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Given all the arguments posted I'm not very convinced that the statement holds for all possible A,B,C.

@levifinkelstein nice

@levifinkelstein I'm surprised that you are not convinced 😂 since you said "proof" I think that most people were betting about a mathematical proof convincing you, and multiple proofs were given. Can you point out why those are wrong? I guess this is a lesson to not bet on markets where the resolution depends on the user being convinced of a fact...

@egroj It's perfectly possible to formally define a more natural meaning of the word "inside", in the way humans tend to use it. It's just much harder, since it's a more complicated concept.

@egroj It seemed like the just assumed insideness would be transitive. I was referencing "inside" in a colloquial sense not in the set-theoretic sense, so any argument would have to satisfy my intuitive understanding of insideness, which I'm now convinced is not transitive in all instances.

Given that the title didn't say anything about using a set-theoretical definition of the term, and the clear ambiguity of the English term has been demonstrated in multiple different ways, I trust you'll make the correct decision.

I will resolve the market when I wake up tomorrow as I will most likely be asleep when it closes.

Okay guys, after having read all the arguments I must say I currently find one side a lot more convincing.

Not sure you’ll adopt a definition for inside-ness that’ll allow one of those general proofs resulting in that it’s transitive, so I won’t try to point-free topoi you. 😎 Moreso that I’ll need either to read up or invent along that way.

@degtorad I'll try my best to have an open mind if you wanna give it a shot 😁

@levifinkelstein Honestly though I’m not sure it could end up being essentially different from pointful arguments, either subobject-like (A ⊂ B ⊂ C) or topologic-y.

Lesotho is inside South Africa. Maseru is inside Lesotho. Maseru is not inside South Africa.

@IsaacKing This is very thought provoking, thank you.

@levifinkelstein Maseru might not be IN South Africa, but it looks INSIDE South Africa when I look at the map

Set.prototype.isInside = function(set) {

return set.has(this);

}

const A = new Set;

const B = new Set;

const C = new Set;

A.add(B);

B.add(C);

console.log(C.isInside(B)) //true

console.log(B.isInside(A)) //true

console.log(C.isInside(A)) //false

@IsaacKing This is interesting, I will take this into consideration.

You have to define what you mean by being inside. If we are talking about sets, your set C is inside of A (or is a subset) if for every element that is in C that element is in A.

If C is empty, then C is in any set including A. The proof is by contradiction: if C is not in A then there must exist an element x that is in C but not in A. But there cannot exist such an element because C is empty and thus doesn't have any elements.

So let's suppose C is not empty. We have to prove that for every element x in C, x is in A.

Let x be in C. Our definition of being inside says that every element of C must be in B, and since x is in C, x must be in B. And since x is in B and every element of B is in A because B is inside of A, then x is in A. And the proof is done, we picked an arbitrary element in C and showed that it is in A, thus every element of C is in A and that is the definition of C being inside of A.

@egroj I hope that I'm not doing your homework 😄

@egroj Thanks, this is helpful

@egroj This would be pretty lame homework anyways.

Let's define inside. Intuitively, A is inside of B of every point inside A is a point inside B.

If every point inside A is a point inside B, and every point inside B is a point inside C, then every point inside A is a point inside C.

@EdwardKmett Maybe you could elaborate on why this is intuitive to you such that I might have a greater chance of understanding it assuming it makes sense.

@levifinkelstein By a dictionary I have on hand: inside refers to "situated with the boundaries or confines of." That is about the limit of work I'm willing to put in for Ṁ1. I wish you luck translating betting markets into proof labor.

@EdwardKmett Thanks for contributing.

That is definitely one potentially useful definition. Another useful definition would be "if any point inside A is a point inside B". Another would be "if the majority of A's volume is inside B".

To give a real-world case study on how definitions can vary: if I'm standing inside a house, and I reach a hand outside a window, I would expect that some people will say I'm still inside the house, and others will say I'm not.

This will absolutely depend on the definition of "inside". There's already sufficient evidence in these comments to show that there exists a definition of "inside" for which the answer is YES, and to show that there exists a definition of "inside" for which the answer is "no".

It is therefore definitely not the case that for every definition of "inside" the answer is YES.

@josh Hmm, yes this is fascinating, I will consider this.