See: 0:00 to 2:12 – Sean Carroll explains Hilbert Space – a mathematical concept which describes “any finite or infinite numbers of dimensions”, which is used in quantum mechanics and within various other subjects.
See: 3:10 to end of video – Lex Fridman asks Sean whether he thinks quantum mechanical systems are infinite – Lex also expresses some of his own thoughts about the idea of infinity.
Mathematical models which utilise infinity must either be using it semantically to represent something other than a numerical value, or any such model is strictly speaking in error, simply because there cannot exist an infinite number of anything – rumours are going around claiming that infinity in mathematical models can be replaced with extremely large values... which is good to hear.
Comparing being infinite to being extremely large is an invalid comparison, because there is no size difference between any finite value and infinity.
The reason for there not being an actual size difference between any finite value and infinity, is simply that infinity does not refer to a specific size – the very reason it is a misconception.
In other words: there cannot exist a difference between something which is defined and something which is undefined, other than the obvious – one is defined, and the other is not.
Contradiction: “the total number of all numbers” (assumed to be infinite) – the idea of infinity cannot exist as a totality – hence, it is a misconception.
It's very easy to confuse potential infinity (a misnomer) with actual infinity.
An “Infinite” value is supposed to (must) refer to a state of existence, i.e., not a finite value that is increasing or decreasing indefinitely.
The meaning of “take an infinite amount of time” is oxymoronic, i.e., infinity is not a value, therefore it cannot be counted to – infinity refers to states of existence that are supposed to exist theoretically (i.e., can be truly represented numerically) and/or physically.
Notation such as “...” does not represent infinity.
Calculus, such as the velocity example explained in: Calculus & Infinity Paradoxes does not represent infinity.
Infinity does not exist – it is a misconception.
The relationship between 0, 1, and infinity, can be considered as follows:
0 Represents 0 elements.
1 Represents 1, or any other integer by adding 1s together.
0 and 1 can be used in some other data system, such as binary.
Infinity = 1 / 0 = division by 0 error (undefined), which does indeed correctly represent infinity – a misconception.
Infinity is not a number and therefore it cannot represent a number – applying infinity to reality results in paradoxical scenarios.
It is impossible for two objects to exist in separate locations to one another that are both infinitely far away.
Any numerical value which is added to infinity becomes insignificant, to the extent that it is completely erased.
Likewise, any value that is divided by infinity becomes paradoxically spread out, to the extent that it does not exist anymore.
In fact, anything that relies on quantitative existence is erased by infinity, such as density – although infinite density is physically impossible of course.
Example: an infinite amount of matter cannot exist within the totality of spacetime – in fact, an infinite amount of matter cannot even exist by numerical value, i.e., to exist it must be part of a totality, regardless of being restricted by some type of container; spacetime – therefore an infinite quantity of matter would mean an infinite density – an impossibility of course.
See: Straight-line distance infinity analysis ... for an explanation of why distance must be finite.
In other words: the finitude of distance itself is effectively a container.