## Abstract

Over the last century, the principle of “continuous induction” has been studied by different authors in different formats. All of these different forms are equivalent to one of the three versions that we isolate in this paper. We show that one of the three forms of continuous induction is weaker than the other two by proving that it is equivalent to the Archimedean property, while the other two stronger versions are equivalent to the completeness property (the supremum principle) of the real numbers. We study some equivalent axiomatizations for the first-order theory of real closed fields and show that some first-order formalization of continuous induction is able to completely axiomatize it (over the theory of ordered fields).

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## Acknowledgements

The authors are very much grateful to the anonymous referee of the Bulletin for carefully reading the article and for many helpful suggestions which improved the results and made the paper much more readable.

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Communicated by Ali Enayat.

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This is a part of the Ph.D. thesis of the second author written under the supervision of the first author at the University of Tabriz, Iran.

## Appendix

### Appendix

Here, we give a proof of Tarski’s theorem on the completeness of the theory of real closed (ordered) fields; this is a slightly modified proof given in [11].

### Theorem 5.1

\(({{{\mathbf {\mathtt{{IVT}}}}}}\) is \({\text {Complete}})\) The theory \({{{\mathbf {\mathtt{{OF}}}}}}+{{{\mathbf {\mathtt{{IVT}}}}}}\) is complete.

### Proof

We will employ the method of quantifier elimination; i.e., we prove that every formula in the language of ordered fields is equivalent to a quantifier-free formula with the same free variables, over the theory \({{{\mathbf {\mathtt{{OF}}}}}}+{{{\mathbf {\mathtt{{IVT}}}}}}\). Since this theory can decide (prove or refute) quantifier-free sentences, then the quantifier elimination theorem will show that \({{{\mathbf {\mathtt{{OF}}}}}}+{{{\mathbf {\mathtt{{IVT}}}}}}\) can decide every sentence in its language; i.e., either proves it or proves its negation.

For that purpose, it suffices to show that every formula of the form \(\exists x\,\theta (x)\), where \(\theta \) is the conjunction of some atomic or negated-atomic formulas, is equivalent (in \({{{\mathbf {\mathtt{{OF}}}}}}+{{{\mathbf {\mathtt{{IVT}}}}}}\)) to a quantifier-free formula (with the same free variables). To see this, suppose that every such formula is equivalent to a quantifier-free formula. Then by induction on the complexity of a formula \(\psi ,\) one can show that \(\psi \) is equivalent to a quantifier-free formula: the case of atomic formulas and the propositional connectives \(\{\lnot ,\wedge ,\vee ,\rightarrow \}\) are trivial, and the case of \(\forall \) can be reduced to that of \(\exists \) by \(\forall x\,\psi (x)\equiv \lnot \exists x\,\lnot \psi (x)\). It remains to show the equivalence of \(\exists x\,\psi (x)\) with a quantifier-free formula, where \(\psi \) is a quantifier-free formula. One can write \(\psi \) (equivalently) in disjunctive normal form \(\bigvee \bigvee _i\theta _i\) where each \(\theta _i\) is a conjunctions of some atomic or negated-atomic formulas. So, \(\exists x\,\psi (x)\equiv \exists x\,\bigvee \bigvee _i\theta _i\equiv \bigvee \bigvee _i\exists x\,\theta _i\), and by the induction hypothesis each of \(\exists x\,\theta _i\) is equivalent to some quantifier-free formula; thus the whole formula is so.

