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It seems like a lot of the difficulties in set theory come from trying to make sense of ordered lists just starting with unordered sets.

Wouldn't it be easier to have ordered lists as the fundamental objects? Or even just an ordered pair?

Then the number 5 could be written as $(0,(0,(0,(0,(0)))))$ for example.

Are there any alternatives to set theory that start with ordered lists or pairs and which derive sets from them instead of the other way round?

Andrés E. Caicedo
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zooby
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    That works as the number 5 even without ordering. – Arthur Jun 24 '19 at 06:15
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    What difficulties do you think arise from using unordered sets compared to ordered ones? As Arthur said, your construction of the number 5 already works with unordered sets. – M. Winter Jun 24 '19 at 06:18
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    Really? I always thought that the difficulties with set theory come from the fact most people do not fully posses the abstraction abilities needed to understand it "immediately", and the whole idea is that they develop it through repeated exposure. After that, the main problem is that infinite things do not behave like finite things, but that's a different story. – Asaf Karagila Jun 24 '19 at 06:34
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    There's nothing difficult about making ordered things from unordered things. We figure out a way of implementing an ordered pair, then we just reason about ordered pairs without needing to look under the hood again (except under very weird circumstances). – Malice Vidrine Jun 24 '19 at 06:50
  • I am not sure that ordered lists would be more intuitive than unordered lists. Note that ordered lists are just functions/finite sequences and set theory handle them pretty well. – Taladris Jun 24 '19 at 08:49

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