# Special Relativity over the Field of Rational Numbers

## Abstract

We investigate the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? The answer to this question depends strongly on the auxiliary assumptions we add to the basic assumptions of special relativity. We show that there is a natural axiom system of special relativity which can be modeled even over the field of rational numbers.

### Keywords

Relativity theory Special relativity Rational numbers Axiomatic theories First-order logic## 1 Introduction

In this paper, we investigate, within an axiomatic framework, the question: what structures of numbers (as physical quantities) are suitable to be used in special relativity? There are several reasons to investigate this kind of questions in the case of any theory of physics. First of all, we cannot experimentally verify whether the structure of quantities is isomorphic to the field of real numbers. Moreover, the fact that the outcome of every measurement is a finite decimal suggests that rational numbers (or even integers) should be enough to model physical quantities. Another reason is that these investigations lead to a deeper understanding of the connection of the mathematical assumptions about the quantities and the other (physical) assumptions of the theory. Hence these investigations lead to a deeper understanding of any theory of physics, which may come handy if we have to change some of the basic assumptions for some reason. For a more general perspective of this research direction, see [4].

To introduce the central concept of our investigation, let Th be a theory of physics that contains the concept of numbers (as physical quantities) together with some algebraic operations on them (or at least these concepts are definable in Th). In this case, we can introduce notation“

What structure can numbers have in a certain physical theory?”

*Num*(Th) for the class of the possible quantity structures of theory Th:

In this paper, we investigate our question only in the case of special relativity. However, this question can be investigated in any other physical theory in the same way.

We will see an axiom on observers implying that positive numbers have square roots. Therefore, we recall that *Euclidean fields*, which got their names after their role in Tarski’s first-order logic axiomatization of Euclidean geometry [18], are ordered fields in which positive numbers have square roots.

*d*-dimensional special relativity (\(\mathsf{SpecRel_{d}} \), see p. 5) captures the kinematics of special relativity perfectly if

*d*≥3, see Theorem 1. Without any extra assumptions \(\mathsf{SpecRel_{d}} \) has a model over every ordered field, i.e.,

*d*≥3 by Theorem 2, i.e.,

*d*≥3, i.e.,

Physical theories over ordered fields different from the field of real numbers are interesting not only within theoretical physics, e.g., [10] is a biology paper developing its results over the field of rational numbers.

An interesting and related approach of Mike Stannett introduces two structures, one for the measurable numbers and one for the theoretical numbers and assumes that the set of measurable numbers is dense in the set of theoretical numbers, see [16].

We chose first-order predicate logic to formulate our axioms because experience (e.g., in geometry and set theory) shows that this logic is the best logic for providing an axiomatic foundation for a theory. A further reason for choosing first-order logic is that it is a well defined fragment of natural language with an unambiguous syntax and semantics, which do not depend on set theory. For further reasons, see, e.g., [3, Sect. Why FOL?], [5], [17, Sect. 11], [19, 20].

## 2 The Language of Our Theories

To our investigation, we need an axiomatic theory of special relativity. Therefore, we will recall our axiom system \(\mathsf{SpecRel_{d}} \) in Sect. 3. To write up any axiom system, we have to choose the set of basic symbols of its language, i.e., what objects and relations between them will be used as basic concepts.

^{1}language of first-order logic parametrized by a natural number

*d*≥2 representing the dimension of spacetime:

*B*(bodies) and

*Q*(quantities) are the two sorts, IOb (inertial observers) and Ph (light signals) are one-place relation symbols of sort

*B*, + and ⋅ are two-place function symbols of sort

*Q*, ≤ is a two-place relation symbol of sort

*Q*, and W (the worldview relation) is a

*d*+2-place relation symbol the first two arguments of which are of sort

*B*and the rest are of sort

*Q*.

Relations IOb(*m*) and Ph(*p*) are translated as “*m**is an inertial observer*,” and “*p**is a light signal*,” respectively. To speak about coordinatization of observers, we translate relation W(*k*,*b*,*x*_{1},*x*_{2},…,*x*_{d}) as “*body**k**coordinatizes body**b**at space-time location* 〈*x*_{1},*x*_{2},…,*x*_{d}〉,” (i.e., at space location 〈*x*_{2},…,*x*_{d}〉 and instant *x*_{1}).