Since every term in the language \(\{+,0,-,\times ,1\}\) with the free variable *x* is a polynomial of *x* whose coefficients are some *x*-free terms, then every atomic formula with *x* is equivalent to \({\mathfrak {p}}(x)=0\) or \({\mathfrak {q}}(x)>0\) for some polynomials \({\mathfrak {p}},{\mathfrak {q}}\). Noting that the negation sign \(\lnot \) can be eliminated by \([{\mathfrak {p}}(x)\ne 0]\equiv [{\mathfrak {p}}(x)>0\vee {\mathfrak {p}}(x)<0]\) and \([{\mathfrak {q}}(x)\not >0]\equiv [{\mathfrak {q}}(x)=0\vee {\mathfrak {q}}(x)<0]\), we can consider atomic formulas only. So, we consider the formulas of the following form for some polynomials \(\{{\mathfrak {p}}_i(x)\}_{i<\ell }\) and \(\{{\mathfrak {q}}_j(x)\}_{j<n}\): \(\exists x\,[\bigwedge \bigwedge _{i<\ell } {\mathfrak {p}}_i(x)=0\;\wedge \; \bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0]\). Finally, it suffices to consider the bounded formulas of the form \(\exists x\in ]a,b[:\psi (x)\) since every formula \(\exists x\,\psi (x)\) is equivalent to \(\exists x<-1\psi (x) \vee \psi (-1) \vee \exists x\in ]-1,1[:\psi (x) \vee \psi (1) \vee \exists x>1\psi (x)\), and also we have \(\exists x<-1\psi (x)\equiv \exists y\in ]0,1[:\psi (-y^{-1})\) and \(\exists x>1\psi (x)\equiv \exists y\in ]0,1[:\psi (y^{-1})\). Let us note that for any polynomials \({\mathfrak {p}}(x)=\sum _{i\leqslant m}a_ix^i\) and \({\mathfrak {q}}(x)=\sum _{j\leqslant l}b_jx^j\) and any \(y>0\) we have

and so any atomic formula \(\theta (y^{-1})\) (and also \(\theta (-y^{-1})\)), for \(y>0\), is equivalent to another atomic formula \(\eta (y)\).

Whence, we show the equivalence of all the formulas in the following form with a quantifier-free formula:

This will be proved by induction on the degree of a formula which we define as follows:

for a term \({\mathfrak {p}}(x)=\sum _{i\leqslant m}a_ix^i,\) let \(\deg _x {\mathfrak {p}}=m\);

for atomic formulas \({\mathfrak {p}}(x)=0\) or \({\mathfrak {q}}(x)>0\), let

\(\deg _x({\mathfrak {p}}(x)=0)=\deg _x{\mathfrak {p}}\) and \(\deg _x({\mathfrak {q}}(x)>0)=1+ \deg _x{\mathfrak {q}}\);

finally, the \(\deg _x\) of a formula is the maximum of the \(\deg _x\)’s of its atomic sub-formulas.

What we prove is:

\((\star )\) for any formula \(\varphi (a,b)\) as (\(\dagger \)) above, there are some formulas \(\{\varPhi _k(y),\varPsi _k(y)\}_{k<m}\) such that \({{{\mathbf {\mathtt{{OF}}}}}}+{{{\mathbf {\mathtt{{IVT}}}}}} +a<b\vdash \varphi (a,b)\leftrightarrow \bigvee \bigvee _{k<m} [\varPhi _k(a)\wedge \varPsi _k(b)]\),

and moreover the \(\deg _y\) of \(\{\varPhi _k(y),\varPsi _k(y)\}\)’s are less than \(\deg _x(\varphi (a,b))\).

This will be shown by induction on \(\hbar =\deg _x\big (\varphi (a,b)\big )\); let us note that here *a*, *b* are treated as (new) variables. If \(\hbar =0\), then *x* appears only superficially in \(\varphi (a,b)\) and so it is equivalent to a quantifier-free formula. Now, suppose that we have the desired conclusion (\(\star \)) for all the formulas with \(\deg _x\) less than \(\hbar \).

