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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.

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Are you just asking for a category which is not concretizable? A famous result of Freyd asserts that the homotopy category of topological spaces has this property. See, for example, amathew.wordpress.com/2012/01/26/homotopy-is-not-concrete . –  Qiaochu Yuan Dec 17 '12 at 11:23
    
I want something where no sets are involved or rather that sets are not a sufficient manner to model this mathematical object. –  cjdi43 Dec 17 '12 at 11:25
    
What do you mean by "involved"? As Peter Smith points out in his answer below, there are various things that you can talk about in a language that makes no explicit reference to set theory even if you can write down models of these things using sets. In what sense does a non-concretizable category not answer your question? –  Qiaochu Yuan Dec 17 '12 at 11:26
    
I have a decent intuition of sets. But I want some exposure to things which "are not sets". I want a mathematical object that cannot be realised using sets and thus has to be realised using "something else". The problem is that the "something else" is alien to me. –  cjdi43 Dec 17 '12 at 11:28
    
That sounds like you want examples of non-concretizable categories to me (but I'm biased; I think of all mathematical objects as living inside categories). –  Qiaochu Yuan Dec 17 '12 at 11:37
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1 Answer 1

\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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Perhaps I should ask for mathematical objects that cannot be constructed from sets. –  cjdi43 Dec 17 '12 at 11:13
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@cjdi43 Ah: that is indeed a different question! But still too vague. "Constructed" in exactly what sense? –  Peter Smith Dec 17 '12 at 11:23
    
That it is not possible to implement/realize/embed/model some mathematical object using sets. –  cjdi43 Dec 17 '12 at 11:24
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