*Quantity terms* are the variables of sort *Q* and what can be built from them by using the two-place operations + and ⋅, *body terms* are only the variables of sort *B*. IOb(*m*), Ph(*p*), W(*m*,*b*,*x*_{1},…,*x*_{d}), *x*=*y*, and *x*≤*y* where *m*, *p*, *b*, *x*, *y*, *x*_{1}, …, *x*_{d} are arbitrary terms of the respective sorts are so-called *atomic formulas* of our first-order logic language. The *formulas* are built up from these atomic formulas by using the logical connectives *not* (¬), *and* (∧), *or* (∨), *implies* (→), *if-and-only-if* (↔) and the quantifiers *exists* (∃) and *for all* (∀).

To make them easier to read, we omit the outermost universal quantifiers from the formalizations of our axioms, i.e., all the free variables are universally quantified.

We use the notation *Q*^{n} for the set of all *n*-tuples of elements of *Q*. If \(\bar{\mathbf{x}}\in \mathit{Q} ^{n}\), we assume that \(\bar{\mathbf{x}}=\langle x_{1},\ldots,x_{n}\rangle\), i.e., *x*_{i} denotes the *i*-th component of the *n*-tuple \(\bar{\mathbf{x}}\). Specially, we write \(\mathsf{W} (m,b,\bar{\mathbf{x}})\) in place of W(*m*,*b*,*x*_{1},…,*x*_{d}), and we write \(\forall\bar{\mathbf{x}}\) in place of ∀*x*_{1}…∀*x*_{d}, etc.

*models*of this language are of the form

*B*and

*Q*are nonempty sets, \(\mathsf{IOb} _{\mathfrak{M}}\) and \(\mathsf{Ph} _{\mathfrak{M}}\) are subsets of

*B*, \(+_{\mathfrak{M}}\) and \(\cdot_{\mathfrak{M}}\) are binary functions and \(\le_{\mathfrak{M}}\) is a binary relation on

*Q*, and \(\mathsf{W} _{\mathfrak{M}}\) is a subset of

*B*×

*B*×

*Q*

^{d}. Formulas are interpreted in \(\mathfrak{M}\) in the usual way. For the precise definition of the syntax and semantics of first-order logic, see, e.g., [7, Sect. 1.3], [8, Sects. 2.1, 2.2].

## 3 Axioms for Special Relativity

Now having our language fixed, we can recall axiom system \(\mathsf{SpecRel_{d}} \), as well as two theorems on \(\mathsf{SpecRel_{d}} \) related to our investigation.

- AxPh
- For any inertial observer, the speed of light is the same everywhere and in every direction (and it is finite). Furthermore, it is possible to send out a light signal in any direction (existing according to the coordinate system) everywhere:
^{2}

To get back the intended meaning of axiom AxPh (or even to be able to define subtraction from addition), we have to assume some properties of numbers.

- AxOField
- The quantity part 〈
*Q*,+, ⋅,≤〉 is an ordered field, i.e.,〈

*Q*,+,⋅〉 is a field in the sense of abstract algebra; and- the relation ≤ is a linear ordering on
*Q*such that- (i)
*x*≤*y*→*x*+*z*≤*y*+*z*and - (ii)
0≤

*x*∧0≤*y*→0≤*xy*holds.

- (i)

Using axiom AxOFiled instead of assuming that the structure of quantities is the field of real numbers not just makes our theory more flexible, but also makes it possible to meaningfully investigate our main question. Another reason for using AxOField instead of ℝ is that we cannot experimentally verify whether the structure of physical quantities are isomorphic to ℝ. Hence the assumption that the structure of quantities is ℝ cannot be empirically supported. The two properties of real numbers which are the most difficult to defend from empirical point of view are the Archimedean property, see [11], [12, Sect. 3.1], [13, 14], and the supremum property.^{3}

*event*occurring for observer

*m*at point \(\bar{\mathbf{x}}\), we mean the set of bodies

*m*coordinatizes at \(\bar{\mathbf{x}}\):

- AxEv
- All inertial observers coordinatize the same set of events:$$ \mathsf{IOb} (m)\land \mathsf {IOb} (k)\rightarrow\exists\bar{\mathbf{y}}\,\forall b \big[ \mathsf{W} (m,b,\bar{\mathbf{x}})\leftrightarrow \mathsf{W} (k,b,\bar{\mathbf{y}})\big]. $$

- AxSelf
- Any inertial observer is stationary relative to himself:$$ \mathsf{IOb} (m)\rightarrow\forall\bar{\mathbf {x}}\big[ \mathsf{W} (m,m,\bar{\mathbf{x}}) \leftrightarrow x_2=\cdots=x_d=0\big]. $$

