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AShepard comments on Why We Can't Take Expected Value Estimates Literally (Even When They're Unbiased) - Less Wrong
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That's precisely why it is arbitrary -- it's a cultural artifact, not an inherently meaningful level.
but because it is the standard value, you can be more confident that they didn't "shop around" for the p value that was most convenient for the argument they wanted to make. It's the same reason people like to see quarterly data for a company's performance - if a company is trying to raise capital and reports its earnings for the period "January 6 - April 12", you can bet that there were big expenses on January 5 and April 13 that they'd rather not include. This is much less of a worry if they are using standard accounting periods.
It's still a pretty significant worry. If you know that some fiscal quarter or year will be used to qualify you for something important, it is often possible to arrange for key revenue and expenses to move around the boundaries to suit what you wish to portray in your report.
That's true. Arbitrary means different things, from "not chosen by nature", to "not chosen by an outside standard".
I think if we tabooed (taboo'd?) "arbitrary", we would all find ourselves in agreement about our actual predictions.
In this case, the fact that it evidently was chosen to conform to scientific culture, and not for some ulterior motive, is bayesian evidence in favor of the validity of the frequentist methodology.