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Can someone give me examples of mathematical objects which do not involve sets? For instance, the category of groups is a concrete category, but I want to consider non-concrete things.

Thanks

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    Properly defined it isn't a quotient – it's a localisation.2012-12-17

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\begin{sermon}

What about the natural numbers? Numbers aren't sets and don't "involve" sets (whatever that means). It is nonsense to ask "what are the members of the number 10?" -- which is why that question is never even raised in a course on elementary number theory.

Of course, you can implement/realize/embed/model [choose your favourite terminology] the natural numbers inside ZFC or NF or MAC [whatever your favourite set theory happens to be]. But so what? That doesn't mean that the natural numbers are, or "involve" sets.

Set theory is like an all-purpose Lego kit. You can build all kinds of models with the Lego kit (from models of pirate ships to models of farmyards); you can build all kinds of models in set theory (from models of the natural numbers to models of Hilbert spaces). But everything is what it is and not another thing: pirate ships aren't to be confused with models of them, natural numbers aren't to be confused with models of them. Likewise for lots of other mathematical objects.

\end{sermon}

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    That it is not possible to implement/realize/embed/model some mathematical object using sets.2012-12-17