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## (Finite) Models of two subtheories of Peano Arithmetic

Consider first-order theory (with identity) of Peano Artithmetic built in the language $\{S,+,\times,0\}$ and with the following set of axioms: \begin{align} \neg Sx&=0\tag{1}\\ Sx=Sy&\rightarrow x=y\tag{2}\\ x+0&=x\tag{3}\\ x+S(y)&=S(x+y)\tag{4}\\ x\times 0&=0\tag{5}\\ x\times S(y)&=(x\times y)+x\tag{6} \end{align} plus the full induction schema.

Let $PA^{(-1)}$ be the subtheory of $PA$ (first-order Peano Arithmetic) which has all other axioms except for (1), similarly $PA^{(-2)}$ let be the subtheory without (2). It is rather easy, but nevertheless interesting, result that both this theories have finite models, $PA^{(-1)}$ even has the degenerate one-element model.

My question is: has any research been made towards characterization of class of models of the theories above? If yes, could please someone provide me with suitable information? I am particularly interested in finite models.

EDIT: Following J.D. Hamkins advice I explicitly stated the language and the axiomatization I am interested in.

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Since there are various standard equivalent formulations of PA, which are no longer equivalent when you drop axioms, it may help to be more explicit about what the axioms of your theories are. In particular, first-order PA is usually axiomatized in the language of ordered rings, in the language with $+, \cdot, 0,1$ and $\lt$, rather than in the language with unary successor $S$. (But of course we can define $S(x)$ in that language as $x+1$.) – Joel David Hamkins Nov 4 at 13:55
I agree with Joel... As stated, without addition and multiplication, the models are easy to characterize: Models of $PA^{(-1)}$ are of the form $(X;0,S)$ where $0 \in X$ and $S:X \to X-\lbrace0\rbrace$. Models of $PA^{(-2)}$ are of the form $(X;S)$ where $S:X\to X$ is an injection (and, if desired, $0$ is any element of $X$). – François G. Dorais Nov 4 at 15:59
Following J.D. Hamkins' advice I stated explicitly what language and what axiomatization I am interested in. – R.G. Nov 4 at 16:01
You should look at Leon Henkin, "On Mathematical Induction," The American Mathematical Monthly, Vol. 67, No. 4 (Apr. 1960), pp. 323-338. You might also find this interesting (in your axiomatiation, you're assuming the functionality and totality of the successor function, but these can also be removed): andrewboucher.com/papers/ga.pdf – abo Nov 4 at 18:07
@abo: Thanks for the references. – R.G. Nov 4 at 19:19