Can we generalize Aleph numbers to non integer values? I'm really new to those kind of arguments so don't call me mad but I was wondering if there is a way to define an infinite set which cardinality is an Aleph-number like: $\aleph_{\frac 12}$. I have a sort of passion to generalization of math objects and I've been really astonished when I learned about fractals, whose dimensions are not an integer number so I was thinking if there is some way to build infinite sets (maybe with an unusual list of axioms) whose cardinality is not an Aleph-number with integer index  and how those numbers and sets would work. Any help or reference is appreciated :)
 A: This is not a real answer, but you could try to see if you can extend the $\aleph$ function from the ordinal surreal numbers onto the cardinal surreal numbers to the whole collection of positive surreal numbers with nice properties, and then study what $\widetilde{\aleph}_{\frac{1}{2}}$ is (for instance, is it an ordinal surreal number?). But with what properties, how, and what for?
A: Beware of over-generalizations! (I loved them, but I was quite wrong on that one.)

I think that your question has a nice interpretation. I'll take your phrase

if there is a way to define an infinite set which cardinality is an Aleph-number like: $\aleph_{\frac{1}{2}}$

to mean if it is reasonable to assume that infinite cardinalities may have the ordering of the rational numbers. Written as such, this is in contradiction to the Axiom of Choice (AC), as stated in the comments. In particular, AC implies that cardinalities are well-ordered.
But perhaps you may object: “Ok, but AC is overkill. I only need that cardinalities are totally ordered; that's enough for having a neat notion of size of sets, so I can compare any two of them”.
This may be called the Hypothesis of Cardinal Comparability. Now the punchline is, cardinal comparability implies AC! So, any universe of set theory (i.e., satisfying ZF) where cardinalities are totally ordered, they must be well-ordered.
So, if your function $r \mapsto \aleph_r$ is (strictly) increasing, its domain must be wellordered.
A beautiful introduction to all this stuff is this book.
A: Although the answers above are completely correct, I think the answerers are possibly missing part of the point of the OP's question, which was motivated by learning about fractals, which have fractional dimension.  The crucial part of that context is that to define the dimension of a fractal, you need an essentially new definition, one which agrees with the more "standard" notion of dimension for more "standard" shapes.
So while it is true that, as long as we are interested in sets and cardinality, there is no need to consider $\aleph_r$ for $r \notin \mathbb N$, I think what the OP is wondering about is:

Is there some generalization of set theory in which we consider "set-like" objects, for which there is a corresponding "cardinality-like" notion, with the properties that (a) for a genuine "set", the cardinality-like notion is equivalent to ordinary cardinality, but (b) for things that are not really sets but only "set-like", the cardinality-like notion might require "pseudocardinalities" that lie strictly between the alephs?

As far as I know no such generalization exists, but that doesn't mean it can't exist.  In particular I think it's worth noting that as far as cardinality goes, even a finite set has to have a cardinality within $\mathbb N$, so from that point of view one does not need non-natural numbers at all, let alone alephs with a rational index -- but of course we have plenty of other uses for rational and negative numbers besides answering the question "How many elements?"
