Is this a proof that the impossible can happen?

Yes there's a lot of fault in your reasoning. First you talk about the odds that it hits any one of the squares. Then you talk about the probability that it hit that 1 particular square....Both have different probability because the first case doesn't care which square it hits on, and then you begin to focus your attention to the dart hitting one particular square for the second case.

Plus I don't think probability deals with these infinite values, at least not the way you put it where you focus on 1/infinite number of squares. Infinity is not even a number like 1,2,3... are anyways. It's the idea that a value never stops increasing. So lets assume that there are 1 trillion square which seem like infinite amount when you look at it. The probability that it lands on the square you chose is still 0, but not a definite 0. Its more like 0.00000000001 or something like that, but realistically, it feels like a definite 0%.

We never run into a wall with infinite values, so its not very practical to calculate odds under those circumstance

alright so i in no way know any philosophy but i'm an applied mathematics, engineering & physics major so i could try using some of that knowledge. ok so if we start with the topic of theory of probability the only things that are considered truly random are certain areas of particle behavior as well as the behavior of people. randomness isn't an actual thing. random number generators on your computer are really just math equations who's input is really just really small increments of whatever the time is. ok now lets apply some physics to explain why this can't be random. depending on the mass, force, gravity, ect, you can determine where its going to hit. BUT if we use the math we know that lim 1 -> infintiy of 1/x (as x gets larger and larger), it can be determined that it will equal 0. using all these different ideas you can adjust your answer to not saying its impossible but its infinite which honestly is pretty close to the same thing. to your other question, no i don't believe thats related at all. this is just a trippy question, doesn't have to do with the universe (at least i think so). now you can technically say that the big bang was impossible (still happened). before the big bang everything was in a sphere that was infinitely small, with infinite temperature, energy, and mass. everything about the big bang defies physics, physics didn't exist. honestly so far no one knows what everything exists, one reason being that you can always ask well what happened before that, what happened before the big bang. idk if i'm answering any of your questions but heres something a little crazy. if the universe is infinitely large are your a finite size (going back to 1/x) do you actually exist? another one is time is never ending, you'll only live for what 80 years, can you say that you ever were alive by that idea

It may seem counter-intuitive, but an event having a probability = 0 of occurring does not automatically imply that the event cannot occur.

For example, if you use a true random number generator to generate a real number between 0 and 1, then the probability of any specific number being generated is zero (because there are an infinite number of possibilities, same as the argument you are making). However, a specific number is in fact generated when you do this, meaning that an event with zero probability of occurring occurs.

Two ways I can answer I'll go with my personal favorite first why do you assume human created math is perfect humans are but a baby in this universe and yet you and everyone else assumes that human math is flawless or always correct ever thought of the possibility that human created math is wrong at its core all it takes is one misplaced mistake to be accepted as truth and built upon to let an entire empire fall now the other one if the squares are all around you which if they are all infinite have to be at your feet at some point and if they are all together with no space then it would have to hit a square no matter how or where you threw the dart

Probability will narrow down which square is most likely to be hit based on several factors such as the trajectory of the dart, opposite force, gravitation etc. Becasue of this, the dart cannot reach an infinite amount of targets.

This is the scientific approach.

If you take a more philosophical approach to the question, then I suppose you can surmise that a nearly infinite amount of factors and formulas can be concluded to yeild an infinite result.

Yes, there is a fault in your reasoning. The probability of the dart landing in a particular square is 0. But it has landed in one of those squares. So an event with 0 probability has just happened. But this does not mean that an impossible event has happened. Impossible events have zero probability but an event with 0 probability is not necessarily impossible.

Suppose you ask me to choose any real number and then you guess it. Since there are an infinite number of real numbers. Your probability of choosing my number is 0. Suppose you guess 2. It is still possible that i couls have chosen 2. Even with 0 probability its not impossible. You can try googling it.

In real number system, infinitessimals are not allowed, so probability of 1/infinity=0 in real number system. But i guess that in some number systems 1/ infinity is equal to an infinitessimally small number. Your probability of gusseing my number is infinitessimally small. But it is >0 , so it can happen.

The flaw is in the concept of infinity itself. Infinity is counter-logical. It generates all kinds of these paradoxes.

That's a definite maybe.

By definition, the impossible is NOT POSSIBLE! Cannot happen!

If it was so labeled prematurely, like airplanes and flying, it was never actually impossible.

If it is truly impossible, there it ends.