What is the definition of belonging in axiomatic set theory? Ffo s
In Hungerford's Algebra it says that
Intuitively we consider a class to be a collection $A$ of objects such that given any object $x$ it is possible to determine whether or not $x$ is a member (or element) of $A$
In Thomas Jech's Set theory it states the extensionality without defining what is belonging.
I just learn that in set theory everything existing is just set. But it doesn't state that set is a collection of something so the natural membership definition of belonging may not exist. For example, red may be a set but nothing is a member of it in a real-life widely-admitted sense.
Then what is belonging, i.e. $\\in$ defined?
You can see also my another question
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8$\\begingroup$ It is primitive, not defined. The axioms govern its behavior. (Also,I assume you mean Thomas Jech?) $\\endgroup$ – spaceisdarkgreen 22 hours ago
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$\\begingroup$ Red is totally a set: it's one of the equivalence classes of the relation "is the same color as". $\\endgroup$ – eyeballfrog 1 hour ago
2 Answers
In axiomatic set (or class/class-set) theory, the membership relation is a primitive relation. It is not defined, but other things are defined from it. The axioms tell us how it behaves.
We can compare to the situation of asking how the successor operation is defined in an axiomatic treatment of arithmetic. You might say $S(0)=1$ and $S(1)=2,$ etc, but that's no definition at all. In fact, the right way to think of it is that $1$ is an abbreviation for $S(0)$ and $2$ is an abbreviation for $S(S(0)),$ etc. In other words, the successor operation is a basic notion with no definition. The axioms tell us its basic properties, like if $S(x) = S(y)$ then $x=y,$ and that there is no $x$ such that $S(x)=0,$ but if $y\\ne 0$ then there is an $x$ such that $S(x)=y.$ Oh and how do we define $0$? Well, if we want, we can take it as an axiom that there is a unique object with no predecessor and define $0$ as that object. But it's really all the same and saves a little ink to take $0$ as primitive, so that's what's usually done.
In set theory, $\\in$ is the only primitive relation other than equality (although we can even define equality in terms of $\\in,$ but this is a little troublesome and not typically done). Other basic notions like the empty set, the subset relation, the union operation, etc, are defined in terms of $\\in$ and $=$ (where in the case of constants and operations, we also must use the axioms to establish existence/uniqueness/well-definedness). It may be convenient in certain modifications to take some of these as primative (e.g. the empty set), but it's customary in a standard treatment of ZF to have them defined.
I think you are reading the phrase "everything existing is a set" a bit too literally. What we mean is that every object in the domain of discourse of our theory is a set. This really has no content: we are just saying that the objects referred to by variables in our theory are called sets. We are certainly not making a sweeping metaphysical statement that every concept in the world (e.g. "red") is somehow a set.
Finally, a word on classes. In set theory, a class $C$ is just an informal notion of a collection of sets satisfying some formula $\\phi_C$. So $x\\in C$ is just an abbreviation for $\\phi_C(x)$. In class theory the membership relation for classes is a primitive relation again. There are a number of ways of slicing and dicing, but one way is to just let all objects be classes, and let sets be a special kind of class. We can define the predicate $Set(x)$ as $\\exists z (x\\in z),$ i.e. a set is a class that is an element of some class.
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$\\begingroup$ For the proposer: A class in the language of ZFC can be considered as a modification to the language to allow some abbreviations. E.g. the formula $\\phi$ in one free variable that defines Godel's constructible class is very long so we write $x\\in L$ rather than writing out $\\phi(x)$ in full. This does NOT assert that there exists a set $L=\\{x: \\phi(x)\\}.$ And in this case, since there does exist some $x$ such that $x\\in L$ but there does not exist a set $\\{x:x\\in L\\}$ we call $L$ a proper class. (An improper class, which is a term you will never see, would be a set.)............+1 $\\endgroup$ – DanielWainfleet 11 hours ago
In axiomatic set theory, $\\in$ is a primitive relation - in fact, the only such in the theory in usual formulations, while the only primitive objects are sets. Primitives are not formally defined: if you want to think of them as having a definition, that definition is not part of the formal system (the system of logical axioms that set theory is, written in symbolic notation) and, instead, must belong to external, natural language. You can think of the formal system then as being something which tells these things how to behave, and in which we conduct arguments and proofs. We should think of these natural-language understandings (i.e. "a set is a collection") as a way to make sense of the formal axioms through intuition, so as to not look arbitrarily-chosen, random rules (as they aren't), and the formal axioms in turn, in a way, as "feeding back" on that in that they help to sharpen this intuition.
Insofar as the idea that "red" is a set, this is not correct. The statement "everything existing is a set" doesn't mean "everything existing in real life", rather it means "everything existing in the world of set theory". The world of set theory, at least Zermelo-Fraenkel set theory, only has sets as primitive objects.