- AxSymD
- Any two inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them; furthermore, the speed of light is 1 for all observers:

*worldview transformation*between observers

*m*and

*k*(in symbols, w

_{mk}) as the binary relation on

*Q*

^{d}connecting the coordinate points where

*m*and

*k*coordinatize the same events:

*P*:

*Q*

^{d}→

*Q*

^{d}is called a

*Poincaré transformation*iff it is an affine bijection having the following property

Theorem 1 shows that our axiom system \(\mathsf{SpecRel_{d}} \) captures the kinematics of special relativity since it implies that the worldview transformations between inertial observers are Poincaré transformations.

### Theorem 1

*Let**d*≥3. *Assume*\(\mathsf{SpecRel_{d}}\). *Then*w_{mk}*is a Poincaré transformation if**m**and**k**are inertial observers*.

For the proof of Theorem 1, see [4]. For a similar result over Euclidean fields, see, e.g., [1, Theorems 1.4 and 1.2], [2, Theorem 11.10], [17, Theorem 3.1.4].

- AxThExp
- Inertial observers can move along any straight line with any speed less than the speed of light:

Theorem 2 below shows that axiom AxThExp implies that positive numbers have square roots if \(\mathsf{SpecRel_{d}} \) is assumed.

### Theorem 2

*If*

*d*≥3,

*then*

### Remark 1

*d*≥2,

*d*≥2. Moreover, \(\mathsf{SpecRel_{d}}\) also has non trivial models in which there are several observers moving relative to each other. We conjecture that there is a model of \(\mathsf {SpecRel_{d}}\) over every ordered field such that the possible speeds of observers are dense in interval [0,1], see Conjecture 1 on p. 7.

- AxThExp
^{−} - Inertial observers can move roughly with any speed less than the speed of light roughly in any direction, see Fig. 1:
^{4}

By Theorem 3, a model of \(\mathsf {SpecRel_{d}} + \mathsf{AxThExp^{-}} \) has a model over the field of rational numbers in any dimension. We use the notation \(\mathfrak{Q}\in\mathit{Num}( \mathsf {Th} )\) for algebraic structure \(\mathfrak{Q}\) the same way as the model theoretic notation \(\mathfrak{Q}\in Mod( \mathsf{AxField} )\), e.g., ℚ∈*Num*(Th) means that ℚ, the field of rational numbers, can be the structure of quantities in theory Th.

### Theorem 3

*For all*

*d*≥2,

For the proof of Theorem 3, see Sect. 4.

*Archimedean field*iff for all

*a*, there is a natural number

*n*such that

### Corollary 1

*For all*

*d*≥2,

The question “exactly which ordered fields can be the quantity structures of theory \(\mathsf{SpecRel_{d}} + \mathsf{AxThExp^{-}} \)?” is open. By Lövenheim–Skolem Theorem it is clear that \(\mathit{Num}( \mathsf{SpecRel_{d}} + \mathsf{AxThExp^{-}} )\) cannot be the class of Archimedean fields since it has elements of arbitrarily large cardinality while an Archimedean field has at most the cardinality of continuum since Archimedean fields are subsets of the field of real numbers by Hölder’s Theorem. We conjecture that there is a model of \(\mathsf {SpecRel_{d}} + \mathsf{AxThExp^{-}} \) over every ordered field in any dimension, i.e.:

### Conjecture 1

*For all*

*d*≥2,

## 4 Proof of Theorem 3

In this section, we are going to prove our main result. To do so, let us recall some concepts and theorems from the literature. The following theorem is well-known, see, e.g., [15, Theorem 2.1].

### Theorem 4

*The unit sphere of* ℝ^{n}*has a dense set of points with rational coordinates*.

*Euclidean length*of \(\bar{\mathbf{x}}\in \mathit{Q} ^{n}\) if

*n*≥1 is defined as:

*norm*of linear map

*A*:ℝ

^{d}→ℝ

^{d}, in symbols ∥

*A*∥, is defined as follows:

*A*is called

*orthogonal transformation*if it preserves the Euclidean distance.

Theorem 4 implies Theorem 5, see [15, Theorem 3.1].

### Theorem 5

*For all orthogonal transformation**T*:ℝ^{n}→ℝ^{n}*and any**ε*>0, *there is an orthogonal transformation**A*:ℚ^{n}→ℚ^{n}*such that* ∥*T*−*A*∥<*ε*.

