Timeline for Plausibility argument for a measurable cardinal
Current License: CC BY-SA 3.0
7 events
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| Jul 11, 2015 at 3:51 | comment | added | Mohammad Golshani | 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. | |
| Jul 10, 2015 at 20:33 | answer | added | Timothy Chow | timeline score: 4 | |
| Jul 10, 2015 at 19:20 | comment | added | Ashutosh | 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. | |
| Jul 10, 2015 at 18:04 | comment | added | Joel David Hamkins | 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. | |
| Jul 10, 2015 at 18:02 | comment | added | cody | 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"? | |
| Jul 10, 2015 at 17:04 | review | First posts | |||
| Jul 10, 2015 at 17:11 | |||||
| Jul 10, 2015 at 17:00 | history | asked | Henry | CC BY-SA 3.0 |