Infinity was the topic of a book by John D. Barrow published in 2005. He wrote of the theologian Nicholas of Cusa’s work on the infinite circle (in radius) amongst other topics and the way Nicholas described any given section of a circle infinite in radius as being indistinguishable from an infinite line segment (straight).
As the infinite circle expands to infinity in size (such an infinite circle ought to be incapable of comparison to other infinite shaped objects (like Euclidean solids) theoretically, for it would then need to be of a given finite size) any local area seems nearly without curvature.
I wanted to remark about a more simple and evident aspect of the infinite circle. Earlier Barrow pointed out Aristotle’s work on potential and actual infinities-especially in mathematical series of numbers. The difference between a potential and an actual infinite circle is more of a paradox than the largest number known to which one may add one more forever (N+1).
A circle of any size must be complete in order to exist. I believe I must be referring to a 2 dimensional circle for perhaps no other may exist (The circular characteristic of a hula hoop is meaningful in just two dimensions of a three dimensional object).
At any rate the infinite circle to actually exist would then no longer be infinite in size, but finite. An infinite circle could exist only in potential, yet a potential circle cannot exist, or does not exist except perhaps as a continually increasing circle expanding like an infinite universe regarded as a spherical construction of an infinite number of simultaneously expanding, contiguous circles. A hyper-inflating universe expanding with space-time might be exemplary.
An infinite circle could never exist as a finite actual circle or even as an increasing in size circle expanding forever from a singularity of infinitely small size without being compared with co-existent additional infinite circles of a larger temporal size making the primary infinite expanding circle a smaller infinity at any given point in the process. Comparison of infinities seems bound to happen if they are actualized within a temporal progression. A forever-increasing-in-size infinite circle would be finite at any given point in time.
Perhaps Bertrand Russell would have regarded the infinite circle as a categorical problem. It is an abstract thought and perhaps without meaning akin to a square circle.
The transition from thought constructions to real world possibilities for existence would seem to require consideration of how an infinite circle might be constructed given the quantum packets or material available in the Universe. It seems unlikely that an infinite circle could ever be made to exist, yet string theorists conjecture that universe sized strings left over from the start of the Universe might exist and be observed one day (I don’t know much about that topic). Perhaps they inflated unbroken expanding with space-time from less than a Planck length in size to the size of the Universe today.
The nature of infinity philosophically considered does appear to support conjecture about the Universe even in light of contemporary cosmology and concepts of God. One may get the notion that God would need to surpass Gödel’s incompleteness theorem criterion. That is God would need to contain all infinities as part of His nature. Containing temporality is a similarly challenging problem; how can an infinite temporality be contained and foreknown by a greater than temporality God? Barrow quotes Augustine on the matter of infinities.
These classic theological and philosophical issues re-examined with the tools of a mathematical astrophysicist are interesting reading.