Defining the membership relation in set theory?

[jordi]

My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory.

Let me describe a method to define the membership relation, and please correct me if I am wrong somewhere in my argument:

A class can be defined as the collection of objects satisfying a given property.

For me, first order predicate theory has no problems in its definitions: first order predicate theory is just a way to put order into our metalanguage (English), such that all statements are true or false. It is a somewhat "limited" language (in comparison to English) but it is useful nonetheless.

So, if x is an object (a "name") and p is a property (an "adjective"), we can define p(x) as "has the object x the property p?" and this is false or true.

Then, a class A is the collection of objects x such that p(x) is true.

We can define x ∈A as "p(x) is true", where p is the property defining A.

To sum up: we have defined ∈ just with the language of first order predicate theory (which is equivalent to classes).

Then, what are sets?

Sets are classes. But not all classes: only those classes satisfying a series of statements in first order predicate theory (which include also the membership relation; but this is not a problem, since we perfectly understand what the membership relation is: we have defined it earlier above in this post), the so called axioms of set theory.

If a class does not satisfy these axioms, it is not a set.

And then, we state that "all mathematics" can be built out of sets (not classes; well, some foundational issues use classes which are not sets, such as the class of all ordinals, but anyway).

If my explanation were right, then everything is fine for my intuition: the membership relation is just DEFINED as a property in first order predicate theory. And sets are just defined as classes (which are a relabeling of properties) which satisfy a series of "strange" axioms (the axioms of set theory). One finds these "strange" axioms are useful, and only those classes that satisfy those axioms (the sets) are necessary to build all mathematics, which is a great feature to have.

Any comments?

 

[Stephen Tashi]

My main question is regarding whether the membership relation is taken as an undefined concept (as is usually hinted in set theory books) or if the membership relation can be defined within the language of first order predicate theory.

Can you do it without subscripting each occurrence of $\in$ with a notation indicating its associated property?

For example:
$P(x):$ x is a Philsopher
$B(x):$ x is a Banker

Define "$x \in_{P} P$" to mean $P(x)$
Define "$x \in_{B} B$" to mean $B(x)$

But how do you define "$\in$" without any subscript?

 

[jordi]

Good point. My answer to your question is that I cannot see how to define the membership relation without any subscript. But in my opinion, this is not necessary: the axioms of set theory are considered to be true for any membership relation. So, for each property, I have a membership relation, as you point out. And the axioms are true for all those properties (and membership relations).

The algorithm to check whether x ∈ A is true or not is obvious then: I observe the class A. This means to observe which is the defining property of A; let us call it p. Then I check if p(x) is true or not. This allows me to check the validity of the original question.

 

[Stephen Tashi]

the axioms of set theory are considered to be true for any membership relation. So, for each property, I have a membership relation

When we introduce the union and intersection of sets, we'd have to deal with membership relations whose subscripts are functions of other subscripts - e.g. $\in_{\cup,A,B}$ from $\in_A$ and $\in_B$.

You can accomplish your idea informally, but it isn't clear that you can implement it in a formal logical system - and perhaps you don't want to!

The difficulty in formal logic is that given a set of axioms, we tend to reason about them using all the mental concepts at our disposal even if the axioms are intended to restrict the available concepts. This leads to some curious situations. ( For example, people who prove things about modal logics and fuzzy sets, employ ordinary logic and ordinary sets.) I'm not a logician, but my exposure to the field indicates that the professional way to proceed is treat the subject matter as if it concerned manipulations of symbols without any interpretations of the symbols. ( Contrast the Wolfram presentation of first order logic http://mathworld.wolfram.com/First-OrderLogic.html which is reasonably formal with the Wikipedia presentation of first order logic, which is informal.)

A set of formal axioms defines some permitted (a.k.a "well formed") symbolic strings and gives ways of manipulating them. When we reason about a set of axioms, we use many mental concepts that are not expressed in the axioms. For example, the notion of a proof (i.e. a "derivation") employs a notion of order - we begin with something "given" and using permitted manipulations, we "derive" a result by an ordered series of manipulations. Yet nothing in the symbols we are manipulating may have anything to do with formalizing the notion of an order relation. As another example, the Wolfram presentation of first order logic refers to an "n-place function" and thus introduces the notion of a natural number and an order of the arguments in the function, but these notions are not defined within the axioms of first order logic.

