(I haven’t read this specific book, but I have read other books about sets.)
For starters, what exactly does the word “naive” mean, in the context of Naive Set Theory? It seems that different authors use this word a little differently, but generally it means: “I allow myself to skip the technical details when they become boring, and also the difficult topics because this is a book for beginners”.
Sets in set theory are not generally assumed to exist—with the sole exception of the axiom of infinity, our existential starting point. Sets in set theory are instead constructed in steps, from the axiom of infinity, using the existential conditionals of the remaining axioms of set theory.
I think the class of sets whose existence can be proved from axioms is called “constructible universe”. This is the minimum that must exist. But the axioms are not saying that other sets do not exist.
So if you can imagine other sets, whose existence is neither provable from the axioms, nor does it contradict them… then such sets may or may not exist. More precisely speaking, now we have multiple possible meanings of the word “set”. (Of course, if you add some new set, you also need to add all those sets whose existence can be proved from the axioms and the existence of this set.) This is basically what it means that some statements are not decidable from the axioms: depending on what meaning you choose, such statements can be either true or false.
I guess a specific example would be useful here. We have a set of all natural numbers. Now suppose that God flips a coin for each of these numbers, and creates a set “G” of those where the coin came up heads. (There were infinitely many coin flips. The coin is fair, so there were infinitely many heads, infinitely many tails, and there is no finite text that could describe the results exactly.) Does the set “G” exist? Your intuition might say yes, but it is not provable from the axioms, because all proofs have finite length, and therefore cannot specify objects that do not have a finite definition.
Observe that there is some method in this apparent madness; the number of elements in the sets 0, 1, or 2… is, respectively, zero, one, or two
Each natural number is represented by a set containing representations of all smaller numbers. This is convenient because it allows to express “a < b” as “the representation of a is a subset of the representation of b”. And it makes the definition of a successor very simple. Furthermore, this also works for (some?) infinite sets; for example ω is the smallest ordinal greater than all integers, and simultaneously a set of all integers (i.e. the smallest set containing all integers). -- In other words, for certain sets, you can use “smaller” and “subset” as synonyms; prove one statement, get the other statement for free.
the role of the axiom [of choice] is to guarantee that possibility in infinite cases...
Generally, it happens many times that something is obvious for finite sets, but for infinite sets it might be tricky to prove, or not true at all. You could see an infinite chain of arguments where each step is a correct proof and yet it does not help you, because proofs cannot be infinitely long.
Example, as an intuition pump: Imagine that there is a function that receives a set of natural numbers, and returns a natural number. We know that f(empty set) = 42, and f(any set with one element) = 42, and also for each two sets A and B if f(A) = 42 and f(B) = 42 then f(A union B) = 42. At this moment you may be tempted to say “okay, obviously the function returns a constant 42 no matter what”. But actually, all we know is that it returns 42 for finite sets; for infinite sets it could be anything.
(I haven’t read this specific book, but I have read other books about sets.)
For starters, what exactly does the word “naive” mean, in the context of Naive Set Theory? It seems that different authors use this word a little differently, but generally it means: “I allow myself to skip the technical details when they become boring, and also the difficult topics because this is a book for beginners”.
I think the class of sets whose existence can be proved from axioms is called “constructible universe”. This is the minimum that must exist. But the axioms are not saying that other sets do not exist.
So if you can imagine other sets, whose existence is neither provable from the axioms, nor does it contradict them… then such sets may or may not exist. More precisely speaking, now we have multiple possible meanings of the word “set”. (Of course, if you add some new set, you also need to add all those sets whose existence can be proved from the axioms and the existence of this set.) This is basically what it means that some statements are not decidable from the axioms: depending on what meaning you choose, such statements can be either true or false.
I guess a specific example would be useful here. We have a set of all natural numbers. Now suppose that God flips a coin for each of these numbers, and creates a set “G” of those where the coin came up heads. (There were infinitely many coin flips. The coin is fair, so there were infinitely many heads, infinitely many tails, and there is no finite text that could describe the results exactly.) Does the set “G” exist? Your intuition might say yes, but it is not provable from the axioms, because all proofs have finite length, and therefore cannot specify objects that do not have a finite definition.
Each natural number is represented by a set containing representations of all smaller numbers. This is convenient because it allows to express “a < b” as “the representation of a is a subset of the representation of b”. And it makes the definition of a successor very simple. Furthermore, this also works for (some?) infinite sets; for example ω is the smallest ordinal greater than all integers, and simultaneously a set of all integers (i.e. the smallest set containing all integers). -- In other words, for certain sets, you can use “smaller” and “subset” as synonyms; prove one statement, get the other statement for free.
Generally, it happens many times that something is obvious for finite sets, but for infinite sets it might be tricky to prove, or not true at all. You could see an infinite chain of arguments where each step is a correct proof and yet it does not help you, because proofs cannot be infinitely long.
Example, as an intuition pump: Imagine that there is a function that receives a set of natural numbers, and returns a natural number. We know that f(empty set) = 42, and f(any set with one element) = 42, and also for each two sets A and B if f(A) = 42 and f(B) = 42 then f(A union B) = 42. At this moment you may be tempted to say “okay, obviously the function returns a constant 42 no matter what”. But actually, all we know is that it returns 42 for finite sets; for infinite sets it could be anything.