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  1. Is there a concept called "concept" defined in a formal system?
  2. Can concepts always be treated as axioms?

    Can axioms always be used to define concepts?

    For example, in ZFC set theory, I think the axioms are used to define what a set is?

Thanks and regards!

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The closest thing I can think of to a concept is a formula. –  André Nicolas Jan 10 '12 at 19:18
    
If there's no such concept, then that's something that formal systems have not yet modeled. –  Michael Hardy Jan 10 '12 at 19:20
    
@AndréNicolas: (1) Are concept, formula, axioms and theorems all elements of the formal language of a formal system? (2) Is concept a character in the alphabet of the formal language, or an element of the formal language? –  Tim Jan 10 '12 at 19:22
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@Tim: There is no formal thing called concept. The simplest advice is to forget about it. The not simple suggestion is that if you feel the idea should be formalized, try to do so. Don't quite know what "elements of the formal language" is supposed to mean. The natural elements of a language $L$ are the formulas. Or perhaps they are the symbols. The axioms, and theorems, are not determined by $L$. –  André Nicolas Jan 10 '12 at 19:32
    
@AndréNicolas: Is a concept instead just a shorthand/notation for what was used to defined it? PS: "elements of the formal language" means elements of the formal language being a set. –  Tim Jan 10 '12 at 19:36
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The short answer to your first question is no: there is no generally used formal notion of concept. It’s an inherently somewhat fuzzy and informal notion.

For the second question, I would not say that axioms are used to define concepts; rather, we start with some informal concept/notion, like that of a set, and attempt to formalize the concept by axiomatizing it. The axioms of ZF are one such attempt, though not the only one. Like all such attempts, they don’t define the everyday notion of set; at best they implicitly define a kind of mathematical object that we might for short call a ZF-set, one that we hope consistently captures our intuition of what a mathematical set ought to be.

Note that even in much simpler settings we can’t really say that axioms define concepts in the everyday sense of the word define. Suppose that I axiomatize a notion of a nice committee system (NCP) by saying that it comprises a set $P$ of persons and a collection $\mathscr{C}$ of subsets of $P$, called committees, such that (1) each person belongs to exactly two committees, and (2) each committee comprises exactly two persons. This describes exactly how NCP-persons and NCP-committees are related, but actual examples of nice committee systems needn’t involve real people and committees at all: if $G=\langle V,E\rangle$ is a finite undirected graph whose components are simple cycles, $G$ is a nice committee system if I interpret $V$ as the set of persons and $E$ as the set of committees. For that matter, I could let $\mathbb{Z}$ be the set of persons and $$\big\{\{n,n+1\}:n\in\mathbb{Z}\big\}$$ the set of committees and have a nice committee system. And of course it’s also possible to have a nice committee system whose persons really are people, and whose committees really are committees. This is a highly simplified and somewhat artificial example, but the point is that a system of axioms typically has many models that don’t look much like the intuitive concept underlying the system.

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Thanks! (1) Is it correct to understand a concept same as a alias/shorthand/notation/alias for what was used to defined it? e.g. in "Define A to be BCD", can the concept A be thought as an alias of BCD? If yes, then concept can exist in formal systems? (2) Do you mean that "concepts" are only defined in models of a formal system, not itself? In other words, it must be semantic and cannot live in a formal/syntax world? –  Tim Jan 11 '12 at 16:11
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