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I'm currently reading about non-principal ultrafilters and their relationship with defining the hyperreals, I am struggling to really get my head around the notion that a non-principal ultra filter allows one 'select the non-necessary elements' of the series on whose back we create the equivalence class that goes on to be used in the sense of a quotient to define the $^*\mathbb{R}$. How is it that this ultrafilter does such a thing? I see that it defines some sort of measure on the sets of $\mathbb{N}$ and this establishes somehow that these are the elements that matter for defining the equivalence, and in essence one can ignore those subsets of $\mathbb{N}$ that aren't in this set. What I can't understand is why? How is it these sets of $\mathbb{N}$ identify the key elements of the series which matter, in particular given they are necessarily infinite, and how is this done in a manner that allows for the transfer theorem?

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    $\begingroup$ You are justified in being confused. The Wikipedia article ultrafilter states "that free ultrafilters therefore exist, but the proofs involve the axiom of choice (AC) in the form of Zorn's Lemma" (where free means non-principal). Thus, free ultrafilters are not constructive. $\endgroup$ – Somos Apr 22 at 21:06
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    $\begingroup$ A ultrafilter is the output of an infinite choice procedure that yields a coherent way to decide, among a subset and its complement, which one matters. I think it is enlightening to see for oneself why the axiomatic properties of ultrafilters are enough to prove the transfer theorem. For instance, try to derive the validity of a sentence of the form $\exists x \forall y \exists z(\varphi[x,y] \rightarrow \psi[x,y,z])$ which is valid in $\mathbb{R}$ using those properties (and countable choice). $\endgroup$ – nombre Apr 23 at 3:36

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