**(1)** First, we consider the case of \(\ell =0\); i.e., the formulas that are in the following form:

By Lemma 4.7, \({\mathfrak {q}}_j\)’s are positive at a point in ]*a*, *b*[ if and only if they are positive in some sub-interval \(]u,v[\,\subseteq \,]a,b[\). So, we have \(\varphi (a,b)\equiv \mathsf{F}(a,b)\vee \bigvee \bigvee _{i<n}\mathsf{G}_i(a,b)\vee \bigvee \bigvee _{i,j<n}\mathsf{H}_{i,j}(a,b)\), where \(\mathsf{F}\), \(\{\mathsf{G}_i\}_{i<n}\) and \(\{\mathsf{H}_{i,j}\}_{i,j<n}\) are as follows:

\(\mathsf{F}(a,b)=\forall x\in ]a,b[:\bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0\),

\(\mathsf{G}_i(a,b)=\big [\exists u\in ]a,b[:{\mathfrak {q}}_i(u)=0\wedge \mathsf{F}(a,u) \big ]\bigvee \big [\exists v\in ]a,b[:{\mathfrak {q}}_i(v)=0\wedge \mathsf{F}(v,b)\big ]\), and

\(\mathsf{H}_{i,j}(a,b)=\exists u,v\in ]a,b[: u<v \wedge {\mathfrak {q}}_i(u)=0 \wedge {\mathfrak {q}}_j(v)=0 \wedge \mathsf{F}(u,v)\).

The formula \(\mathsf{F}(a,b)\) is equivalent to \(\bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(a)\geqslant 0\wedge \bigwedge \bigwedge _{j<n}\lnot \exists x\in ]a,b[:{\mathfrak {q}}_j(x)=0\), whose \(\deg _x\) is less than \(\hbar \), and so by the induction hypothesis (\(\star \)) is equivalent to the formula \(\bigvee \bigvee _{k<m} [\varPhi _k(a)\wedge \varPsi _k(b)]\) for some *x*-free formulas \(\{\varPhi _k(y),\varPsi _k(y)\}_{k<m}\) whose \(\deg _y\) are less than \(\hbar \). So, for any \(i<n\), \(G_i(a,b)\) is equivalent to the following formula:

Since the \(\deg _u\) of \({\mathfrak {q}}_i(u)=0\wedge \varPsi _k(u)\) and the \(\deg _v\) of \({\mathfrak {q}}_i(v)=0\wedge \varPhi _k(v)\) are less than \(\hbar ,\) the induction hypothesis (\(\star \)) applies to all \(\mathsf{G}_i\)’s. Finally, for \(\mathsf{H}_{i,j}(a,b),\) we note that it is equivalent to the disjunction (over \(k<m\)) of the following formulas:

Now, the formula \({\mathfrak {q}}_j(v)=0\wedge \varPsi _k(v)\) has \(\deg _v\) less than \(\hbar \); so by the induction hypothesis (\(\star \)), there are formulas \(\{\varTheta _{j,k,\iota }(z),\varUpsilon _{j,k,\iota }(z)\}_{\iota <l}\) with \(\deg _z\) less than \(\hbar \) such that the formula \(\exists v\in ]u,b[:[{\mathfrak {q}}_j(v)=0\wedge \varPsi _k(v)]\) is equivalent to \(\bigvee \bigvee _{\iota <l} [\varTheta _{j,k,\iota }(u)\wedge \varUpsilon _{j,k,\iota }(b)]\). Thus, \(\mathsf{H}_{i,j}(a,b)\) is equivalent to the disjunction of \(\exists u\in ]a,b[:\big ({\mathfrak {q}}_i(u)=0\wedge \varPhi _k(u) \wedge \varTheta _{j,k,\iota }(u) \big )\wedge \varUpsilon _{i,j,\iota }(b)\), for \(k<m,\iota <l\), to which the induction hypothesis (\(\star \)) apply, since the \(\deg _u\) of the all the formulas \({\mathfrak {q}}_i(u)=0\wedge \varPhi _k(u) \wedge \varTheta _{j,k,\iota }(u)\) are less than \(\hbar \).

