I will try to answer #3, but full disclosure, in set theory, I am just a mathematically gifted autodidact.
How can the Continuum Hypothesis be independent of the ZFC axioms? Why does the lack of “explicit” examples of sets with a cardinality between that of the naturals and that of the reals not guarantee that there are no examples at all? What would an “implicit” example even mean?
It seems to me that the literature on set theory focuses on the technical details and ignores the meaning of what they are doing. Which makes sense from certain perspective: it is easy to bullshit when you talk about the meaning of stuff, so you can signal seriousness by only doing the equations. Also, for the experts, the meaning is probably so obvious that it doesn’t seem to them like it needs to be communicated.
Unconcerned with expert status, here is my attempt to communicate what the set theory seems to be about, to me.
First, we have some intuition of set as a collection. Set of all natural numbers. Set of 4, 8, 17. Set of sets, as in “all possible pairs of natural numbers between 1 and 10”. So it seems like we talk about the same thing. Yet I bet that if you interrogated hundred people about their intuitions, you would get many different answers.
Can a set contain objects of a different type, such as the number 5, but also the set of 7, 9, and 11? The official ZF answer is “yes”, but probably many mathematicians do not actually have a practical use for such sets, so they would not mind a negative answer. Can a set contain itself? Some people would say “why not”, others would object that in their intuition, a set is an afterwards created collection of pre-existing objects, therefore no (ZF also says no). But it is also possible to answer positively. Can there be an infinity so insanely large that it cannot even be expressed in the terms of smaller infinities? (Technically: Can there be K which is larger than a limit of any sequence of numbers smaller than K where the length of the sequence is also smaller than K? Well, there is the omega, which is larger than a “limit” of any finite sequence of finite numbers, but other than that?) Some people would go like “why the heck would I need such infinity”, while others would be like “yeah, if we can describe them consistently, the richer the set theory, the better”. You can keep inventing questions like this, so you can have dozens of mutually different definitions of “a set” what comply with someone’s intuition.
Second, there is an ambition to express the axioms in the language of first-order logic. There are some good reasons to do that, but the problem is that first-order logic cannot even define what “a natural number” means. The ZF set theory is supposed to have well-founded ordinals, but you can’t express “well-founded” in first-order logic either! The Axiom of Foundation/Regularity, translated to first-order logic, is just not the same thing.
Thus, if we stick to the first-order logic, we are not actually talking about “sets” (in the sense of: collections compatible with some sane person’s intuition of a set), but about “mathematical structures that technically follow the axioms of the set theory, even if no sane person would actually call them sets”. The latter is usually abbreviated to “sets”. But it includes (mostly) monstrosities beyond imagination which technically happen to obey the dozen axioms that we chose.
Generally, “X independent of ABC axioms” means that the axioms are ambiguous, there are many possible structures that obey them, in some of them X is true, in others X is false. Among the monstrosities that follow the ZFC axioms, and no sane person would call most of them sets, in some of them the technical interpretation of Continuum Hypothesis is true, in others it is false.
The idea of “cardinality” itself does not mean what you might naively expect it to mean. Consider Skolem’s paradox: “there are countable models of uncountable sets”. In other words, sets that are obviously countable, but also technically uncountable. That’s because, technically, “same cardinality” is defined as “there is a bijection”, and “bijection” is defined as “a set of pairs”… so if you cleverly redefine “set” to exclude those inconvenient sets of pairs—tada! -- you have an obviously countable set which, technically, now does not have a bijection to the natural numbers, and therefore, technically, is uncountable.
So the “set with a cardinality between that of the naturals and that of the reals” could technically be some set of reals defined by a really good lawyer. (Unfortunately, I am not that good lawyer, yet, so I cannot give you the exact definition.)
Actually, this is also related to #35:
Does truth transcend proof? In what sense?
I think “proof” is a concept from first-order logic, and “truth” is a concept from second-order logic. But I am confused about this, so this is just a vague pointer towards the answer, not the answer itself.
Interesting questions!
I will try to answer #3, but full disclosure, in set theory, I am just a mathematically gifted autodidact.
It seems to me that the literature on set theory focuses on the technical details and ignores the meaning of what they are doing. Which makes sense from certain perspective: it is easy to bullshit when you talk about the meaning of stuff, so you can signal seriousness by only doing the equations. Also, for the experts, the meaning is probably so obvious that it doesn’t seem to them like it needs to be communicated.
Unconcerned with expert status, here is my attempt to communicate what the set theory seems to be about, to me.
First, we have some intuition of set as a collection. Set of all natural numbers. Set of 4, 8, 17. Set of sets, as in “all possible pairs of natural numbers between 1 and 10”. So it seems like we talk about the same thing. Yet I bet that if you interrogated hundred people about their intuitions, you would get many different answers.
Can a set contain objects of a different type, such as the number 5, but also the set of 7, 9, and 11? The official ZF answer is “yes”, but probably many mathematicians do not actually have a practical use for such sets, so they would not mind a negative answer. Can a set contain itself? Some people would say “why not”, others would object that in their intuition, a set is an afterwards created collection of pre-existing objects, therefore no (ZF also says no). But it is also possible to answer positively. Can there be an infinity so insanely large that it cannot even be expressed in the terms of smaller infinities? (Technically: Can there be K which is larger than a limit of any sequence of numbers smaller than K where the length of the sequence is also smaller than K? Well, there is the omega, which is larger than a “limit” of any finite sequence of finite numbers, but other than that?) Some people would go like “why the heck would I need such infinity”, while others would be like “yeah, if we can describe them consistently, the richer the set theory, the better”. You can keep inventing questions like this, so you can have dozens of mutually different definitions of “a set” what comply with someone’s intuition.
Second, there is an ambition to express the axioms in the language of first-order logic. There are some good reasons to do that, but the problem is that first-order logic cannot even define what “a natural number” means. The ZF set theory is supposed to have well-founded ordinals, but you can’t express “well-founded” in first-order logic either! The Axiom of Foundation/Regularity, translated to first-order logic, is just not the same thing.
Thus, if we stick to the first-order logic, we are not actually talking about “sets” (in the sense of: collections compatible with some sane person’s intuition of a set), but about “mathematical structures that technically follow the axioms of the set theory, even if no sane person would actually call them sets”. The latter is usually abbreviated to “sets”. But it includes (mostly) monstrosities beyond imagination which technically happen to obey the dozen axioms that we chose.
Generally, “X independent of ABC axioms” means that the axioms are ambiguous, there are many possible structures that obey them, in some of them X is true, in others X is false. Among the monstrosities that follow the ZFC axioms, and no sane person would call most of them sets, in some of them the technical interpretation of Continuum Hypothesis is true, in others it is false.
The idea of “cardinality” itself does not mean what you might naively expect it to mean. Consider Skolem’s paradox: “there are countable models of uncountable sets”. In other words, sets that are obviously countable, but also technically uncountable. That’s because, technically, “same cardinality” is defined as “there is a bijection”, and “bijection” is defined as “a set of pairs”… so if you cleverly redefine “set” to exclude those inconvenient sets of pairs—tada! -- you have an obviously countable set which, technically, now does not have a bijection to the natural numbers, and therefore, technically, is uncountable.
So the “set with a cardinality between that of the naturals and that of the reals” could technically be some set of reals defined by a really good lawyer. (Unfortunately, I am not that good lawyer, yet, so I cannot give you the exact definition.)
Actually, this is also related to #35:
I think “proof” is a concept from first-order logic, and “truth” is a concept from second-order logic. But I am confused about this, so this is just a vague pointer towards the answer, not the answer itself.