If by cardinal you mean an initial ordinal (an ordinal not equinumerous with any smaller ordinal), then your new scheme is merely an instance of the scheme for ordinals. Indeed, you can see it clearly as a special case, if you realize that under AC every cardinal is $\aleph_\alpha$ for some ordinal $\alpha$, and so your proposed cardinal scheme is asserting that if $P(\aleph_\beta)$ for all $\beta\lt\alpha$, then $P(\aleph_\alpha)$ holds. (And actually, the new scheme is equivalent to the old scheme, if you consider replacing $\alpha$ with $\aleph_\alpha$.)
Without AC, however, there is a more general concept of cardinal, by which is meant something like the equinumerosity class of a set. If $Y$ is a set, then a smaller cardinality amounts to a set $X$ such that $X$ injects into $Y$ but not conversely. Without AC, these cardinals are not necessarily well-founded. Thus, the transfinite induction scheme fails for these more general types of cardinals.
For example, it is consistent with ZF that there are infinite sets that are Dedekind finite, so that they are not bijective with any proper subset of themselves. Let $P(X)$ assert that $X$ is not infinite Dedekind finite. If $Y$ is any set and is infinite Dedekind finite, then for any $a\in Y$ the set $X=Y-\{a\}$ is strictly smaller in size (else there is a countable subset of $Y$ by iterating the bijection) and $X$ is also infinite Dedekind finite. In other words, the property $P(X)$ satisfies the induction scheme for cardinalities, but does not hold of all cardinalities (if there are some infinite Dedekind finite sets). So transfinite induction for cardinalities can fail without the Axiom of Choice.
As part of his question, Bell Crowell correctly observes:
"Section 9 of the Ehrlich paper discusses the relationship between R∗ and No within NBG. He presents Keisler's axioms for the hyperreals, which basically say that they're a proper extension of the reals, the transfer principle holds, and they're saturated. At the end of the section, he states a theorem: "In NBG [with global choice] there is (up to isomorphism) a unique structure ⟨R,R∗,∗⟩ such that [Keisler's axioms] are satisfied and for which R∗ is a proper class; moreover, in such a structure R∗ is isomorphic to No.""
At that time I made it absolutely clear that the first part of the result is due to H.J. Keisler (1976) and that my modest contribution is to point out the relation (as ordered fields) between R* and No. The work of Keisler and the relation of my work to it seem to be lost in the remarks of Vladimir.
Of course, attributing the result to Keisler, as I remain entirely confident I correctly did, does not diminish the subsequent important contributions of others.
Edit: Readers interested in reading the paper including the discussion of Keisler's work may go to: http://www.ohio.edu/people/ehrlich/
EDIT: Since Vladimir appears to insist in his comment below that Keisler DOES NOT discuss proper classes in 1976, I am taking the liberty to quote Keisler and some of the relevant discussion from my paper. I will leave it to others to decide if I am giving Keisler undue credit.
Following his statement of his Axioms A-D of 1976--the function axiom, the solution axiom, and the axioms the state that R* is proper ordered field extension of the complete ordered field R of real numbers--Keisler writes:
“The real numbers are the unique complete ordered field. By analogy, we would like to uniquely characterize the hyperreal structure ⟨R,R∗,∗⟩ by some sort of completeness property. However, we run into a set-theoretic difficulty; there are structures R* of arbitrary large cardinal number which satisfy Axioms A-D, so there cannot be a largest one. Two ways around this difficulty are to make R* a proper class rather than a set, or to put a restriction on the cardinal number of R*. We use the second method because it is simpler.” [Keisler 1976, p. 59]
With the above in mind, Keisler sets the stage to overcome the uniqueness problem by introducing the following axiom, and then proceeds to prove the subsequent theorem.
AXIOM E. (Saturation Axiom). Let S be a set of equations and inequalities involving real functions, hyperreal constants, and variables, such that S has a smaller cardinality than R*. If every finite subset of S has a hyperreal solution, then S has a hyperreal solution.
KEISLER 1 [1976]. There is up to isomorphism a unique structure ⟨R,R∗,∗⟩ such that Axioms A-E are satisfied and the cardinality of R* is the first uncountable inaccessible cardinal.
