I have started working through the thread picking out some bits which seem important to me, and adding some comments relevant to my understanding. Lack of time has prevented me from getting very far, but I would really appreciate some comments.

Well then find the radius of the circle whose circumference is infinite.

This is a mathematical question, so mathematical infinities would be appropriate. If the circumference is infinite, then the radius is also infinite, but they must be different sized infinities, and different infinities are acceptable in mathematics. We cannot assign any finite value to either radius or circumference, effectively, infinite means so large that we might as well consider it infinite.

The point here is that any system that can normally be considered as bounded cannot include an infinite component. So if the mathematics of a formerly finite system go infinite something is terribly wrong.

The first sentence is true if it refers to what we might need to call “absolute” infinity. If we are using mathematical infinities, then it can be argued that, for example, there is an infinite number of rational numbers between any two integers. However, Zeno’s paradox leads us to suspect that that this is a “fact” only in principle.

The second sentence must be true, as any finite thing would approach infinity infinitely, thus it would never “arrive”.

Let me make this very clear first; {infinity} is not a number.

As far as I can tell, everyone in this thread agrees with this statement.

There is an infinite number of integers 1,2,3,.... because we can always add one more

There are rational numbers between the integers 1, 3/2, 7/4, 2, 9/4, 19/8, 3....

Indeed there is an infinite number of rational numbers between any two integers

So the number of rational numbers must be greater than the number of integers

As so often happens when talking to scientists about infinity, we return here to mathematical infinities. Although these are boundless, their “infinite” nature can never be physically demonstrated, because it is not possible to count to infinity, nor to enumerate all the rational numbers between two integers.

IMHO if ever you bump into an infinity in physics, then something is wrong somewhere.

Nonsense. It's beginning to become clear that the universe is flat and boundless and as such goes on forever, never ending. That's what it means to be infinite.

Infinite answers are certainly not right, so they are a sign that your theory is not very good. A theory needs to fit the data, but it also needs to make mathematical sense.

Do we lack reasonable consensus between scientists as to exactly what is meant by “infinity” in individual cases? It seems so, but that presents a problem for non-experts. How does one choose whom to believe?

…… there can never be an infinite amount of distance between any two particles as that would place a boundary on infinity.

That makes perfect sense to me, as long as we accept that mathematicians will always be able to fit mathematical infinities into finite spaces.