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Intuitively, the more time we spend exploring ZFC without finding a contradiction, the higher the (subjective) probability that ZFC is consistent. Is this intuition sound?

If not, why not?

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Yes, your intuition is correct. –  André Nicolas Jan 30 '13 at 18:06
I'd think there might be a huge bias, though. It's like saying that the probability of finding fossils in the middle of the highway is zero, because so many people haven't found any fossil lying on the highway yet. But then again, what do I know about probability? –  Asaf Karagila Jan 30 '13 at 18:19
that's not rigorous. your reasoning is philosophically sound but it's not in any sense mathematical. –  user58512 Jan 30 '13 at 18:30
There is a practical component. There are some people looking (not many!). Other things being equal, the longer a contradiction is not found, the more reasonable it is to bet one will not be found next year. –  André Nicolas Jan 30 '13 at 18:50
Yes, but it need not get strictly higher---for example, in the case that my subjective probability is already 1, or the case that it's less than 1 but I am almost certain that the shortest proof of a contradiction is longer than any human could have found in the time since humans began doing mathematics. –  Trevor Wilson Jan 30 '13 at 18:51

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