Set theoretic definition of a singleton.

[matheinste]

Hello all.

While reading about ordered pairs in the context of the formal language of axiomatic set theory I came across the following in a list of abbreviations. The book is The Foundations of Mathematics by Simpson. Page 93, although I have seen this definition elsewhere.

(singleton) z={x} is an abbreviation for z={x,x}

What does this mean. I always thought that a set with a single member was just a set with a single member. Does singleton have some other meaning.

Thanks.

Matheinste.

 

[rasmhop]

Why don't you think that {x,x} has a single element? When people are informally introduced to set theory they often don't see why {x} or {x,y,z} may not be sets (they turn out to be though, but only because of a number of axioms which are rarely mentioned or even thought of by beginners). In the normal axiomization of set theory we formally state how to construct sets. Basically all sets are constructed from the following axioms:
1. There exists a set.
2. There exists an inductive set.
3. Given a set S and a predicate p(x) we can construct the subset of S satisfying p.
4. Given two sets x,y we can construct the set {x,y}.
5. Given a set X we can construct the union of its elements.
6. Given a set S we can construct its power set.
7. (possibly the axiom of choice)
Nowhere does it state that singletons exists. However we can construct pairs, so we can construct the pair P={x,x} which is a singleton since two sets are equal if and only if they have the same elements, but P only have x as an element so it's indifferent from what you call {x}. Thus we usually introduce the notation {x} to mean {x,x} since that matches our intuitive understanding of it. Similarly we haven't defined {x,y,z} and frankly we're not sure that a set with x,y and z exists. We can however show that it exists since pairs exists we know that {x,y} and {y,z} exists, but then {{x,y},{y,z}} exists, and since we can take the union of the elements of a set we can take the union of {x,y} and {y,z} which is what we often write as {x,y,z}.

 

[matheinste]

Thanks rasmhop.

Its my bedtime now but a quick glance at your reply sems very enlightening. I will study it in deail later.

Thanks.

Matheinste

 

[CRGreathouse]

I imagine a singleton is defined as a pair {x, x} because you have an axiom of pairing (or the like).

 

[matheinste]

Why don't you think that {x,x} has a single element? When people are informally introduced to set theory they often don't see why {x} or {x,y,z} may not be sets (they turn out to be though, but only because of a number of axioms which are rarely mentioned or even thought of by beginners). In the normal axiomization of set theory we formally state how to construct sets. Basically all sets are constructed from the following axioms:
1. There exists a set.
2. There exists an inductive set.
3. Given a set S and a predicate p(x) we can construct the subset of S satisfying p.
4. Given two sets x,y we can construct the set {x,y}.
5. Given a set X we can construct the union of its elements.
6. Given a set S we can construct its power set.
7. (possibly the axiom of choice)
Nowhere does it state that singletons exists. However we can construct pairs, so we can construct the pair P={x,x} which is a singleton since two sets are equal if and only if they have the same elements, but P only have x as an element so it's indifferent from what you call {x}. Thus we usually introduce the notation {x} to mean {x,x} since that matches our intuitive understanding of it. Similarly we haven't defined {x,y,z} and frankly we're not sure that a set with x,y and z exists. We can however show that it exists since pairs exists we know that {x,y} and {y,z} exists, but then {{x,y},{y,z}} exists, and since we can take the union of the elements of a set we can take the union of {x,y} and {y,z} which is what we often write as {x,y,z}.

Hello rasmhop,

I have read more closely your reply and am aware of the axioms except that I am not sure of the definition of an inductive set.

You say that people often don't see why {x} or {x,y,z}may not be sets. Perhaps that is where my misunderstanding comes from because I cannot see why either, if the objects concerned are just simple objects and not some sort of undefinable sets.

I am aware that {x,x} and {x,x,x}and so on are sets with only one member and so are equal to {x}. You say that nowhere do the axioms state that a singleton exists. But by the same reasoning can we also say that they do not state that sets containing two objects exist. I just cannot grasp why we have to define a singleton with reference to a pair. I want to understand because it seems fundamental to the whole idea of sets.

Just to clarify, I am assuming that a singleton is a set containing only one object and that this object may be “atomic” or a set.

Matheinste

 

[Hurkyl]

I just cannot grasp why we have to define a singleton with reference to a pair.

We have two options:
(1) Include a "singleton set operator" amongst the primitive terms of set theory, include an "axiom of the singleton set", and derive the its other properties as a theorem, including how the singleton set operator relates to the pair set operator.

(2) Define the "singleton set operator" in terms of the pair set operator, and prove its properties as a theorem.

Both amount to the same work, but the latter gives a smaller presentation of set theory, in the sense that it requires fewer primitive terms and fewer axioms.

 

[matheinste]

Thankyou Hurkyl,

I think your reply gives me an idea of what I should be looking for in the books that I have. I will get back when I have understood or need more help.

Thanks.

Matheinste.

 

[matheinste]

Thanks to all responders.

I see it now.

I think more attention to the axioms and less/no reliance on intuition is required. I suppose the clue is in the name, Axiomatic Set Theory.

Matheinste.