Question for WetWired regarding omega

WetWired, in an earlier thread you stated that omega was the largest integer. I didn't think such a quantity existed. The reason I bring this up now is because of a question in one of my classes.

If we have a potential vector space with addition of two vectors defined as x+y = max(x, y) then according to my understanding of the number system this would not constitute a vector space. The reason being is that there is no singular 0 vector for the entire vector space, meaning there is no value of y I can think of where max(x,y) will always give x because y can always be x-k. However, if a number such as omega exists, then negative omega should satisfy the conditions and constitute a 0 vector. I was just wondering if you could clarify, since I was always taught that the idea of the "largest integer" is non-sensical.

It was something I read in an issue of Discover many many years ago. They were talking about a system of representing numbers that could represent both finite and infinite numbers. The system was composed entirely of up and down arrows and a means of indicating infinite repeating. They stated that a single infinite up is omega, the largest integer, and that an up followed by a single infinite down is iota, its reciprocol. The system could actually count beyond these.

http://discovermagazine.com/1995/dec...typlusonea599/

Edit:

The notation is messed up in the online version...

Thanks a lot!