![]() In Combinatorial Game Theory, there are infinitesimals like " up" that don't reside in a field, but that's a pretty niche area/application. But these are not often useful for analysis purposes. Now, you can change the arithmetic on the ordinals to get the surreal numbers, or look at other non-archimedean fields, perhaps in a more general/abstract way. And for the others, an infinitesimal would break the (Dedekind) completeness property of the reals that is critical for usual analysis to work. But none of these contexts directly lend themselves to an infinitesimal.įor ordinals and cardinals, we don't even have something positive but less than $1$. And if we broaden out view to complex analysis, the Riemann sphere is fundamental and has a point labeled $\infty$. And $\pm\infty$ in the extended reals help to give a tidy account of limits and measure. ![]() Ordinals were discovered when Cantor was working on real analysis, and cardinals (especially the countable-uncountable distinction) are often useful when dealing with infinite sets, both in and outside of analysis. Infinitesimals don't arise in common contexts This has knock-on effects for, say, how math curricula are designed in universities, the level of general awareness of mathematicians which affects their ability to spread ideas, etc. I think a large portion of the reasons come down to the fact that there are more contexts in which "infinities" would be useful than "infinitesimals". This is a tough question that's hard to answer definitively because of the different "infinities" (for an overview, see Understanding infinity), the history and popularity of different branches of math involved, etc.
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