**The need to distinguish between languages of adequate expression and those of unambiguous communication**

We now highlight the need to distinguish between the essentially different roles and requirements of mathematical languages of rich and adequate expression (such as ZF) vis-à-vis those of effective and unambiguous communication (such as PA)—a distinction that may be significant for determining logical limits on the interpretation of number-theory in set-theory.

A particular limitation is reflected in the argument that PA does not admit a non-standard model of PA ^{[1]}.

This challenges the impression given by informal arguments in standard texts which suggest that we may trivially infer the existence of non-standard models of PA from standard set-theoretic theorems of first-order logic.

Another limitation is reflected in the argument that although every PA-theorem relativises to a corresponding ZF-theorem that holds over the finite ordinals if ZF is consistent and has a model, we cannot presume that if a ZF-theorem holds over the finite ordinals in a putative model of ZF, then we may conclude that a corresponding PA-theorem holds over the natural numbers similarly.

We show below that such unqualified extension of set-theoretic reasoning to number-theory can be both misleading and invalid.

**Why PA cannot admit a set-theoretical model**

Let denote the PA-formula:

where is the `successor’ of under any sound interpretation of PA.

If we assume that the standard interpretation of PA is sound, then this translates under any unrelativised interpretation of PA as:

If denotes an element in the domain of an unrelativised interpretation of PA, then either is , or is a `successor’.

Further, in every such interpretation of PA, if denotes the interpretation of :

(a) is true;

(b) If is true, then is true.

Hence, by Gödel’s completeness theorem:

(c) PA proves ;

(d) PA proves .

**Gödel’s Completeness Theorem:** In any first-order predicate calculus, the theorems are precisely the logically valid well-formed formulas (*i.e., those that are true in every model of the calculus*).

Further, by Generalisation:

(e) PA proves ;

**Generalisation in PA:** follows from .

Hence, by Induction:

(f) is provable in PA.

**Induction Axiom Schema of PA:** For any formula of PA:

In other words, except , every element in the domain of any unrelativised interpretation of PA is a `successor’. Further, can only be a `successor’ of a unique element in any such interpretation of PA.

**PA and ZF have no common model**

Now, since Cantor’s first limit ordinal, , is not the `successor’ of any ordinal in the sense required by the PA axioms, and if there are no infinitely descending sequences of ordinals ^{[2]} in a model—if any—of set-theory, PA and Ordinal Arithmetic ^{[3]} cannot have a common model, and so we cannot consistently extend PA to ZF simply by the addition of more axioms.

**Notes**

Return to 1: See this earlier post

Return to 2: Elliott Mendelson. 1964. *Introduction to Mathematical Logic*. Van Norstrand, Princeton, p.261.

Return to 3: ibid., p.187.

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