But in addition to historic reasons to do with battling with hidebound modes of assumed (And that i am knowledgeable about Cantor's difficulties with Kronecker and people of his way of contemplating), does there exist a concrete definition that one can drop by that claims: "
$begingroup$ Inside your opinions, you requested how summing up finite items (or products and solutions of finite items with other finite points) can provide you with some thing infinite. This shouldn't be too counterintuitive; The reasoning is that you are summing up infinitely a lot of Individuals finite issues. If something, It is really counterintuitive you can often sum up infinitely many things and obtain a finite outcome. I do agree, however, that the thought of a recreation similar to the St. Petersburg Match would seem counterintuitive. Section of it might be due to phrase "expectation." In frequent usage, after we assume some thing to occur, we think It really is much more possible to happen than not.
It is exclusive as many as isomorphism (so we frequently take a look at "the" algebraic closure of $K$), and we compose $L=overline K $, or at times $L=K^ text alg $.
without working with Taylor series. far more explicitly without employing calculus. how can we know if a operate is usually expressed as sequence or not ?
$begingroup$ Then it's possible I ought to request how can We all know when to make use of Taylor Sequence to determine a perform ? $endgroup$
It can be akin to asking, if John runs 2 times as fast as Jack and both of those run off far from me, am i able to divide John's ultimate place by Jack's remaining place, which are both of those even more far from me than I am able to ever go, and get $2=one$? (Naturally neither John nor Jack on their own can arrive at their "ultimate position", but the procedure by which they 'strategy' it clarifies the specific situation quite very well.)
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I'm unsure if there are actually other methods to demonstrate it. Probably You will find there's way with what are generally known as Fourier sequence, as loads of series might be stumbled on in this way, but it isn't that instructive. $endgroup$
Far more suitable should you utilised Conway's surreal numbers. While in the surreals, It will be all-natural to associate $one+1+ldots$ with $omega$, Despite the fact that there remains an ambiguity as pointed out by Karolis. $endgroup$
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We confer with the extension as just one entity by the expression $L/K$ (it's not a quotient similar to this or this, however). An extension $L/K$ is undoubtedly an algebraic extension when each and every