Nouns are categories into which we put other nouns. Sets are collections of objects. Sets can also be elements of bigger sets. There is nothing inventive about me saying so other than that thinking about one can help us feel more warmly about the other.
In logic, when we say: ‘The horse is an animal,’ we are simply saying that for any individual that is a horse it belongs to a class of animals. Complex sentences and action verbs can be reduced to this talk: ‘The apples have dropped onto the ground,’ is ‘Something that belongs to the class of apples is such that it belongs to things that have dropped onto the ground.’ The efficiency of such language cannot compete with the efficiency of natural language, but this is quite irrelevant for the purposes for which sentences are analysed in this manner. A lot of our speech is simply set talk.
Nouns are countable and uncountable. So are sets. The parallels are more intriguing than they are exact, but they are there nevertheless. A set is countable when it is a finite set or a countably infinite set, that is, it has the same size as the set N of natural numbers. Nouns are countable when we can count the objects they refer to.
Some people get confused about this business of counting. As in grammar so in mathematics, ‘to count’ does not necessarily mean to say how many, although this is indeed what it means in the case of finite sets. We say we can count the elements of an infinite set A if we can find a function from N (set of natural numbers) to A that is both one-to-one and onto. In other words the sets are the same size, A = N, or have the same cardinality. A very close equivalent of a countably infinite set in language is the plural form of a noun, for example:
apples
When we use it like that, without the definite article or any other restricting phrase, we are talking about an infinite collection of apples, yet one for which we can easily find a one-to-one and onto function of the type f: N → A. Suppose we want to name the finite set. In that case, we say:
the apples
There is then a natural number n such that f: n → A. The expression:
apple sauce
is an example of an uncountable noun, matching closely the definition of an infinite set which is not countable. That is, we can’t even begin to think of how we would go about finding a function from the set of natural numbers N to the set S (set representing apple sauce). However, since we can also say:
the apple sauce
as in ‘I spilled the apple sauce,’ it would follow that there must be some kind of finite dimension of this otherwise uncountable infinite set. Well, that’s where the parallel is not exact in my view, because as far as I know there are no such things in set theory, but we can, since we have defined S as a set, using Cantor’s axiom of existence and axiom of equality, construct the set {S}, whereby X = {S}, which is equivalent to a container with one element in it. Hence, we can find a function from N to X. And that is not a far cry from what we mean when we say ‘I spilled the apple sauce,’ – the container we spilled it from defines the apple sauce.
Enough about apple sauce. This rambling has focused (if rambling can be focused) on the property of nouns and sets involving countability, but the operations of union and intersection of sets, subsets, and power sets also have parallels in the realm of nouns.
In logic, when we say: ‘The horse is an animal,’ we are simply saying that for any individual that is a horse it belongs to a class of animals. Complex sentences and action verbs can be reduced to this talk: ‘The apples have dropped onto the ground,’ is ‘Something that belongs to the class of apples is such that it belongs to things that have dropped onto the ground.’ The efficiency of such language cannot compete with the efficiency of natural language, but this is quite irrelevant for the purposes for which sentences are analysed in this manner. A lot of our speech is simply set talk.
Nouns are countable and uncountable. So are sets. The parallels are more intriguing than they are exact, but they are there nevertheless. A set is countable when it is a finite set or a countably infinite set, that is, it has the same size as the set N of natural numbers. Nouns are countable when we can count the objects they refer to.
Some people get confused about this business of counting. As in grammar so in mathematics, ‘to count’ does not necessarily mean to say how many, although this is indeed what it means in the case of finite sets. We say we can count the elements of an infinite set A if we can find a function from N (set of natural numbers) to A that is both one-to-one and onto. In other words the sets are the same size, A = N, or have the same cardinality. A very close equivalent of a countably infinite set in language is the plural form of a noun, for example:
apples
When we use it like that, without the definite article or any other restricting phrase, we are talking about an infinite collection of apples, yet one for which we can easily find a one-to-one and onto function of the type f: N → A. Suppose we want to name the finite set. In that case, we say:
the apples
There is then a natural number n such that f: n → A. The expression:
apple sauce
is an example of an uncountable noun, matching closely the definition of an infinite set which is not countable. That is, we can’t even begin to think of how we would go about finding a function from the set of natural numbers N to the set S (set representing apple sauce). However, since we can also say:
the apple sauce
as in ‘I spilled the apple sauce,’ it would follow that there must be some kind of finite dimension of this otherwise uncountable infinite set. Well, that’s where the parallel is not exact in my view, because as far as I know there are no such things in set theory, but we can, since we have defined S as a set, using Cantor’s axiom of existence and axiom of equality, construct the set {S}, whereby X = {S}, which is equivalent to a container with one element in it. Hence, we can find a function from N to X. And that is not a far cry from what we mean when we say ‘I spilled the apple sauce,’ – the container we spilled it from defines the apple sauce.
Enough about apple sauce. This rambling has focused (if rambling can be focused) on the property of nouns and sets involving countability, but the operations of union and intersection of sets, subsets, and power sets also have parallels in the realm of nouns.