# Is there any real difference between 'unbounded' and 'bounded by infinity'?

When treated as a limit, 'infinity' is essentially synonymous with 'never' - saying that 'a process terminates after an infinite amount of time' means exactly the same thing as 'the process does not terminate'.

So is there a diffierence between 'unbounded' and 'bounded by infinity'?

Or, more generally, if $$S\subset X$$ is a partially ordered set and $$\exists x\in X\setminus S:\forall s\in S.s\ll x$$ ($$x$$ is infinitely greater than $$s$$, or $$x\notin\text{gal}(s)$$, or $$\nexists t\in S:\Vert s-x\Vert=\Vert t\Vert$$, etc...)$$^1$$ then is '$$S$$ bounded by $$x$$' equivalent to '$$S$$ not bounded'?

Ex. $$\mathbb{C}\subset\mathbb{C}\cup\{\infty\}$$ and $$\forall z\in\mathbb{C}.z\ll\infty$$, so $$\mathbb{C}$$ is bounded by $$\infty$$

$$^1$$ It is assumed that $$x$$ is infinite.

$$\text{gal}(s)$$ is the galaxy of $$s$$ from nonstandard analysis. The galaxy of an element $$s\in S$$ is the subset of elements of $$S$$ that differ from $$s$$ by a finite amount - e.g. $$\text{gal}(0)=\mathbb{R}$$

Note that $$\nexists t\in S:\Vert s-x\Vert=\Vert t\Vert$$ does not imply that $$S$$ is not bounded; however if $$S$$ is not bounded, then $$\nexists t\in S:\Vert s-x\Vert=\Vert t\Vert$$.

• I could not understand the loaded notation, although I did not vote to close. I have never heard the phrase "bounded by infinity" but I often write "assume $E[|X|]<\infty$" to mean $E[|X|]$ is a (finite) real number, and I use notation "$5 \leq x <\infty$" to mean that $x$ is in the interval $[5, \infty)$. – Michael Apr 9 at 14:51
• @Michael Thanks, which notation are you referring to? – R. Burton Apr 9 at 14:53
• $\overline{S}$, $gal(\cdot)$, $||S||$, "immeasurably greater," etc. – Michael Apr 9 at 14:54
• @Michael Corrected. I should really try to be less lazy with my notation. $gal(s)$ is galaxy of $s$ from nonstandard analysis. The galaxy of an element $s$ is the set of elements that differ from $s$ by a finite amount - e.g. $\text{gal}(0)=\mathbb{R}$ – R. Burton Apr 9 at 15:05
• In the context of a partially ordered set such as $X$, it makes no sense to say "$x \in X$ is infinite", and it makes no sense to subtract two elements of $X$. You cannot simply important operations of numbers into set theory in this fashion. – Lee Mosher Apr 9 at 15:27