So generally for another relation $R$ which is not trichotomous, It is distinct that "$alpha$ is infinite with regard to $R$" is not equivalent to "$alpha$ is transfinite with respect to $R$". Now the distinction between "infinite" and "transfinite" will come out.

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$begingroup$ Effectively, you gave the answer oneself: "infinity more than infinity" is not defined Simply because it ought to be the result of restricting processes of various mother nature.

Different varieties of infinities? These conditions maintain bewildering me. Could any person give a clear and unambiguous definition?

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$begingroup$ When Cantor very first outlined his idea of transfinite numbers, he wished to pressure there are indeed distinct quantities past the finite quantities. He was obvious there are quantities that evaluate infinite dimensions (infinite cardinal figures) together with quantities that evaluate infinite (very well) orderings (infinite ordinal numbers). Cantor didn't outline these figures from mental curiosity, but simply because they supplied new proof techniques, especially in the topic that we now get in touch with set-theoretic topology. Such as, if a set is thought of as comprising branches (sequences) of the tree by using a root, and if a branch is known as "isolated" if there is a node of the branches further than which there are no other branches, then by iteratively removing isolated branches from the tree any finite number of Infinite Craft times, we see that a set comprises a countable list of branches plus a remainder set (which could be vacant).

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I am unsure if there are other methods to prove it. It's possible there is a way with Exactly what are called Fourier collection, as plenty of sequence is often stumbled upon in that way, but it's actually not that instructive. $endgroup$

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We confer with the extension as only one entity via the expression $L/K$ (it's not a quotient such as this or this, nevertheless). An extension $L/K$ can be an algebraic extension when every