An intuition is that an inverse of an infinite quantity is an infinitesimal, but can there be other approaches?
Can there be a field or integral domain with infinitely large quantities but without infinitesimals?
Asked
Active
Viewed 54 times
0
-
1How do you define infinitely large quantity and how do you define infinitesimal? – José Carlos Santos Dec 30 '19 at 10:57
-
2I imagine the spirit of the question is to consider various definitions. – Ciarán Ó Raghaillaigh Dec 30 '19 at 11:00
1 Answers
3
Consider $\mathbb{Z}[x]$. It is an intergral domain and you can put an ordering on it so that $x$ is bigger than any integer.
Fields won't work if you want an ordering compatible with field operations so you would need to think of a different notion of big. Maybe some comparison to the characteristic field? Maybe the size of an element is the degree of the polynomial which it solves with transendentals being infinitely big.
Ciarán Ó Raghaillaigh
- 1,424
-
"Fields won't work if you want an ordering compatible with field operations" - what about an integral domain? – Anixx Dec 30 '19 at 11:11
-
Also, what about if inverses of infinite quantities are not ordered (ordering is only defined for reals and infinite quantities)? – Anixx Dec 30 '19 at 11:12
-
1The integral domain I gave is an example where the ordering is compatible with ring operations. – Ciarán Ó Raghaillaigh Dec 30 '19 at 12:26
-
@Anixx See the last paragraph here for more on this ring of (germs of) polynomials at $\infty$. To learn more search on Hardy fields, which are used in algebraic asymptotic analysis. – Bill Dubuque Dec 30 '19 at 21:02
-
-
@Anixx Yes, but they have subrings without such- which is what you are interested in. – Bill Dubuque Dec 31 '19 at 08:48
-
@BillDubuque actually I am interested in how to build a field or integral domain with infinite quantities but without infinitesimals. – Anixx Dec 31 '19 at 11:40
-