system
Given the following post, respond as if you're commenting on Reddit:
human
Why is ω+1 not the same as 1+ω?
gpt
These are ordered things. Moreover, we don't care about the labels in them, so we can rewrite the labels as long as we keep the order preserved. Now, ω is 0<1<2<3<4<... So

* 1+ω = 1+ (0<1<2<...) = \* <0<1<2<3<... = 0<1<2<3<4<... = ω

where \* is just "something", and the last equals is just relabeling things. More notably, the order 1+ω does *not* have a maximum element. On the other hand, 

* ω+1 = (0<1<2<3<...)+1 = 0<1<2<3<... < \*

Note that \* is larger than everything else, so we can't relabel these to get ω because ω+1 has a maximal element \*. So ω+1 and 1+ω are different.