Using Theorem 5, let us prove that its statement also holds for Poincaré transformations.

### Theorem 6

*For every Poincaré transformation**L*:ℝ^{d}→ℝ^{d}*and positive real number ε*, *there is a Poincaré transformation**L*^{∗}:ℚ^{d}→ℚ^{d}*such that* ∥*L*−*L*^{∗}∥<*ε*.

*Lorentz boost*corresponding to velocity

*v*∈[0,1), in symbols

*B*

_{v}, is defined as the following linear map:

### Lemma 1

*For any Lorentz boost**B*_{v}:ℝ^{d}→ℝ^{d}*and positive number**ε*, *there is a Lorentz boost**B*_{w}:ℚ^{d}→ℚ^{d}*such that* ∥*B*_{v}−*B*_{w}∥<*ε*.

### Proof

*δ*>0 and

*v*∈[0,1), there is a

*w*∈ℚ∪[0,1) such that |

*v*−

*w*|<

*δ*and \(\sqrt{1-w^{2}}\in\mathbb{Q}\), i.e.,

*B*

_{w}takes rational point to rational ones. So we have to show that ∥

*B*

_{v}−

*B*

_{w}∥<

*ε*if

*δ*is small enough. Since in a finite-dimensional vector space all norms are equivalent, see [6, Sect. 8.5], it is enough to show that the norm of

*B*

_{v}−

*B*

_{w}can be less than any positive real number according to the Euclidean norm, which is

*B*

_{v}−

*B*

_{w}is less than any fixed positive real number if |

*v*−

*w*| is small enough. Therefore, there is a Lorentz boost

*B*

_{w}such that

*B*

_{w}maps rational points to rational ones and ∥

*B*

_{w}−

*B*

_{v}∥<

*ε*. □

### Lemma 2

*Let**A**and**B**be linear bijections of *ℝ^{d}. *Let**A*′ *and**B*′ *linear maps such that* ∥*A*−*A*′∥<*ε*_{1}*and* ∥*B*−*B*′∥<*ε*_{2}. *Then* ∥*BA*−*B*′*A*′∥≤*ε*_{1}∥*B*∥+*ε*_{1}*ε*_{2}+*ε*_{2}∥*A*∥.

### Proof

### Proof of Theorem 6

Every Poincaré transformation is a composition of a translation, a Lorentz-boost *B*_{v} and an orthogonal transformation. Therefore, Lemmas 1 and 2, together with Theorem 5 imply our statement. □

*identity map*of ℚ

^{d}. We denote the

*origin*of

*Q*

^{n}by \(\bar{\mathbf{o}}\), i.e.,

*time-axis*be defined as the following subset of

*Q*

^{d}:

*H*be a subset of

*Q*

^{d}and let

*f*:

*Q*

^{d}→

*Q*

^{d}be a map. The

*f*-

*image*of set

*H*is defined as:

*worldline*of body

*b*according to observer

*m*is defined as follows:

### Proof of Theorem 3

*Q*,+, ⋅ ,≤〉 be the ordered field of rational numbers. Let

*B*=IOb∪Ph. First we are going to give the worldview of observer Id. Let

*m*, let

*p*∈Ph, let

*m*as follows:

*b*∈

*B*, see Fig. 2.

Now we have given the model. Let us see why the axioms of \(\mathsf{SpecRel_{d}} \) and \(\mathsf {AxThExp^{-}} \) are valid in it.

*m*and

*k*are inertial observers, then

*m*∈IOb and

*p*∈Ph, then

*m*and Id is

*m*, i.e., w

_{mId}=

*m*by equation (24). Therefore, the worldview transformation between inertial observers

*m*and

*k*is

*k*

^{−1}∘

*m*, i.e.,

_{mk}=w

_{Idk}∘w

_{mId}and w

_{Idk}=(w

_{kId})

^{−1}by the definition of the worldview transformation (5). In particular, the worldview transformations between inertial observers are Poincaré transformations in these models (as Theorem 1 requires it). Hence

Axiom AxPh is valid for observer Id by the definition of Ph and that of his worldview. It is also clear that the speed of light is 1 for observer Id. Axiom AxPh is valid for the other observers since Poincaré transformations take lines of slope one to lines of slope one. This also show that the speed of light is 1 according to every inertial observer, which is the second half of AxSymD.

