So the idea that a "set" is a thing in and of itself such that it can even be empty means that a "set" is more than the things in the "set" collectively considered. Without this concept of an "empty set", if we just considered a set a collection of things, what would math be missing and would calculus and other such things still hold?
Light and how it interacts with stuff seems very fortuitous to me. It just so happens to pass through the atmosphere rather than bouncing off of it or being absorbed by it thus lighting up everything we can see. My question is given the multitude of particles in the atmosphere why does enough light pass through to see instead of just bouncing off of the air molecules and going back into space? Or even getting absorbed by the molecules? To clarify the kind of answer I'm looking for what is the quantum mechanical picture whereby photons are passing through instead of interacting with the air molecules?
edit: what I'm looking for is something similar to how Feynman talked in QED, for example "the atmosphere's translucency is something like 90% whereby a photon only has a 10% chance to reflect or be absorbed. Thus 90% of photons pass through." something along those lines
I'm no mathematician so maybe I misunderstand but it seems to me like something in zfc might be arbitrary. I think I understand the concept of a set, where the quantity of 5 is a set of 5 thus numbers are sets. However, let's take the idea of an empty set.
Now my understanding of what an empty set is, is a box of chocolates w/o any chocolate. It's purely a mental overlay of reality when we say the box is an empty set. But the question is does nature deal in empty sets outside of the one's invented by our minds?
It seems to me that if mathematics may be said to exist in some capacity, such as if math is merely the laws or rules of existence, that it would not be meaningful to have an "empty set". As that's saying there is something ontologically more to a set than it being the collection of things in a set. In one instance your saying a set is a thing in and of itself, in the other "set" just refers to the things collectively considered such that an absence of the things leaves you with no set rather than something that's empty.
This "something" that is called a "set" such that it can even be empty seems like something that has no ontological reality and things that have no ontological reality can't be said to exist.
I guess the question is if mathematics exists mind independently can an empty set actually exist also or is it merely invention and if so how can the concept be said to be a "foundation" of math? Thoughts?