Is 1 + infinity > infinity? [closed]

The short reason that your debate partner's argument is invalid is that $1+x>x$ is only true in certain contexts— in other words, you can't just substitute anything for $x$ and expect it to be true, or even meaningful. Such subtleties don't usually bother us because, for instance, this inequality holds for all $x$ that come from an ordered field (such as the real numbers $\Bbb{R}$), but infinity as it is usually understood cannot exist in an ordered field.

However, there is some truth to what they are saying. For instance, the equation $x+1>x$ is true in ordinal arithmetic. (Amusingly, the equation $1+x>x$ is not true using the standard definition of $+$ for ordinals; such is the weirdness that arises when we try to make infinite things precise.)

Like many paradoxes in mathematics at this level, this one arises because we assume that we can use our informal understanding of objects (in this case, $+$, $>$, and $\infty$). Once one formally defines what one means, these problems tend to go away. Thus the question becomes what sort of definitions one should accept, but this is often not in the scope of mathematics.