be in motion, but is itself not in motion.
Mathematics is an abstraction, which has the ability to be held within the mind as a way of measuring or comparing patterns of information across an infinity of possible orthogonal dimensions at will. By this is meant that the mind can instantaneously create a measurement across any structure or representation held within it, and then begin "counting" that measurement, developing mathematical sets and categories and types across boundaries which literally do not exist in the world of physics, which itself operates upon a foundation that is another sort of sea, a sea of highly energetic sub-quantum foam which arises and descends out of "nowhere" into and out of three-dimensional space which is never completely empty nor completely full, but also always in motion, with nothing perfectly still for long anywhere ever.
Ontologically, this means that motion precedes volume, or units of timespace, and volume precedes measurement across volumes.
Therefore what we call static, or "unmoving," is imaginary -- not real -- an abstraction which cannot be ab initio, from the beginning, since what is primordial, the quantum sea, is measureably and intrinsically intensely in motion, by all accounts, and measurement itself is just as equally intrinsically static. Because mathematics is an abstraction above measuring (arising from division), it is a tertiary level of abstraction away from physical reality, and thus it is only in mathematics that a static idea, or the concept of perfect stillness, unchanging, unmoving, exists. [note, Longo talks about the certainty of mathematics, and how it provides more certainty than any other field, and this unchanging aspect of math is where that certainty derives from.]
[original draft of these ideas Dec 6, 2020, updated January, Feb, and March 2021, then edited and expanded for publishing on blog May 13, 2021]