Ok so I'm arguing with this one person that says that $1 + \infty > \infty$ is true, and I disagree.
But I can't disprove their points.
My argument is that if $1 + \infty > \infty$ then there exists a number greater than $\infty$, disproving the concept of infinity, because you can't simply add $1$ to infinity, because infinity is ever increasing.
So new_infinity would just become "1 + infinity".
They argue that you can just substitute in $x$ for infinity and have the statement $1 + x > x$ which is true (but I don't think you can substitute a variable in for infinity).
I asked my math professor about this question and he said $1 + \infty > \infty$ is false, but I don't really remember the explanation.
Could someone explain it in layman's terms (and maybe i misheard my professor and it is true idk at this point)?
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.