The general scenario for formal logic is that that it studies manipulations of strings of symbols and reasons about these manipulations using "meta-language" and "meta-logic" - for example, English and "ordinary" logic. It's a delicate question how to limit the concepts used in the meta-logic. For example, if we are reasoning about strings of symbols, are we allowed to make assertions about infinite strings of symbols? It would take a real logician to explain what restrictions apply to meta-logic. ( We can note that the presentations of first order logic speak of a finite "universe of discourse".)

If we want to carry out your idea ( to define $\in$ formally within first order logic) our first quandry is:
"In the study of manipulating strings of symbols, what is a definition?"!

One concept of a (formal) definition is that it is a rule that permits us to substitute one string of symbols for another in any string of symbols where one or the other of the strings occurs. We'll have to ask an expert if that's a good definition of "definition".

 

[jordi]

Very interesting comments, thank you.

At least for A ∪ B, I have a definition that allows me to define ∈ for the union set:

If A is the class defined by the proposition p, and B is the class defined by the proposition q (and then, x ∈ A iif p(x) is true, resp. for B), A ∪ B is the class defined by the proposition p ∨ q.

So, I have been able to define ∈ for the union set. Granted, I believe here I have just been lucky: the union is a famous way to create sets, so it seems reasonable to assume there is a famous way to find the corresponding proposition, in this case, using ∨.

But what happens to other ways to construct classes? Will there always be to create the corresponding proposition? I do not know. In fact, it seems unlikely to me. But at least, it is encouraging that for the union, I could find it.

About "defining the concept of definition": luckily, I do not have this conceptual problem. I believe I understand the metalanguage (with all its pitfalls and contradictions) and I believe first-order logic is just a way to refine the metalanguage to make useful statements. But I consider the concepts of SET, NUMBER FOUR, FUNCTION and similar to have a "platonic" meaning outside their mathematical definitions (and then, to the mathematical concept I will call it set, not SET, and the same for 4 and for function). Mathematical definitions are just the "refinement" of a concept it "exists out there". It is a little bit like the Euclid axioms: points and lines exist. The Euclid axioms are just a refinement of the "stylized behaviour" of points and lines, which allows to extract its main characteristics (not color or width, though; but this does not matter much), and even more importantly, it allows to prove statements we may believe are true by looking at the "real" points and lines, but we are not sure about them.

In fact, that the concept called NUMBER FOUR exists, it can be "proven" (not a mathematical proof, but a philosophical proof): apply "cogito, ergo sum". In the same way, if we think about a concept called NUMBER FOUR, this necessarily means this concept exists, since we think about it: so, cogito, ergo sum. So, NUMBER FOUR exists.

My main question regarding this is your statement: "my exposure to the field indicates that the professional way to proceed is treat the subject matter as if it concerned manipulations of symbols without any interpretations of the symbols" ... but I believe you interpret what ∈ means, in an intuitive way: the way ∈ behaves for finite collections of members ... no?

 

[Stephen Tashi]

My main question regarding this is your statement: "my exposure to the field indicates that the professional way to proceed is treat the subject matter as if it concerned manipulations of symbols without any interpretations of the symbols" ... but I believe you interpret what ∈ means, in an intuitive way: the way ∈ behaves for finite collections of members ... no?

Like most students of pure mathematics, I personally do not have a consistent way of interpreting mathematical concepts. I mix intuitive notions with formal notions in my thinking. However, in proving things rigorously, I can attempt to ignore the intuitive interpretations.

You said you can define a class in the meta-language and this leads to a definition of the relation "$\in$" in the meta-language. However, using meta-language does not answer the question of whether "class", "$\in$" ,and the rest of elementary set theory can be developed (formally) within first order logic. We can think of any familiar mathematical concept using meta-language. Such thinking does not demonstrate a result in formal logic.

Are you interested in whether elementary set theory can be defined and developed formally using only first order logic? - or are you only interested in finding a personal way of thinking about such a development in meta-language?

 

[jordi]

No, no. Probably I have not expressed myself correctly. What I mean is that I am comfortable with first order logic. What I am not comfortable with (due to my ignorance) is with set theory: when I see the axioms of set theory in a book such as Jech, I just wonder what this ∈ symbol means. It is not defined in that book, at least, as far as I know.

Of course, one could take the approach that ∈ is just a symbol without meaning, and start making symbolic manipulations. But I believe most mathematicians in the end give to the symbol ∈ the interpretation of "belonging". I am uncomfortable with this approach.

So, I tried to see if I could define the symbol ∈. And I tried to do it the way I have described above: x ∈ A is defined as p(x), where p is the proposition defining the class A.

