After asking a question about how natural numbers can be defined, I became curious whether natural numbers can be defined without inductive definition. Is this impossible?
Also, is it possible to define a cardinal number without reference to ordinal numbers?
I really appreciate your help.
Thanks.
The strength of the natural numbers are the fact we can use them for inductions. Indeed the common theory for the natural numbers today is Peano Arithmetic which is pretty much based on induction.
In the language of rings and the theory of the real numbers by Tarski you cannot define the natural numbers internally, nor you can define them in the theory of algebraically closed fields (e.g. $\mathbb C$).
This means that any definition would require some sort of induction, furthermore induction is the basis for our mathematics. Even if you avoid it in one place you will still find it somewhere else.
Now as for defining cardinality, in theory you don't need the ordinals for defining cardinality. For a given set $A$ you can consider the class: $$|A|=\{B\mid\exists f\colon A\to B\text{ a bijection}\}$$
We can show that these are equivalence classes in an equivalence relation. However these are not sets, but proper classes which are too big for ZFC to handle. To overcome this we again have to resort to ordinals and the Foundation Axiom (known as Regularity sometimes) of ZFC. In the case of ZFC the whole thing essentially reduces to ordinals and the usual definition as well because we there is an ordinal in every class of this form and we can choose the minimal one.
You can read on that a bit more here: Defining cardinality in the absence of choice.
Even if you manage to define what a natural number is without induction (e.g. as the Dedekind-finite ordinals), you cannot define the set of natural numbers in ZFC without some form of induction somewhere.
This is because the Axiom of Infinity is explicitly about inductive sets, and you can't construct any infinite set without using it somehow. There are models of ZFC$-$Inf where all sets are finite.
It is possible to define the cardinals as follows. There is a class of sets $A$, with the property that any set $X$ is equinumerous to exactly one $X' \in A$. The class $A$ would be defined to be the set of cardinals. In ZFC, such as a cardinal assignment exists -- take $A$ to be a subclass of the ordinals.
