I forgot the point out which i wish to Specific the defition without implementing Taylor sequence, as it makes use of calculus, which that I will not want to take into consideration in the meanwhile. $endgroup$

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You'll be able to incorporate 'infinity' to this list of figures, but after that conventions has to be created to get an extending of this multiplication. This in this kind of way that The principles of multiplication continue to be legitimate as considerably as you can. $endgroup$

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$begingroup$ Effectively, you gave the answer on your own: "infinity more than infinity" is just not described just because it should be the results of restricting processes of different character.

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quantity, within a amount procedure $E$ extending $mathbb R $, can be a number more compact than every single favourable serious $rinmathbb R $. An considerable

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Considering that the word "transfinite" suggests "past finite" (Observe that the prefix "trans" suggests "past") which happens to be similar to "larger than or equal to many of the finite organic quantities", we Infinite Craft define "transfinite" as follows.

one $begingroup$ @sos440: In NSA, infinite numbers don't have specifiable sizes, and you may't uniquely identify a sum like $1+1+one+ldots$ with a certain hyperreal. Hyperreals might be defined as equivalence classes of sequences less than an ultrafilter. Considering the fact that ultrafilters cannot be explicitly produced, you can't, generally speaking, get infinite sums $sum a_i$ and $sum b_i$ and say whether they consult with the exact same hyperreal.

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user21820user21820 60.1k99 gold badges106106 silver badges270270 bronze badges $endgroup$ two $begingroup$ Observe that you are also assuming "there is only one quantity with greater magnitude than any finite amount" as a way to have the ability to use $infty$ as a proper noun. $endgroup$

– Qiaochu Yuan Commented Aug 21, 2012 at 23:forty six six $begingroup$ @saadtaame, I imagine that what Qiaochu is trying to speak you is the fact if you can find some facts in Wikipedia, not to mention in many hundreds of Countless other web pages in the online, then you shoud first try to learn there one or two things regarding your have dilemma, lest persons Assume you happen to be far too lazy to do this by by yourself . $endgroup$