The acceptance of multiple infinities and some concepts around continuum (Dedekind cuts etc) are questionable due to this assumption.
We prove this by contradiction. Consider any function f: A -> P(A) , i.e. it takes elements of A and outputs subsets of A. Suppose this function is surjective: i.e. for all y in P(A) there is some x in A such that f(x) = y. But let q = { a in A | a is not in f(a) }. Clearly this is a valid set. And if f is surjective, there must be some x in A such that f(x) = q. Is x in q? If x were in q, then x would be in f(x), so that's a contradiction. If x were not in q, then by the definition of q x would be in q, which is also a contradiction. Thus we have a contradiction, so f cannot be surjective.
As you can see, nowhere do we make any logical jumps that would only make sense in the case of finite sets. This proof is as straightforward as the proof that there is no set of all sets. We don't use the axiom of choice, the argument is even valid in constructive mathematics (though you have to make some adjustments).
Now, we define one set X as being "less than" another set Y if and only if there is no surjective function from X to Y. You'll see that this definition corresponds exactly to the usual notions of size for finite sets, and makes intuitive sense (if for every y we have an x, there must be at least as many xs as ys). Now, just plug in the set of natural numbers into Cantor's theorem and you get: the power set of natural numbers is larger than the set of natural numbers.
fn contain(u0:(universal, set), u1:(universal, set)){ return false }
fn contain(u:(universal, set), s:set){ return true }
but don't how much logical sense it will make
But there is a way to get around Russel's paradox, the standard way is to define a new kind of object called a class. See https://en.wikipedia.org/wiki/Class_(set_theory) . Basically a class is like a set, a set can be a member of a class, and you can do most operations with sets on classes as well (union, intersection, etc). And you can create a class which is all sets satisfying some property. From this you can't have a set of all sets, but you can have a class of all sets. Russel's paradox doesn't go away, you still can't have the set of all sets (and you can't have the class of all classes), but this still gives you enough to talk about properties of all sets. See https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Bernays%E2... for an example of how to define classes. That particular example is a conservative extension of ZFC: any statement about sets that can be proved using NBG can also be proved using ZFC and vice versa (assuming ZFC is consistent). Your statement can't involve classes, because there is no definition of classes in ZFC so it wouldn't translate, but your proof in the NBG system can make use of it
This has always been my problem with the idea of some infinities being "bigger" than others. This definition does not correspond to the usual notion of size for finite sets, because finite sets actually have size. So rather than making "intuitive sense", it makes no sense (to me).
Having no surjective function from one set to another makes sense, but defining one set necessarily as "less than" the other does not, as the word "less" cannot, in such a context, mean what its definition implies.
> (if for every y we have an x, there must be at least as many xs as ys)
The lack of a surjective function does not mean that we cannot have an X for every Y, it just means we can't map an X for every Y with a surjective function. It doesn't mean a pairing cannot exist (at least not abstractly, which is how all infinite sets exist anyway). The pairing can simply be random and undefinable. There, now a pairing exists and both sets are the same "size". (It's not a surjective function, but if the point is simply to compare "sizes", then what does it matter?)
If sizes were sensible then for a set A composed of "every other integer" would be smaller than the a set B composed of "2 times every integer". If we were using calculus to take the limit of the ratio of sizes of the generating functions for A and B as the source set size goes to infinity then we find that A is in fact half the size of B. If we try to simply apply cardinality to A and B we find that the sets are exactly equal, since cardinality doesn't care about source sets or limits!
This can a big deal if say, you're calculating probabilities across an infinite number of possible events/event configurations. There is a big difference between a 50% chance, a 0.000001% chance and a 100% chance.
I'm not saying there's no use for cardinality and infinity classes, but they can easily be misapplied to allow you to be wrong, with confidence.
Sorry, but they simply don’t meet your (non-standard) definition of sensible. Both of the sets you mentioned can be interpreted as the set of even integers. This set is in bijection with itself (trivially), and thus is not considered to be strictly smaller than itself.
if this were true it would be trivial for you to give devastating examples
I strongly believe that statement to be false.
Can you provide a contradiction that is provable using Cantor's definition of the cardinality of sets?
If we accept infinity as a participant in the mathematical logic, the infinitesimal also deserves a role as a counter-party.
Infinitesimals have a much bigger role in widely-known mathematics than infinities do; they are the basis for calculus.
This is the magic that makes calculus work (from a formal standpoint).
I do think that infinitesimals are a much more intuitive way to understand calculus. It’s a shame it took so long to find positive numbers less than any real.