Any Poincaré transformation *P* preserves the spatial distance of points \(\bar{\mathbf{x}}\), \(\bar{\mathbf{y}}\) for which *x*_{1}=*y*_{1} and \(P(\bar{\mathbf{x}})_{1}=P(\bar{\mathbf{y}})_{1}\). Therefore, inertial observers agree as to the spatial distance between two events if these two events are simultaneous for both of them. We have already shown that the speed of light is 1 for each inertial observers in this model. Hence axiom AxSymD is also valid in this model.

*h*IOb(

*h*) part of axiom AxThExp is valid, since there are Poincaré transformations (e.g., Id is one). To show that the rest of axiom \(\mathsf{AxThExp^{-}} \) is valid, let

*m*be an inertial observer and let us fix an

*ε*>0 and a \(\bar{\mathbf {v}}\in\mathbb{Q}^{d}\) for which \(v_{2}^{2}+\cdots+ v_{d}^{2}<1\) and

*v*

_{1}=1. Let \(\bar{\mathbf{1}}\) be vector 〈1,0,…,0〉. Let

*L*be a Lorentz transformation (i.e., linear Poincaré transformation) for which

*δ*<1 be such that

*x*for which \(|L(\bar{\mathbf{1}})_{1}-x|<\delta\). By Theorem 6, there is a Lorentz transformation

*L*

^{∗}which takes rational points to rational ones and ∥

*L*−

*L*

^{∗}∥<

*δ*. Then

*λ*∈ℚ such that \(\bar{\mathbf{y}}-\bar{\mathbf{x}}=\lambda\bar{\mathbf {w}}\). To finish the proof of \(\mathsf{AxThExp^{-}} \), we have to show that there is an inertial observer

*k*such that \(\mathsf{W} (m,k,\bar{\mathbf{x}})\) and \(\mathsf{W} (m,k,\bar{\mathbf{y}})\), i.e., \(\bar{\mathbf{x}},\bar {\mathbf{y}}\in \mathsf{wl} _{m}(k)\). Let \(P^{*}=L^{*}+\bar{\mathbf{x}}\).

*P*

^{∗}is a Poincaré transformation taking rational points to rational ones. Therefore, there is an inertial observer

*k*such that w

_{km}=

*P*

^{∗}. Since \(\mathsf{wl} _{m}(k)= \mathsf{w} _{km}\ [\mathsf{t\text{-}axis}]\), we have that \(\mathsf{w} _{km}(\bar{\mathbf{o}})=\bar {\mathbf{x}}\in \mathsf{wl} _{m}(k)\) and that \(\bar{\mathbf{y}}=a L^{*}(\bar{\mathbf{1}})+\bar{\mathbf{x}}= \mathsf {w} _{km}(a\bar{\mathbf{1}})\in \mathsf{wl} _{m}(k)\), where \(a=\lambda/L^{*}(\bar{\mathbf{1}})_{1}\). This shows that \(\mathsf{AxThExp^{-}} \) is also valid in our model. □

## Footnotes

- 1.
That our theory is two-sorted means only that there are two types of basic objects (bodies and quantities) as opposed to, e.g., Zermelo–Fraenkel set theory where there is only one type of basic objects (sets).

- 2.
That is, if

*m*is an inertial observer, there is a is a positive quantity*c*_{m}such that for all coordinate points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) there is a light signal*p*coordinatized at \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) by observer*m*if and only if equation \(\mathsf{space}^{2}(\bar{\mathbf{x}},\bar{\mathbf{y}})= c_{m}^{2}\cdot\mathsf{time}(\bar{\mathbf{x}},\bar{\mathbf{y}})^{2}\) holds. - 3.
The supremum property (i.e., that every nonempty and bounded subset of the numbers has a least upper bound) implies the Archimedean property. So if we want to get ourselves free from the Archimedean property, we have to leave this one, too.

- 4.
That is, for every vector \(\bar{\mathbf{v}}=\langle v_{1},\ldots,v_{d}\rangle\) determining a spatial direction and a slower than light speed, there is another vector \(\bar{\mathbf{w}}\) in the

*ε*-neighborhood of \(\bar {\mathbf{v}}\), such that, if points \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\) are on a line parallel to \(\bar{\mathbf{w}}\) (i.e., on a line corresponding to a uniform motion with the speed and in the direction determined by \(\bar{\mathbf{w}}\)), then there is an inertial observer*k*moving through \(\bar{\mathbf{x}}\) and \(\bar{\mathbf{y}}\), see Fig. 1.

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