Previous section : Non-standard
models of Arithmetic

We shall use previously defined concepts of meta
(1.3), predicate
(1.4), invariant (1.5, i.e. definable),
«ground»
(1.8), and the notation *T*⊢*F* to mean that *F*
is provable in *T*.

The idea is that, in any theory able to describe expressions, any expression can also take, as an argument designating an expression, a quotation of itself, this quotation included.

**Theorem.** Let *T* a theory able to describe
itself, in the sense that it defines in (every one of) its
model M an *image of itself*, thought of
as «a theory» *T '* where :

- Each expression
*F*in the language of*T*, can be quoted, i.e. we can write a ground term*t*_{F}also in the language of*T*, whose value is the copy [*F*] of*F*in*T*′, - It can describe the quotation process in
*T '*, by a unary term*J*of*T*, taking argument in some class of expressions of*T '*, and values in the class of ground terms of*T '*: for any expression*F*we have*T*⊢*J*(*t*_{F})=*t*._{tF} - It can operate formal substitutions (compositions) in
*T '*: given a variable*A*in a class of expressions of*T '*with a distinguished free variable, and*u*in the class of ground terms of*T '*, the operation (*A*:*u*) gives the expression in*T '*obtained by replacing the variable of*A*by*u*. So, for any unary formula*F*and ground term*K*of*T*we have*T*⊢ (*t*:_{F}*t*) =_{K}*t*_{F(K)}.

Proof.

LetH(A) be the expressionF(A:J(A)) with variableAranging over the class of expressions ofT 'with the same type (inT ') asF(in T), and thus asH(without specifying the range of their distinguished variable).

Then letXbe the term (t_{H}:J(t_{H})). ThusG, that isF(t_{H}:J(t_{H})), can also be abbreviated asH(t_{H}). ThusT⊢ (t_{H}:J(t_{H})) = (t_{H}:t) =_{tH}t_{G}∎

Proof : by the self-quotation theorem, taking for

Intuitively, if we read

**Corollary. ***If M is explicitly built
(meta-invariant) determining values of ground formulas F
of T (or if M is any model agreeing with specific
invariant determinations of these values), then the computation
v[F] in M of these values by the same
rules (if possible) is incorrect. *

In particular, the construction in the proof of the Completeness theorem necessarily makes a «wrong choice» at some step. Namely, it is the provability predicate (whether each formula is refutable from previous axioms) used in each step of its construction, that is the weakly defined concept: it is a non-algorithmic operation, differently interpreted between models. Of course it will keep its unique true interpretation as long as the set of natural numbers remains standard. From this we deduce that any model of a founding theory (arithmetic or set theory) obtained by such a construction, necessarily contains non-standard numbers, that modify the first-order properties of the set of natural numbers (with addition and multiplications).

Finally, any construction of a model should be understood as implicitly depending on some (arbitrary) preexisting universe, with its set of natural numbers whose character of "being standard" cannot be defined in the absolute.

**Theorem. ***If T ′ is
built like T which can express the provability p
in T ′, and F is a
ground formula, *⊢

1) (T

2) (

3) (

Proofs:

1) The proof of *F* in *T* can be copied into *T* ′, or converted to a proof of existence of a
proof.

2) Let *T*⊢ (*G* ⇎ *p*[*G*]). The proof (*T*⊢
*G*) ⇔ (*T*⊢ *G*∧*p*[*G*]) ⇔ (*T*⊢
0) formalized in *T* gives *T*⊢(*p*[*G*]⇔*p*[0]).
Thus (*T*⊢ ¬*p*[0]) ⇔ (*T*⊢ ¬*p*[*G*]) ⇔
(*T*⊢ *G*) ⇔ (*T*⊢ 0).

3) from 2) applied to the theory (*T*+¬*F*). Finally
take *F*=«*F* is provable», ie *T*⊢ (*F* ⇔ *p*[*F*]).
∎

No converse for 1): a proof in

One might object that the countable model made by the
construction of the completeness theorem, only simulates a
powerset, not of the true ℕ, but of a non-standard model of ℕ.
Indeed, the models provided by the precise construction we
described in the proof of the Completeness theorem, all contain
non-standard natural numbers.

However, the argument of the Skolem's paradox still holds in 2
ways.

On the one hand, by the fact that it is still anyway a countable
simulation of the powerset of a countable set : as (inside a fixed
model of set theory) bijections between sets provide bijections
between their powersets, the oddity of this being lost (thus
getting a "powerset" that cannot be the "true" one) when comparing
the interpretations of countability between different models,
remains intact. We can also consider the correspondance by the embedding of the
standard (external) ℕ into the internal (non-standard) one : being
externally countable, the internal "*P*(ℕ)" is insufficient
not only to exhaust the external powerset of the internal ℕ, but
also (by restriction of the ∈ predicate) that of its (external)
subset of standard numbers.

On the other hand, for example, in the framework of set theory
with the axiom of choice, there is another construction of a
countable model of (the first-order theory of) second-order
arithmetic, with the "true" (standard) ℕ but a different
(countable) *P*(ℕ), with an elementary embedding from this
model (ℕ, *P*(ℕ)) to the "true" one.

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