On the 5-point scale current in Estonia, and surely in nearby nations, and familiar to observers of the academic arrangements of the late, unlamented, Union of Soviet Socialist Republics (applying the easy and lax standards Kmo deploys in his grubby imaginary "Aleksandr Stepanovitsh Popovi nimeline sangarliku raadio instituut" (the "Alexandr Stepanovitch Popov Institute of Heroic Radio") and his grubby imaginary "Nikolai Ivanovitsh Lobatshevski nimeline sotsalitsliku matemaatika instituut" (the "Nicolai Ivanovich Lobachevsky Institute of Socialist Mathematics") - where, on the lax and easy grading philosophy of the twin Institutes, 1/5 is "epic fail", 2/5 is "failure not so disastrous as to be epic", 3'5 is "mediocre pass", 4.5 is "good", and 5/5 is "excellent"): 4/5. Justification: Kmo had time to do a reasonably complete and (within the framework of the version 1.0.1, 1.0.2, .. process) reasonably polished job.
- UTC=20160830T0233Z/version 2.0.0: Kmo managed to get rid of the point-form outline and to a proper upload. He reserved the right to do tiny cosmetic, nonsubstantive, tweaks over the coming 48 hours, as here-undocumented versions 2.0.1, 2.0.2, 2.0.3, ... .
- UTC=20160830T0003Z/version 1.0.0: Kmo lacked time to be thorough, and so uploaded mere point-form outline. He resolved to get this (admittedly short) blog posting correctly polished by 20160830T0401Z .
[CAUTION: A bug in the blogger software has in some past weeks shown a propensity to insert inappropriate whitespace at some late points in some of my posted essays. If a screen seems to end in empty space, keep scrolling down. The end of the posting is not reached until the usual blogger "Posted by Toomas (Tom) Karmo at" appears.]
Here is a mere public-service announcement.
I had no time this week for writing up "Part C" of my essay praising Moise, but I hope to write it up next week (with upload in the normal four-hour interval, in this case the four-hour interval UTC=20160906T0001Z/20160906T0401Z).
In the mean time, readers may wish to follow me in proving an easy general result pertinent to Moise's first chapter, on fields. (The result emerges quickly from thinking about Moise, although Moise does not himself include it in the chapter problem sets.) Suppose F is any field - finite or infinite. F might, for instance, be the tiny two-element field which Prof. Edelstein drew to the notice of his 1971-spring-semester special Dalhousie University analysis class. Or F might be the rational numbers with standard addition and multiplication. Or F might be the real numbers with standard addition and multiplication. No matter what F is, we can construct a further field, whose elements are pairs (a,b) of F-elements, as follows: addition is defined in terms of F-addition, as (a, b) schplus (c,d) = (a plus c, b plus d), with (0,0) as an (and, we can quickly prove, as the unique) additive identity element; multiplication is defined in terms of F-multiplication, and the additive inverse "minus" of F, as (a, b) schtimes (c, d) = ((a times c) minus (b times d), (a times d) plus (b times c)), and with (1,0) as the identity element. (The task is to prove that the set of such pairs, with schplus and schtimes, is itself a field.) - So given the reals, we can construct the complex numbers as pairs of reals; and given Prof. Edelstein's tiny two-element field, we can construct a four-element field as pairs of Edelstein-tinies; and given the just-constructed four-element field, we can construct an eight-element field; and so on.