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prontoday at 1:35 PM1 replyview on HN

I don't think anyone minds this. The purpose of a formal foundation is to prove useful theorems. Junk theorems are just a side effect. But I'm always tickled by logicians who think type theories are philosophically superior to set theories because in some of the latter you might be able to prove something like 1 ∈ 2, whereas type theories, even without their own junk theorems, have a pragmatic difficulty with division (hence they tend to define 1/0 = 0). Junk theorems just come with the territory, and foundations need to be considered based on their utility, not philosophical purity, which is never achieved anyway (at least not without a cost to utility).


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falcor84today at 1:45 PM

> But I'm always tickled by logicians who think type theories are philosophically superior to set theories because in some of the latter you might be able to prove something like 1 ∈ 2

Note that this is actually how the basic ZF construction works, where 0 = {} and successor(n) = n ∪ {n}, so you immediately get 2 = {0, 1} and thus 1 ∈ 2 , without any need for a proof.

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