Need for rigorous meaning in English for definitions from my reference book

Question

I need verification for some expression in definitions which I can't completely get. Although English isn't my natal language, I still find a confusion with these words, with which just a little bit of punctuation or tonnality can flip upside down the meaning of the mathematical concept.

1. (...) Let $ S $ be the class of all sets; (...)
2. (...) for $ A,B \in S $, $ hom(A,B) $ is the set of all functions $ f $ such that (...)
3. (...) Let $ G $ be the category whose objects are all groups. (...)

The expression in bold keep appearing, and as long as I'm aware of the subject, sometimes meanings are like contradicting. So I wonder if:

1. They mean by the (1) that "all this class contains is some sets" OR "this is the class constructed by all possible sets",

2. "this set is of all about function-type objects inside" OR "this is the set containing every theoretically possible function"

3. "whose objects are all some elements qualified as groups" OR "is the category as the whole group-universe's elements as objects"

I tried above to reformulate the subtilities for the same expression different times.

Thank you for a helping hand.

Answer 1

Consider some other sentences of the same form:

1. The sets of all integers is denoted by $\mathbb{Z}$.

2. Then $\mathbb{N}$ is the set of all $x \in \mathbb{Z}$ with $x \ge 0$.

3. Let $P$ be the set whose elements are all prime numbers.

To me 1. and 2. are clear as "of all" will always mean that your consider the totality of things with a given property.

Or consider for example the list of all questions asked on this site.

The construct in 3. is less clear.

One can say "These are all questions of this site." to express both the entirety of the questions of this site and the fact that an otherwise specified collection of questions all have the property of being questions on this site.

1. "Do you have a backup of the questions?" "Yes, it's file Q. These are all questions of this site."

2. "These questions are not about mathematics." "You think so? These are all questions of this site."

In this case one has to infer from context what is meant. In your case it says "Let $G$ be the category" thus $G$ is determined uniquely by what follows, which is only the case under the interpretation that we consider the entirety of groups.

If somebody says "let $F$ be the set of integers whose elements are all integers divisible by $5$" it is clear they mean the set formed by the entirety of integers divisible by $5$.

Yet, if somebody would say "let $F$ be a set of integers whose elements are all integers divisible by $5$", I'd tend to assume that the mean whatever set of integers with the property that each number in the set is divisible by $5$ but I'd find this a bit ambiguous.

Answer 2

Looking just at the language aspect:

"The $A$ of all $B$ 's" is unambiguous as long as you know the mathematical meaning of an $A$ and a $B$. There is only one $A$ (because of the), and every possible instance of a $B$ belongs to it.

"The category whose objects are all groups" is awkwardly worded. Grammatically, it could mean either of:

• the category all of whose objects are groups.
• the category which has all possible groups as its objects.

The first interpretation is a much more natural way to read the words than the second, but both are possible. So the issue is which one makes mathematical sense when read in context.

The ambiguity arises since all can be read as applying either to groups or to objects: the same sequence of words fits two different grammatical structures, namely

• [objects are all] groups
• [objects are] [all groups].

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Edit for linguists: strictly speaking, objects are all and objects are aren't proper sentence constituents. But I wanted to use brackets to indicate the ambiguity, without changing the word order. The way are is sandwiched between objects and all prevents using strictly correct constituents.