To add to the confusion, there are normally only two symbols for infinity to go around - ω and ∞
This post is the first I recall seeing ω used to refer to a cardinal. (The smallest infinite cardinal is usually called א₀.)
And I think hyperreals are probably a much less common topic than the other three, so at least we have different notation for the top three of these four notions (ω for the ordinal, א₀ for the cardinal, ∞ for the limit).
I’m not confident in this, but also: It looks to me like ω can be considered the same number in the ordinals, surreals, and hyperreals (since the surreals contain both of the other two). So I guess that’s probably why we use the same notation, and why it’s not really a serious ambiguity.
This post is the first I recall seeing ω used to refer to a cardinal. (The smallest infinite cardinal is usually called א₀.)
And I think hyperreals are probably a much less common topic than the other three, so at least we have different notation for the top three of these four notions (ω for the ordinal, א₀ for the cardinal, ∞ for the limit).
I’m not confident in this, but also: It looks to me like ω can be considered the same number in the ordinals, surreals, and hyperreals (since the surreals contain both of the other two). So I guess that’s probably why we use the same notation, and why it’s not really a serious ambiguity.