Exploring Axiom of Extension: Naive Set Theory by P.R Halmos

[StatOnTheSide]

I am reading Naive set theory by P R Halmos. He says that "The axiom of extension is not just a logically necessary property of equality but a non-trivial statement about belonging."

The example for that is

"Suppose we consider human beings instead of sets, and change our definition of belonging a little. If and are human beings, we write whenever is an ancestor of . Then our new (or analogous) axiom of extension would say if two human beings and are equal then they have the same ancestors (this is the “only if” part, and it is certainly true), and also that if and have the same ancestors, then they are equal (this is the “if” part, and it certainly is false"

How does this example substantiate the fact that axiom of extension is not a logically necessary property of sets but a non-trivial statement about belonging? I'd greatly appreciate it if someone can explain the above.

 

[Stephen Tashi]

You should fix your quotation. It is mangled because the symbols were left out. This blog has the quotation: http://topologicalmusings.wordpress.com/2007/12/14/section-1-the-axiom-of-extension/

 

[MLP]

I wouldn't have been able to reply without Stephen's assistance because I am not in proximity to my copy of Halmos. I find his example a little contorted as well. But as long as you understand the point that set membership is distinct from identity (Halmos says 'equality'), I wouldn't sweat the example.
Regards
Michael

 

[Stephen Tashi]

Regarding the example. I haven't read the Halmos book and perhaps it doesn't encourage you to think formally. But let's think formally.

When you define things like sets, membership, equality, you don't assume that they already have a meaning and that your are merely generating some words to summarize it. Your definition must be the meaning. However, things must be defined using symbols or words that are undefined. (Otherwise you get into a endless regression like A is defined in terms of B and B is defined in terms of C and C is ... or else you end up with a circular definition.)

Most people are willing to take "set" and "is a member of" as undefined concepts and proceed from there to do higher mathematics. However, if you want to do set theory, you need to define these things or at least state their properties.

Suppose I make the following somewhat obscure descriptions:

There are things of type X and things of type Y. There is a relation R between things of type X and things of type Y. If x is a thing of type X and y is a thing of type Y then either is true that x R y or it is not true.

At this point, if you wanted to make an example of what I described, you could think of X as the real numbers, Y as the rational numbers and x R y meaning $x \le y$.

You could also think of the example in the blog: Let things of type X be people and things of type Y be people and let x R y mean x is the ancestor of y.

However, assume I add the statement: If y1 and y2 are things of type Y then y1 = y2 if and only if for each x of type X , x R y1 if an only if x R y2.

Then you can't think of the example of people where x R y as means "x is the ancestor of y" because of the objection given in the blog.

You still might be able to think of the other example where x R y means $x \le y \$. We'd have to contemplate that closely.

This shows that the statement about y1 = y2 puts some limitations on the examples you can use for x R y.

The approach in Halmos may be to make more and more statements about the relation x R y until just about the only example you can think of for it is $x \in y$. So when he develops the properties of $\in$ you aren't supposed to think of it as a relation whose meaning is already "commonly understood". You are supposed to think about it as being as generic a relation as the symbol R. You are supposed to note as he adds more assumptions and definitions involving $\in$ how the variety of examples you can use for it get fewer and fewer.

 

[StatOnTheSide]

Thanks very much Stephen and MLP. Very nice descriptions.

I do understand that We cannot keep defining everything as it will go on for ever; however, I felt that the example given is somewhat confusing the way it is worded as in I am still not sure as to what point Halmos is trying to make.

Two things (may be sets, may be people) if and only if condition "VAGUE" is satisfied for some condtion called "VAGUE". In the case of sets and the property of belonging, it is fairly obvious to anyone as to why the axiom of extension is true.

If it is desired to make a formal definition of sets, belonging and equality without STRICTLY stating that it is obvious, it is probably impossible to have a definition. Any number of examples will still not result in concreteness.

I just wanted to make the above points purely to express my views; however, I wish to just clarify once more as to what MLP and Stephen have said. Axiom of extension (AE) is just not about equality but it also states an important property of belonging. It is a relationship between two objects. Here, by belonging, it is meant belonging of elements to a set. It is a relation between an element to a set. Axiom of Extension states that two sets are equal if and only if every element that belongs to one set belongs to the other and every element which belongs to the second set belongs to the first one.

I suppose "have the same extension" means "every element that belongs to one set belongs to the other and ever element which belongs to the second set belongs to the first one".

Please do correct me or confirm the validity of my viewpoint as the case may be.

 

[SteveL27]

ake.

Two things (may be sets, may be people) if and only if condition "VAGUE" is satisfied for some condtion called "VAGUE". In the case of sets and the property of belonging, it is fairly obvious to anyone as to why the axiom of extension is true.

I don't think it's true at all. It's a property that we specify so that we can say what we mean by a set. Let me give an example.

