Formalizing infinity sets

[antaris]

I have a question about a mathematical problem of infinity sets.

It concerns a delicate issue, namely how infinite sets can be formalized in a manageable way.

Let us consider an infinite set (A), from which I approximate a finite region (B), that is, I more or less cut out a patch from (A) so that (B) becomes finite and I can continue working with (B). In itself, this seems to me logically completely unproblematic, since the object of interest ((B)) is finite.

My actual question, however, is how to formalize the complement of (B), i.e. (B' = A \setminus B), without running into any pitfalls related to infinity. It is completely unclear to me how infinity in this example can be understood formally.

I have read a little about complement systems/families and limiting procedures, since that seemed like a natural direction to me, but I am far from sure which lever I actually need to pull, or whether the problem can even be reduced to a single lever at all.

I would therefore be very grateful if someone could point me in the right direction, for example by suggesting primary sources or offering an explanation.

Thanks in advance!

 

[FactChecker]

When you talk about a "infinite", are you talking about a distance/metric, or are you talking about the number of elements in the set A?
If you are talking about the infinite number of elements in set A, you should read about Cantor's work and cardinal numbers.

 

[Dale]

Let us consider an infinite set (A), from which I approximate a finite region (B), that is, I more or less cut out a patch from (A) so that (B) becomes finite and I can continue working with (B). In itself, this seems to me logically completely unproblematic, since the object of interest ((B)) is finite.

So, for a concrete example you could have $A = \{ a \in \mathbb{N} \}$ the set of all natural numbers. Then you could define $B=\{ b \in A | b\le 10 \}$. A is infinite and B is finite.

My actual question, however, is how to formalize the complement of (B), i.e. (B' = A \setminus B), without running into any pitfalls related to infinity. It is completely unclear to me how infinity in this example can be understood formally

You could write $B'=\{ c \in A | c \notin B \}$. In this case B' is also infinite, like A.

 

[PeroK]

In simple terms, if the set of natural numbers is finite, then there is a largest whole number. If not, then the set must be infinite (that is to say, not finite).

More formally, the set is defined to be infinite by the Peano axioms. And, within set theory, there is the axiom of infinity, which fundamentally allows the existence of infinite sets.