But to do that, I am making a strong assumption, which I do not know if it is true or not:

Can I assume that a class is defined by a proposition, in all cases, and that a proposition defines a class? Or not?

 

[Stephen Tashi]

Can I assume that a class is defined by a proposition, in all cases, and that a proposition defines a class? Or not?

If you're asking whether this is useful intuitive way of thinking about classes (and sets), yes, you can assume that.

If you are asking a question about the technical details of logic and set theory, no, you can't assume that. For example, we read (e.g. https://en.wikipedia.org/wiki/Class_(set_theory)) that in the usual formulation of set theory (Zermelo-Fraenkel), the notion of "class" is not formally defined.

 

[Ronald Channing]

In most books I read it was considered as an axiom or "too primitive to be proved", but of course, two sets are equal if and only if they contain the same members.

 

[yossell]

"So, if x is an object (a "name") and p is a property (an "adjective")"

While names are objects, not all objects are names. Similarly, a property is not an adjective. Of course, we may use a name to pick out an object; we may also think of adjectives as picking out properties.

"We can define x ∈A as "p(x) is true", where p is the property defining A."

I don't think so - not with the generality you need to give us set theory, at least, not in a way that stays within first order logic. In set theory, we can form sentences which quantify over sets. `Every object is contained in some set', `for any set x, there is another set y whose sole member is x.' etc. etc. If we really had a definition of `∈', we should be able to analyse these sentences. But that would seem to involve quantifying into predicate position. AxEX("X(x)) is true)". That is no longer a sentence of first order logic.

Perhaps that's ok by you -- but be aware that some have argued that second order logic is unsatisfactory unless the second order variables are interpreted as ranging over all the subsets of the first order domain -- and that will then reintroduce problems of membership again.

Finally, I'm not sure whether you're serious about properties or whether you really are thinking in linguistic terms -- that it really is the adjectives that do the heavy lifting here. If the latter, then, as actual languages contain only denumerably many adjectives, you'll have a hard time getting `enough' sets. If the former then the following issue arises:

'p(x)' is really treated as expressing a relation that holds between the property p and the object x: we might say that x has p, or that it instantiates p, or that it possesses p. But, whatever we call it, it's a relation holding between two kinds of things: objects and properties. It's as much an unanalysed primitive as set-membership was in set-theory. You've defined membership, but only at the cost of another primitive. And notice that this primitive is not needed in simple first order logic.

 

[FactChecker]

You need to define the domain, D, that property p applies to. Then before you can talk about p(x), you need to say x∈D. But how can you do that before you have defined '∈'?

 

[jordi]

You need to define the domain, D, that property p applies to. Then before you can talk about p(x), you need to say x∈D. But how can you do that before you have defined '∈'?

I do not agree. The concept of p(x) is a first order predicate theory concept (in other words, we are using the metalanguage concept of FUNCTION, instead of the set theoretic concept of function). In first order predicate theory, one does not need to define the membership relation.

 

[splinter3d]

I am suffering with the same moments as you.

I always felt something unnatural in set theory as the foundation of everything. I mean they are called SETS. Common. Set by the defenition is something difficult. How can it be primitive concept? it is redicolus.

Actually I think that we can define set as some sort of implication. What does it mean that a banana is a fruit? It means that every time we have a banana we also have a fruit, it is impossible to have a banana and not to have a fruit, because banana is a fruit. But if we have a fruit it doesn't mean we have a banana. It can be an orange or an apple. Isn't it classical implication?

The topic is over a year old so maybe I will need to creat a new one.

 

[jordi]

I am suffering with the same moments as you.

I always felt something unnatural in set theory as the foundation of everything. I mean they are called SETS. Common. Set by the defenition is something difficult. How can it be primitive concept? it is redicolus.

Actually I think that we can define set as some sort of implication. What does it mean that a banana is a fruit? It means that every time we have a banana we also have a fruit, it is impossible to have a banana and not to have a fruit, because banana is a fruit. But if we have a fruit it doesn't mean we have a banana. It can be an orange or an apple. Isn't it classical implication?

The topic is over a year old so maybe I will need to creat a new one.

Well, the reason of this thread remains, after a year. So, it would be great to continue here.

 

[stevendaryl]

To form a class requires two things: (1) an underlying type, and (2) a predicate on that type. So classes lack the property of "extensionality", where classes with the same elements are equal. Sets in contrast are completely defined by their elements.