Can we define a collection of all things?

[Unknown Soldier]

Intuitively, a set of all things seems very possible if not actual. After all, if there are things, then there is all things. All we need to do is define a set that we designate the set of all things, and we are done. Easy enough.

But I'm afraid our intuition is failing us here. To see why we cannot define a set of all things, even if that set contains an infinite number of things, let's take a look at a simple set and apply some set theory to it.

A = {b, c}

As you can see, set A contains two elements: b and c. Mathematicians define for any set its subsets. The "family" of these subsets, the set of all set A's subsets, is called the power set of set A and is denoted, "P(A)." So in this case we have

P(A) = {{}, {a}, {b}, {a, b}}

Note that P(A) has more elements than set A has. In fact, P(A) has twice as many elements as set A has. For any finite set A of cardinality n (the number of elements), the cardinality of P(A) is 2^n. Even if set A is an infinite set, P(A) will always have more elements than set A has. So what this all means is that any set U we define as the set of all things must necessarily lack some things. Therefore, we cannot define a set of all things. The term "all things" is nonsensical because it cannot be defined.

 

[Swammerdami]

The Wikipedia article  Universal set discusses this; the first paragraph of the article is a summary:

In set theory, a universal set is a set which contains all objects, including itself.
In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.

This first paragraph alludes to  Russell's paradox, which is a stumbling block to the definition of a universal set. That paradox is usually avoided by forbidding a set to be a member of itself, but -- again as alluded to in the quote -- alternate axioms can dismiss that as a problem!

Set theory can get interesting, and ridiculously complicated, cf inaccessible cardinals.

One of the 20th century's premier mathematicians, Alexander_Grothendieck, explored the concept of universal set.

At the "other end" let's worry about the smallest sets. There is only one set of size zero, but what about size 1? Are {cat}, {dog}, {my dog Adolf} all sets of size 1? Or are their elements too vague; and we should stick to clear-cut mathematical objects?

Most approaches implicitly adopt the latter approach. We may want to work with sets of counting numbers, e.g.

Primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, ... }​

Or better yet, represent each counting number k as a set of size k built up exclusively from the null set:

0 = {}​

1 = { 0 } = { {} }​

2 = { 0, 1 } = { {}, { {} } }​

k = { 0, 1, 2, ... k-1 }​

Starting from these counting numbers we can build up all of the usual mathematical sets: the rational numbers, the reals, and so on. One example of the elegance that results is  Ordinal number. The ordinal numbers start with the 0, 1, 2, 3, ... just mentioned but after the first infinitely many, they get more interesting. An ordinal number is usually defined as the set of all "smaller" ordinal numbers.

 

[Unknown Soldier]

The Wikipedia article  Universal set discusses this; the first paragraph of the article is a summary:

In set theory, a universal set is a set which contains all objects, including itself.
In set theory as usually formulated, it can be proven in multiple ways that a universal set does not exist. However, some non-standard variants of set theory include a universal set.

This first paragraph alludes to  Russell's paradox, which is a stumbling block to the definition of a universal set. That paradox is usually avoided by forbidding a set to be a member of itself, but -- again as alluded to in the quote -- alternate axioms can dismiss that as a problem!

In the set theory I've studied, a universal set isn't necessarily a set of all objects but a set that contains at least all the elements of the sets I'm working with. So for example if I'm studying two sets
A = {1, 2}
and
B = {3, 4}
then the universal set might be
U = {1, 2, 3, 4}.
Of course, the universal set can affect set properties. For instance, if the universal set is the set of all natural numbers, then the set of upper bounds of A = {1, 2} is {2, 3, 4, ...}, all natural numbers greater than or equal to 2. If U = {1, 2, 3, 4} is the universal set, then the set of upper bounds of set A is {2, 3, 4}, a finite set.

Set theory can get interesting, and ridiculously complicated...

We have Georg Cantor to thank for that.

, cf inaccessible cardinals.

One of the 20th century's premier mathematicians, Alexander_Grothendieck, explored the concept of universal set.

At the "other end" let's worry about the smallest sets. There is only one set of size zero, but what about size 1? Are {cat}, {dog}, {my dog Adolf} all sets of size 1? Or are their elements too vague; and we should stick to clear-cut mathematical objects?

I've heard of this problem. For example, does it make sense to have a set of two apples when we know that no two apples are actually the same?

Most approaches implicitly adopt the latter approach.

One obvious exception is statistical research. When statisticians study apples, they would call a set of apples a "sample." Much of their work involves assuming that apples differ from each other in many ways which is the whole point of their analysis.

We may want to work with sets of counting numbers, e.g.

Primes = { 2, 3, 5, 7, 11, 13, 17, 19, 23, ... }​

Or better yet, represent each counting number k as a set of size k built up exclusively from the null set:

0 = {}​

1 = { 0 } = { {} }​

2 = { 0, 1 } = { {}, { {} } }​

k = { 0, 1, 2, ... k-1 }​

Starting from these counting numbers we can build up all of the usual mathematical sets: the rational numbers, the reals, and so on. One example of the elegance that results is  Ordinal number. The ordinal numbers start with the 0, 1, 2, 3, ... just mentioned but after the first infinitely many, they get more interesting. An ordinal number is usually defined as the set of all "smaller" ordinal numbers.

Hmmm. I haven't seen much of that before.