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In Tarski's system of geometry, he describes his continuity axiom as follows:

To obtain an appropriate set of axioms, we start with an axiom system which is known to provide an adequate basis for the whole of Euclidean geometry and contains β and δ as the only non-logical constants. Usually the only non-elementary sentence in such a system is the continuity axiom, which contains second-order variables X, Y,... ranging over arbitrary point sets (in addition to first-order variables x, y,... ranging over points) and also an additional logical constant, the membership symbol $\in$ denoting the membership relation between points and point sets. The continuity axiom can be formulated, e.g., as follows:

$ \forall XY \{ \exists z \forall xy[x \in X \land y \in Y \to β(zxy)] \to \exists u \forall xy [x \in X \land y \in Y \to β(xuy)] \} $

He describes the function β as:

[T]he formula β(xyz) is read y lies between x and z (the case when y coincides with x or z not being excluded)...

This continuity axioms appears to exemplify the idea of a Dedekind cut, which Dedekind applied to the geometry of lines:

If all points of the straight line fall into two classes such that every point of the first class lies to the left of every point of the second class, then there exists one and only one point which produces this division of all points into two classes, this severing of the straight line into two portions. (Richard Dedekind. Essays on the Theory of Numbers)

According to my interpretation, this principle would exclude the number line consisting of only the integers, for example, because $X$ and $Y$ could be defined as $X = \{x | x \leq 2\}; Y = \{y | y \geq 3\}$, leaving more than one possible value between 2 and 3. For that reason it seems that any formalization of the principle would have to exclude the possibility for more than one value between the upper bound of $X$ and lower bound of $Y$.

For that reason, I would expect Tarski's axiom to be something like the following:

$ \forall XY \{ \exists z \forall xy[x \in X \land y \in Y \to β(zxy)] \to \exists u \forall xy [(x \in X \land y \in Y \to β(xuy)) \color{maroon}{ \land \forall w[β(xwy) \to w=u]}] \} $

Is there something I'm misunderstanding about this?

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    I think that it is simply a question of "generality"; if we consider e.g. the segments $[0,4]$ and $[8,12]$ we cannot impose that there must be only one point in between... The antecedent of the axiom says : For **every** sets of points (segments) $X$ and $Y$ such that $X$ is to the right of $Y$ with respect to an "origin" $z$, there is **at least** a point $u$ "separating" $X$ and $Y$". The axiom is aimed at avoiding the well-known case (see Dedekind) : $X = \{ x \mid x^2 < 2 \} ; Y = \{ y \mid y^2 > 2 \}$. – 2017-01-17
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    @MauroALLEGRANZA. I don't understand your point. If there is more than one value between the segments, they would *not* be continuous. Therefore, to establish continuity, it has to be shown that there is "one and only one point" as Dedekind said. The segments [0,4] and [4, 12] would be continuous on the real number line because 4 is the only number that can be said to be between them. – 2017-01-17
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    Tarski's theory is FOL and the individual variables stay for *points*; a *line* is a particular set of points. In the meta-theory we can use set variables like $X$ to "describe" set of points but in the formal theory we have only formulae with a parameter $\phi(x)$. Thus, in the meta-) we can say that : $x \in X$ **iff** $\phi(x)$. – 2017-01-18
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    Thus, in the meta-, we can abbreviate with $X_{\phi} = \{ x \mid \phi(x) \}$. Having said that, a line is a set of collinear points; thus, again in the meta-, we may abbreviate $\text {line} (\phi) = \text {line} (X_{\phi})$ **iff** $\forall x,y,z [x,y,z \in X_{\phi} \to \beta(x,y,x)]$. – 2017-01-18
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    But this relies on the "intended" interpretation... In general, the axioms consider "sets" of points whatever. If e.g. we consider two sets of points "scattered" on the $x$-axis : e.g. $X= \{ 1,2,4 \}$ and $Y= \{ 3,5,6,7 \}$, it is simply not true that "there exists $z$ (like e.g. the "origin" $0$) such that : for **all** $x \in X$ and $y \in Y$ : $\beta(z,x,y)$". – 2017-01-18
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    Correct to speak of "ray of points"; thus, for every "ray" $r$ from $z$ and subsets $X$ and $Y$ of $r$ such that every point in $Y$ is to the right of every point in $X$ (with respect to $z$), there exists **at least one** point $u$ "separating" $X$ and $Y$. The key-point is "at least one": if we consider the $x$-axis with only rational coordinates and the sets $X= \{x \mid 0 < x \land x^2 < 2 \}$ and $Y = \{ y \mid y^2 > 2 \land y < 4 \}$ it is not true that there is a "point" (i.e. a rational $= \sqrt {2}$) that "separates" them. – 2017-01-18
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    @MauroALLEGRANZA. The only reason I would see that there exists no such $z$ is because of the fact that your set $X$ overlaps $Y$. Otherwise, all $x \in \{1,2,4\}$ would fall between the origin and $Y$ if it were changed to $Y = \{5,6,7,8\}$. There are only three values for x, and they all fall between the origin and the smallest value of $Y$. – 2017-01-18
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    @MauroALLEGRANZA. Concerning your last comment, why are you specifying only rational coordinates? For an adequate axiom of continuity, there has to be the possibility that $u$ can have values other than those contained in $X$ and $Y$, because otherwise the axiom would be tautologous. There is *always* at least one $u$, such as the maximum value $X$ or the minimum of $Y$, that would satisfy $β(xuy)$. You say that the key point is "a least one", but I just don't see how. I believe the key point is "one and only one" as Dedekind said, because, otherwise, there's a gap between $X$ and $Y$. – 2017-01-18

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