There seems to be a paradox which shows up when defining set theory and logic. On one hand, set theory appears to be grounded in logic. However, on the other hand, sets are required to define necessary properties of logic. Have I misunderstood something? Because from my perspective, it seems that a set would need to exist before it can be defined...
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Related: are-the-natural-numbers-implicit-in-the-construction-of-first-order-logic-if-so. – Mauro ALLEGRANZA Apr 10 '17 at 19:48
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This is a very good question, but is treated elsewhere on this site. – Noah Schweber Apr 10 '17 at 20:08
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Also, http://math.stackexchange.com/questions/121128/when-does-the-set-enter-set-theory – Asaf Karagila Apr 11 '17 at 00:29
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1In a nutshell, you can do logic without set theory, but you cannot do set theory without logic. – Dan Christensen Apr 11 '17 at 02:49
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@DanChristensen I disagree to a certain extent - to do logic you need to be able to talk about arbitrary finite strings, and this is equivalent to a very small amount of set theory. So it's not quite true that set theory can be purged from logic, in my opinion. – Noah Schweber Apr 11 '17 at 04:22
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@NoahSchweber A bit like saying a shepherd is a number theorist because he or she must on occasion count sheep by some means. – Dan Christensen Apr 11 '17 at 19:58