**2. Elementarity Axiom**

Let S be a topos of sets, i.e., an elementary topos with a natural number object satisfying well-pointedness and the axiom of choice (See [1], which is based on the idea in [6]). We make use of an endofunctor U : S −→ S with a natural transformation *υ* : *Id*S −→ U satisfying two axioms, "elementarity axiom", and "idealization axiom".

Elementarity Axiom: U preserves all finite limits and finite coproducts.

**Remark 1.** U *does not necessarily preserve power sets. This is the reason for the name of "elementarity".*

It is easy to see that "elementarity axiom" implies the preservation of many basic notions, such as elements, subsets, finite cardinals (in particular, the subobject classifier 2), and propositional calculi. Moreover, the following theorem holds.

**Theorem 1.** U *is faithful.*

**Proof.** It preserves diagonal morphisms and complements.

**Theorem 2.** *For any element x* : 1 −→ *X, <sup>υ</sup>X*(*x*) = U(*x*) ◦ *υ*1*.*

**Proof.** By naturality of *υ*.

**Corollary 1.** *All components of υ are monic.*

From the discussion above, a set *X* in S is to be considered as a canonical subset of U(*X*) through *υX* : *X* −→ U(*X*). Hence, U(*f*) : U(*<sup>X</sup>*0) −→ U(*<sup>X</sup>*1) can be considered as "the function induced from *f* : *X*0 −→ *X*1 through *<sup>υ</sup>*."

**Definition 1.** *Let A*, *B be objects in* S*. The function evA*,*<sup>B</sup>* : *A* × *B<sup>A</sup>* −→ *B satisfying*

$$ev\_{A,B}(a,f) = f(a)$$

*for all* (*a*, *f*) ∈ *A* × *B<sup>A</sup> is called the evaluation (for A*, *B) . The lambda conversion g*ˆ : *Z* −→ *Y<sup>X</sup> of g* : *X* × *Z* −→ *Y is the function satisfying*

$$\mathfrak{g} = ev\_{X,Y} \circ (1\_X \times \mathfrak{g})\_Y$$

*where* 1*X* × *g denotes the function satisfying* ˆ

$$(1\_X \times \mathfrak{g})(x, z) = (x, \mathfrak{g}(z)).$$

*We define a family of functions <sup>κ</sup>A*,*<sup>B</sup>* : U(*B<sup>A</sup>*) −→ U(*B*)<sup>U</sup>(*A*) *in* S *by the lambda conversion of* <sup>U</sup>(*evA*,*<sup>B</sup>*) : U(*A* × *B<sup>A</sup>*) ∼= U(*A*) × U(*B<sup>A</sup>*) −→ U(*B*)*.*

The theorem below means that *<sup>κ</sup>A*,*<sup>B</sup>* ◦ *<sup>υ</sup>B<sup>A</sup>* represents "inducing U(*f*) from *f* through *υ*" in terms of exponentials.

**Theorem 3.** *Let f* : *A* −→ *B be any function in* S*. Then,*

$$
\kappa\_{A,B} \circ \upsilon\_{B^A}(\widehat{f}) = \widehat{\mathcal{U}(f)}.
$$

*(Here, denotes the lambda conversion operation.)*

**Proof.** Take the (inverse) lambda conversion of the left hand side of the equality to be proved. It is *ev*U(*A*),U(*B*) ◦ (*id*U(*A*) × *<sup>κ</sup>A*,*<sup>B</sup>*) ◦ (*id*U(*A*) × *<sup>υ</sup>B<sup>A</sup>* ) ◦ (*id*U(*A*) × *f* ). By the naturality of *υ* and functorial properties of U, it is calculated as follows:

**Corollary 2.** *<sup>κ</sup>A*,*<sup>B</sup>* ◦ *<sup>υ</sup>B<sup>A</sup> is monic.*

**Notation 1.** *From here, we omit υ and κ.* U(*f*) : U(*X*) −→ U(*Y*) *will be often identified with f* : *X* −→ *Y and denoted simply as f instead of* U(*f*)*.*

**Theorem 4.** *Let P* : *X* −→ 2 *be any proposition (function in* S*). Then,*

$$\forall\_{x \in X} P(\mathbf{x}) \Longleftrightarrow \forall\_{x \in \mathcal{U}(X)} P(\mathbf{x}) \dots$$

**Proof.** *P* : *X* −→ 2 factors through "*true* : 1 −→ 2 if and only if *P* : U(*X*) −→ 2 factors thorough "*true* : 1 −→ 2.

Dually, we obtain the following:

**Theorem 5.** *Let P* : *X* −→ 2 *be any proposition (function in* S*). Then,*

$$\exists\_{\mathbf{x}\in\mathcal{X}}P(\mathbf{x}) \Longleftrightarrow \exists\_{\mathbf{x}\in\mathcal{U}(\mathcal{X})}P(\mathbf{x})\dots$$

The two theorems above are considered as the simplest versions of "transfer principle". To treat with free variables and quantification, the theorem below is important. (The author thanks Professor Anders Kock for indicating this crucial point.)