Say there's a house. The house might have some people in it; or it might not have those people in it. It might have different people in it. But it's still the same house.

So when we stipulate the axiom of extension, we're saying what we mean by a set. A set is completely characterized by its members. If a member leaves the set, then it's NOT the same set. So sets are not like houses.

The axiom of extension serves to define what we mean by a set. But it's not necessarily "true" about sets. We could have defined sets like houses. So there's a set A, and today A contains these elements, and tomorrow it contains some other elements. In "house-like set theory," a set is a set regardless of its members. It's characterized by some other property, such as its address,

But a set is characterized completely by its members; it's not characterized by any other property it may have.

So the axiom of extension is not necessarily true; it's true about sets because that's how we wish to view sets; and for no other reason.

 

[StatOnTheSide]

Hi SteveL27. It makes complete sense. When I made the statement regarding the property "VAGUE", I meant to say that a statement can be made but I did not add the statement sying "the property my not be true". In case of sets, I felt that it is obvious to say two sets are equal when they have the same elements. I am still confused as to why it needs a formal definition.

I suppose set theory is dealt that way in general.

I now completely understand that a set is characterized completely by its members. Thanks very much for making that point clear to me. :).

 

[Hurkyl]

In case of sets, I felt that it is obvious to say two sets are equal when they have the same elements. I am still confused as to why it needs a formal definition.

We could be talking about a concept where {a,b} and {b,a} are different.
We could be talking about a concept where {a,a} and {a} are different.
We could be talking about a concept where {a,{b,c}} and {a,b,c} are the same.
We could be talking about a concept where any two 'sets' satisfying x={x} and y={y} must be equal.
We could be talking about a concept where elements have colors, and two 'sets' are different if their elements are colored differently.

But we're not, and we know that because we wrote down the axiom of extensionality which tells us that there is no more or less information in a set S beyond what objects satisfy $x \in S$.

Three of the reasons why a formal definition are good is:

• It makes things explicit. This has all sorts of benefits; it helps us organize the way we think, and to see what might be missing from our definition.
• It removes ambiguity. It makes it clear that we don't mean a variety of other things we might mean.
• It is something we can compute with, in the sense that formal logic does computation with statements and theories and such.

 

[StatOnTheSide]

Makes a lot of sense Hurkyl. I am getting exposed to this subject for the first time and hence, I have a lot of these confusions. Thanks very much for the clarification. I really appreciate it :).

 

[StatOnTheSide]

Hi all. I still can't help but not stop thinking about the particular example Halmos gives. He says

"It is valuable to understand that the axiom of extension is not just a
logically necessary property of equality but a non-trivial statement about
belonging"

and then, he goes on to give the example of human beings and ancestors which is

"Suppose, for instance, that we consider human beings
instead of sets, and that, if x and A are human beings, we write x e A
whenever x is an ancestor of A. (The ancestors of a human being are his
parents, his parents' parents, their parents, etc., etc.) The analogue of the
axiom of extension would say here that if two human beings are equal,
then they have the same ancestors (this is the "only if" part, and it is
true), and also that if two human beings have the same ancestors, then
they are equal (this is the "if" part, and it is false)"

How does this example make it more clear the axiom of extension says something non-trivial about belonging? I am trying very hard to understand his point as I am going to read more of Halmos's material and I need to understand his way of thinking.

 

[SteveL27]

Hi all. I still can't help but not stop thinking about the particular example Halmos gives. He says

"It is valuable to understand that the axiom of extension is not just a
logically necessary property of equality but a non-trivial statement about
belonging"

and then, he goes on to give the example of human beings and ancestors which is

"Suppose, for instance, that we consider human beings
instead of sets, and that, if x and A are human beings, we write x e A
whenever x is an ancestor of A. (The ancestors of a human being are his
parents, his parents' parents, their parents, etc., etc.) The analogue of the
axiom of extension would say here that if two human beings are equal,
then they have the same ancestors (this is the "only if" part, and it is
true), and also that if two human beings have the same ancestors, then
they are equal (this is the "if" part, and it is false)"

How does this example make it more clear the axiom of extension says something non-trivial about belonging? I am trying very hard to understand his point as I am going to read more of Halmos's material and I need to understand his way of thinking.

Others may have different answers, but I think this quote's a bit of a puzzler. Perhaps your best tactic is to just forget about it for now and move on. There's a lot of great stuff in Halmos. This paragraph seems a little convoluted and not really worth pursuing.

Just accept that a set is characterized by its membership and not by anything else.

 

[StatOnTheSide]

Thanks Steve. I am very happy to hear that there is one other person who feels the same way. I am pretty sure that a person like Halmos had an intention behind giving this example but since that is not very clear, I am going to move on.

Thanks much once again.