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$begingroup$ I haven't appropriately acquired my head spherical what precisely the real difference is amongst "transfinite" and "infinite".
1 $begingroup$ @MSIS: Take into consideration an infinite industry including $mathbb Q$. Just about every industry is often a Euclidean domain. If there is a more elaborate set up on your problem, inquiring a brand new Issue (with that context) could be constructive. $endgroup$
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one $begingroup$ @Zev: You may use quad as a protracted space; and Bigg
I Individually prefer Technique 1 since it is quicker and even more intuitive, as we do not have to multiply by $r$.
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These decisions/conventions need to be taken in such a way that the rules of multiplication (e.g. $xtimes y=ysituations x$) remain valid as much as you possibly can. Fairly a occupation! Your intuition claims that for $(two,infty)$ it is a good factor to pick $infty$ as solution. That confirms to me that the intuition will be to be revered. And keep in mind: intuition is critical in arithmetic!
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2 $begingroup$ Two points that I believe a freshman calc scholar needs to absorb: (one) Issues we'd generate as $infty/infty$ are termed indeterminate sorts, and calculus gives unique strategies for learning them. (2) Is infinity is often a quantity? See this problem: math.
drhabdrhab 153k1111 gold badges8686 silver badges219219 bronze badges $endgroup$ one $begingroup$ Well, very clear remedy. $endgroup$
Asaf Karagila♦Asaf Karagila 402k4747 gold badges635635 silver badges1.1k1.1k bronze badges $endgroup$ 1 $begingroup$ To me "transfinite" strongly connotes ideas relevant to ordinals, so I discover it a lousy option in contexts like nonstandard Investigation where it pitfalls contributing to misconceptions folks have that hyperreals have Infinite Craft some particular connection to infinite ordinals. $endgroup$