To prove (2,2) is an interval in R

Question

The definition of an interval in an ordered field is a set that always contains the entire closed interval between any two of its elements. How do we prove (2,2) is an interval which is an empty set by the above definition of interval? As it does not contain any element how do I show that it has a closed interval[a,b] whenever a,b belong to (2,2)?

Answer 1

Here's an interesting fact that you might not be aware of. There's a set $S$ of natural numbers such that every single element in $S$ is even (divisible by $2$) and every single element in $S$ is odd (not divisible by $2$). Surprising? This set actually goes by a different name (symbol?) usually: $\varnothing$, the empty set.

As it turns out, the empty set fulfills all postulates -- all are "vacuously true" for the empty set. More precisely, for every element in the empty set, any postulate $P$ holds, because there's nothing to check! The empty set is an interval in $\mathbb{R}$, but it's also a closed set in $\mathbb{R}$ and an open set in $\mathbb{R}$ and everything else you could think of!

So what you need to prove is that $(2,2)$ is really just the empty set. This isn't too hard: suppose $x \in (2,2)$. Then $2 < x < 2$, but any strict total ordering (like $<$ on $\mathbb{R}$) is transitive, so $2 < 2$. But any strict total ordering is also irreflexive, so this cannot happen: a contradiction. Thus no such $x$ exists.

Does this make sense?

Answer 2

Remember: if FALSE then anything is always a true statement.

"If $2 + 2 = 5=$ then dragons are living in my garage" is a true statement. It's true because the hypothesis is always false and there can not be any counterexample where $2+2=5$ but dragons are not in my garage.

"The moon is a green cat then I like beer, is a true statement.

And

"If $a,b\in (2,2)$ then a purple dinosaur will win the noble prize is 2027" is a true statement.

And

"If $a,b \in (2,2)$ then $[a,b]\subset (2,2)$" is true.

These are vacuously true statements. And although it seems like I'm being a smartass they are logistically sound.

.... or ....

you can think of it this way.

For $(2,2)$ to NOT be a segment there must be some $a,b \in (2,2)$ with $a < b$ where $[a,b] \not \subset (2,2)$.

But there aren't any such $a,b \in (2,2)$ because there aren't any $a,b\in (2,2)$.

So there are not any $a,b\in (2,2)$ where $[a,b]\not \subset (2,2)$.

So for every $a,b; a< b \in (2,2)$ (all zero of them) we do have $[a,b]\subset (2,2)$.

So by the definition, $(2,2)$ is an interval.

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"As it does not contain any element" BECAUSE it does not contain any elements, everything is vacuously true for every element BECAUSE there are none.