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JoshuaZ comments on Open thread, Oct. 12 - Oct. 18, 2015 - Less Wrong Discussion
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What you are doing in many ways amounts to the 18th and early 19th century arguments over whether 1-1+1-1+1-1... converged and if so to what. First formalize what you mean, and then get an answer. And a rough intuition of what should formally work that leads to a problem is not at all the same thing as an inconsistency in either PA or ZFC.