Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with?
If so, what does it mean to speak of such a set if it is not really one single concept with specific properties? What is then the justification to identfy "two" statements, which use such a set in two respective situations, in which in fact the underlying sets properties differ?
This is a somewhat philosophical question (especially the second part of it). The answer is a little bit yes, a little bit no and yes.
The real numbers can be used to index the number of subsets of $\mathbb N$. In set theory it is often useful to think of the real numbers as subsets or sequences of natural numbers. This much is provable from ZF, so whenever we say that $M$ is a model of ZFC we know that in $M$ there is a bijection between the set which $M$ thinks is $P(\mathbb N)$ and the set which $M$ thinks is $\mathbb R$. I'm using "thinks" because if we somehow extend $M$ or take another model contained in $M$ then these other models might have more or less sets in them.
It turns out that different models of ZFC could have a different amount of sets of natural numbers, and they can think differently on "how to well-order the sets". Namely there are models in which there are different numbers of sets and there are models in which the ordinals which are in bijection with the real numbers are different ordinals.
Confused? We'll get to that in a moment. Let us consider an analogy to this situation.
Imagine that you are living in $\mathbb Q$, and you ask yourself "How many numbers have the property $x^3=2$?" it turns out that in $\mathbb Q$ there are none. If we consider $\mathbb R$ then there is only one and if we consider $\mathbb C$ then there are three!
See how the number of solutions to a simple polynomial can change? Well, in set theory this is a lot more fundamental and complicated to grok because "how can a set has different number of subsets?" well, it can have different number of subsets in different models much like this polynomial has a different number of solutions in different models of the theory of fields.
So what we have is that in a given model there is a fixed cardinality, but between the models this cardinality might change and the contents of the set $P(\mathbb N)$ might change as well.
This game is really "internal" vs. "external" point of view. Given a model of ZFC that model, the internal point of view is that this model is the whole universe and there are no other sets. Much like the analogy of the rational numbers and the solutions to $x^3=2$, from within a model we are given a fixed number of solutions but we can always prove these solutions have something in common. From an external point of view we might know of properties that the model has which it cannot see or know about. If we have a countable model of set theory then there are only countably many real numbers, but this model does not know about a bijection between these sets, it "exists outside the model".
When we say $|\mathbb N|<|\mathbb R|$ we say every model knows that there is no function in the universe of the model which is a bijection between what the model thinks is the natural numbers and what the model thinks is the real numbers. Since this above fact is provable from ZFC it is true internally in every model of the theory.
This is why when we talk about the real numbers we are either adding "Assume that $|\mathbb R|=\aleph_2$" (or another axiom which tells us exactly how big the continuum will be) or that we do not refer to this possible $\aleph_\alpha$ but only to what we can ensure about it (e.g. $\alpha>0$).
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Asaf certainly covered the formal parts of it; I'll try to answer what I believe is the underlying issue in understanding.
For real numbers, we have a rather "intuitive" way of thinking what "all" of them are - every possible combination of digits in decimal, for instance.
But when we do formal mathematics and set theory - which we need to if we want to talk about concepts such as infinite cardinalities, then things are more complicated than that. What does the above mean when formalized?
There's a simple fact to be considered here: Our language - natural language, formal language of logic and pretty much anything else we can actually work with - is finite in nature. This means that no statement that we can make can distinguish every single element of an uncountable set. It's not a matter of the "right" language - whatever we humans can do and understand is finite in nature, fundamentally.
This also applies to set theory, and so what happens is that certain models of set theory leave some (or rather most) of these intuitive "real numbers" out simply because there's no formula to describe them (or more precisely, distinguish them from all other "real numbers"). And the same is, of course, true for bijections between (infinite) sets.
And since cardinalities basically are all about the existance of bijections (with the natural numbers, the reals or whatever else), the model of set theory we choose can change which sets and bijections "exist" and thus the answers to your questions.
What does that mean for the "real reals"? That, in the end, is a matter of belief.
Many mathematicians will take the pragmatic approach and work with what they need. It may seem worrying that this might mean their "reals" miss numbers - but consider that they are only missed because there is no way whatsoever to describe them. No formula, no definable function, not a zero of those or anything else. The fact that we have no way of describing them means they cannot cause any problems when missing, too - because if they did, we'd have a way to describe what is missing and thus the number in question.
Of course, there also is the realist's view that the "reals" as intuitively described exist, and in fact, many mathematicians will likely take that point of view unless they need set theoretic formalism for something.
Does the cardinality of sets, like for example the real numbers, depend on the fundamental axioms one is working with?
Well, yes, but as far as I am aware, cardinality, set and real numbers each only have one specific meaning, so I'm not sure what this question really means.
If so, what does it mean to speak of such a set if it is not really one single concept with specific properties? What is then the justification to identfy "two" statements, which use such a set in two respective situations, in which in fact the underlying sets properties differ?
Do you have an example in mind? I'm not sure exactly what you mean by this...