**(2)** Second, we consider the case of \(\ell >0\); let us note that we can assume \(\ell =1\) since we have \(\bigwedge \bigwedge _{i<\ell } \alpha _i=0 \iff \sum _{i<\ell }\alpha _i^2=0\). So, we may replace all the \({\mathfrak {p}}_i(x)\)’s with a single polynomial \(\sum _{i<\ell }{\mathfrak {p}}^2_i(x)\); but this will increase the \(\deg _x\) of the resulted formula. There is another way of reducing \(\ell \) (the number of polynomials \({\mathfrak {p}}_i\)’s) without increasing the \(\deg _x\) of the formula:

__(I)__For polynomials \({\mathfrak {p}}(x)=\alpha x^{d}+\sum _{i<d}a_ix^i\) and \({\mathfrak {q}}(x)=\beta x^{e}+\sum _{j<e}b_jx^j\) assume that \(d\geqslant e\). Put \({\mathfrak {r}}(x)={\mathfrak {q}}(x)-\beta x^e\) and \({\mathfrak {s}}(x)=\beta {\mathfrak {p}}(x)-\alpha x^{d-e}{\mathfrak {q}}(x)\); then we have \(\big [{\mathfrak {p}}(u)= {\mathfrak {q}}(u)=0\big ] \iff \big [\beta =0\wedge {\mathfrak {p}}(u)= {\mathfrak {r}}(u)=0\big ]\vee \big [\beta \ne 0\wedge {\mathfrak {s}}(u)={\mathfrak {q}}(u)=0\big ]\).Continuing this way, at least one of the two polynomials will disappear and we will be left with at most one polynomial, and the \(\deg _x\) of the last formula will be non-greater than the \(\deg _x\) of the first formula.

So, we can safely assume that \(\ell =1\); thus \(\varphi (a,b)=\exists x\in ]a,b[:{\mathfrak {p}}(x)=0\wedge \bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0\) and \(\deg _x(\varphi (a,b))=\hbar \). We can still transform the formula to an equivalent one in which we have \(\deg _x{\mathfrak {q}}_j<\deg _x{\mathfrak {p}}\) for all \(j<n\):

__(II)__If, say, \(\deg _x{\mathfrak {q}}_1\geqslant \deg _x{\mathfrak {p}}\), then write \({\mathfrak {p}}(x)=\alpha x^{d}+\sum _{i<d}a_ix^i\) and \({\mathfrak {q}}_1(x)=\beta x^{e}+\sum _{j<e}b_jx^j\) with \(d\leqslant e\). Put \({\mathfrak {r}}(x)={\mathfrak {p}}(x)-\alpha x^d\) and \({\mathfrak {s}}(x)=\alpha ^2{\mathfrak {q}}_1(x)- \alpha \beta x^{e-d}{\mathfrak {p}}(x)\); then we have \(\big [{\mathfrak {p}}(u)=0 \wedge {\mathfrak {q}}_1(u)>0\big ] \iff \big [\alpha =0\wedge {\mathfrak {r}}(u)=0\wedge {\mathfrak {q}}_1(u)>0\big ] \vee \big [\alpha \ne 0\wedge {\mathfrak {p}}(u)=0\wedge {\mathfrak {s}}(u)>0\big ]\). Continuing this way, either the equality (\({\mathfrak {p}}(x)=0\)) will disappear (and so we will have the first case) or the degree of the inequality \(\deg _x({\mathfrak {q}}_1(x)>0)\) will be non-greater than the degree of the equality, \(\deg _x({\mathfrak {p}}(x)=0)\).