If ⟨R,R∗,∗⟩ satisfies Axioms A-D, then R* is of course real-closed. It is also evident that, if ⟨R,R∗,∗⟩ further satisfies Axiom E, then R* is an $\eta_{\alpha}$-ordering of power $\aleph_{\alpha}$, where $\aleph_{\alpha}$ is the power of R*. Accordingly, since (in NBG) No is (up to isomorphism) the unique real-closed field that is an $\eta_{On}$-ordering of power $\aleph_{On}$, R* would be isomorphic to No in any model of A-E that is a proper class (in NBG).
Motivated by the above, in September of 2002 we wrote to Keisler, reminded him of his idea of making “R* a proper class rather than a set”, observed that in such a model R* would be isomorphic to No, and inquired how he had intended to prove the result for proper classes since the proof he employs, which uses a superstructure, cannot be carried out for proper classes in NBG or in any of the most familiar alternative class theories.
In response, Keisler offered the following revealing remarks, which he has graciously granted me permission to reproduce.
"What I had in mind in getting around the uniqueness problem for the hyperreals in “Foundations of Infinitesimal Calculus” was to work in NBG with global choice (i.e. a class of ordered pairs that well orders the universe). This is a conservative extension of ZFC. I was not thinking of doing it within a superstructure, but just getting four objects R, R*, <*, * which satisfy Axioms A-E. R is a set, R* is a proper class, <* is a proper class of ordered pairs of elements of R*, and * is a proper class of ordered triples (f,x,y) of sets, where f is an n-ary real function for some n, x is an n-tuple of elements of R* and y is in R*. In this setup, f*(x)=y means that (f,x,y) is in the class *. There should be no problem with * being a legitimate entity in NBG with global choice. Since each ordered triple of sets is again a set, * is just a class of sets. I believe that this can be done in an explicit way so that R, R*, <*, and * are definable in NBG with an extra symbol for a well ordering of V." [Keisler to Ehrlich 10/20/02]
Moreover, in a subsequent letter, Keisler went on to add:
I did not do it that way because it would have required a longer discussion of the set theoretic background. [Keisler to Ehrlich 5/14/06]
Best Answer
Andrej Bauer asked in a comment, "What is a model of open induction?" Let me give the most modern and relevant answer and then explain the term "open induction", which is somewhat archaic.
Let $F$ be an ordered field. An (ordered) subring $R$ of $F$ is called an "integer part" of $F$ if
$R$ is discretely ordered: This means the inequality $x < y < x+1$ has no solution in $R$. Equivalently, nothing in $R$ lies between 0 and 1.
For all $x\in F$ there is some $r\in R$ such that $r\le x < r+1$.
Item 2 is equivalent to saying that every element of $F$ is a finite distance from some element of $R$, where "finite" means bounded by an element of $\mathbb{Z}$. Remember that $F$ can and usually will be nonarchimedian.
Items 1 and 2 together imply that there is a unique function $\lfloor\cdot\rfloor$ from $F$ to $R$ given by the inequality $\lfloor x \rfloor \le x < 1+\lfloor x \rfloor$. Think of this as an abstract analog of the integer part operator.
A "model of open induction" is an integer part of a real closed field. Mourges and Ressayre proved that every real closed field has an integer part. The term "open induction" comes from a paper of Shepherdson, who started this whole topic by proving that an ordered ring $R$ is an integer part of its real closure if and only if the positive semiring of $R$ satisfies the axioms of Peano Arithmetic, with the induction axioms restricted to quantifier-free (i.e. open) formulas. Shepherdson also gave recursively presentable nonstandard models of open induction, which is interesting because according to Tennenbaum's Theorem there are no such models of Peano Arithmetic.
Since then there have been many successful attempts to build recursive nonstandard models of theories a little stronger than open induction. The champ, to-date, is a recursive nonstandard normal model of open induction, given in a paper by Otero and Berarducci. Here "normal" means integrally closed in its quotient field.
The main unsolved problems concerning open induction, in my opinion, are (1) Is the universal theory of open induction decidable (i.e. is it decidable whether a diophantine equation has a solution in some model of open induction) and (2) Is there a recursive non-standard diophantine correct model of open induction, where "diophantine correct" means "extends to a model of true arithmetic".