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The following question is not mathematically precise but perhaps of some philosophical interest.

A typical plausibility argument for assuming the existence of inaccessible cardinals goes as follows: There is no reason why there should not be a cardinal so large that it cannot be reached from smaller cardinals via the power set operation.

Are there similar "unreachability from below" arguments for measurable cardinals? In particular, is there a way in which we can view the existence of a non trivial definable elementary embedding of the class of all sets as an assertion of this kind?

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  • $\begingroup$ I'm afraid I'm not a set theorists, but in what way do these "plausibility" arguments differ from "we haven't found them to be contradictory yet" arguments? I.e. how is this different from "there is no reason why there shouldn't be an odd perfect number"? $\endgroup$
    – cody
    Commented Jul 10, 2015 at 18:02
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    $\begingroup$ Henry, do you find that argument for inaccessible cardinals convincing? If so, the bar would seem to be fairly low. Usually, however, the argument for inaccessible cardinals and other smallish large cardinals is not just "there is no reason why", but rather is combined with reflection ideas, asserting that truths about the universe itself will reflect down to a level. Thus, as Ord itself is definably regular and closed under power set, we expect such cardinals to exist at a level. This may be better, but my view is that one then would want justification for the reflection feature itself. $\endgroup$
    – Joel David Hamkins
    Commented Jul 10, 2015 at 18:04
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    $\begingroup$ In the "unreachability from below" argument for inaccessible $\kappa$, to the $\theta \mapsto 2^{\theta}$ cannot reach $\kappa$ criterion, we should also add for every $\theta < \kappa$, $f: \theta \to \kappa$, supremum of range of $f$ is not $\kappa$ (to disallow something like $\beth_{\omega}$ from being unreachable from below). It seems to me that $V = L$ is compatible with the existence of any large cardinal that has an "unreachability from below" sort of justification. This would mean that it would be difficult to justify measurable cardinals in a similar simplistic way. $\endgroup$
    – Ashutosh
    Commented Jul 10, 2015 at 19:20
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    $\begingroup$ The paper Believing in strongly compact cardinals might be also useful. It's abstract is: The classical argument in favor of the existence of strongly compact cardinals is the principle of uniformity. Here we give another argument based on a principle of maximal diversity of reflections. This principle is motivated by Maddy’s set-theoretical naturalism and is inspired by some maximization principles of Leibniz. $\endgroup$
    – Mohammad Golshani
    Commented Jul 11, 2015 at 3:51

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The standard discussion of the justification of measurable cardinals is Penelope Maddy's article Believing the axioms. Regarding the "unreachability from below" argument, which she calls inexhaustibility, Maddy cites several authors saying that no such argument is available.

She does mention Reinhardt's claim that the existence of elementary embeddings is a "reflection principle" of some kind, and discusses this in more detail in Believing the axioms II, but even if you buy that, it's still not really an "unreachability from below" argument.

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