So, assume that in the formula

we have that \(\deg _x(\varphi (a,b))=\hbar =\deg _x{\mathfrak {p}}\) and also \(\bigwedge \bigwedge _{j<n} \deg _x{\mathfrak {q}}_j<\hbar \). Now, the formula \(\varphi (a,b)\) is equivalent to \(\varphi _1(a,b)\vee \varphi _2(a,b)\vee \varphi _3(a,b),\) where

\(\varphi _1(a,b)=\exists x\in ]a,b[:{\mathfrak {p}}(x)=0\wedge {\mathfrak {p}}'(x)=0\wedge \bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0\),

\(\varphi _2(a,b)=\exists x\in ]a,b[:{\mathfrak {p}}(x)=0\wedge {\mathfrak {p}}'(x)>0\wedge \bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0\), and

\(\varphi _3(a,b)=\exists x\in ]a,b[:{\mathfrak {p}}(x)=0\wedge {\mathfrak {p}}'(x)<0\wedge \bigwedge \bigwedge _{j<n} {\mathfrak {q}}_j(x)>0\).

Since, \(\deg _x{\mathfrak {p}}'<\deg _x{\mathfrak {p}}\) (see Definition 4.4), then by applying __(I)__ to \(\varphi _1\) we get an equivalent formula with \(\deg _x\) less than \(\hbar \) and then can apply the induction hypothesis (\(\star \)). For \(\varphi _2,\) we note that, by \({{{\mathbf {\mathtt{{IVT}}}}}}\) and Theorem 4.6, all the \({\mathfrak {q}}_j\)s and also \({\mathfrak {p}}'\) are strictly positive at some point in ]*a*, *b*[ in which \({\mathfrak {p}}\) vanishes, if and only if all the \({\mathfrak {q}}_j\)s and \({\mathfrak {p}}'\) are strictly positive on some open sub-interval \(]u,v[\,\subseteq \,]a,b[\) such that (\({\mathfrak {p}}\) is monotonically increasing and so) \({\mathfrak {p}}(u)<0<{\mathfrak {p}}(v)\). Whence, \(\varphi _2(a,b)\) is equivalent to the disjunction of the following formulas:

- (i)
\({\mathfrak {p}}(a)<0<{\mathfrak {p}}(b) \wedge \mathsf{F}(a,b)\),

- (ii)
\(\bigvee \bigvee _{i<n} \Big (\big [{\mathfrak {p}}(a)<0\wedge \exists u\in ]a,b[:\big ({\mathfrak {q}}_i(u)=0\wedge {\mathfrak {p}}(u)>0\wedge \mathsf{F}(a,u)\big )\big ]\bigvee \)

\(\big [{\mathfrak {p}}(b)>0\wedge \exists v\in ]a,b[:\big ({\mathfrak {q}}_i(v)=0\wedge {\mathfrak {p}}(v)<0\wedge \mathsf{F}(v,b)\big )\big ]\Big )\), and

- (iii)
\(\bigvee \bigvee _{i,j<n}\Big ( \exists u,v\in ]a,b[:\big [{\mathfrak {q}}_i(u)=0\wedge {\mathfrak {q}}_j(v)=0\wedge {\mathfrak {p}}(u)<0<{\mathfrak {p}}(v) \wedge \mathsf{F}(u,v)\big ]\Big )\).

The formula (i) has been treated before (it is equivalent to a formula with \(\deg _x\) less than \(\hbar \)). The formulas (ii) and (iii) can be equivalently transformed to formulas with \(\deg _x\) less than \(\hbar \) by __(I)__ and __(II)__ above. So, the whole formula \(\varphi _2\), and very similarly \(\varphi _3\), can be written in equivalent forms in such a way that the induction hypothesis (\(\star \)) applies to them. \(\square \)

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Salehi, S., Zarza, M. First-Order Continuous Induction and a Logical Study of Real Closed Fields.
*Bull. Iran. Math. Soc.* **46**, 225–243 (2020). https://doi.org/10.1007/s41980-019-00252-0

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DOI: https://doi.org/10.1007/s41980-019-00252-0

### Keywords

- First-order logic
- Complete theories
- Axiomatizing the field of real numbers
- Continuous induction
- Real closed fields