An axiomatic foundation of relativistic spacetime

Abstract

An ab-initio foundation for relativistic spacetime is given, which is a conservative extension of Zermelo’s set theory with urelemente. Primitive entities are worldlines rather than spacetime points. Spacetime points are sets of intersecting worldlines. By the proper axioms, they form a manifold. Entities known in differential geometry, up to a metric, are defined and have the usual properties. A set-realistic point of view is adopted. The intended ontology is a set-theoretical hierarchy with a broad base of the empty set and urelemente. Sets generated from the empty set are mathematically interpreted, all other sets are physically interpreted.

Appendix

Zermelo’s set theory Z is extended to a set theory with urelemente ZU. We write

The usual extensionality axiom is modified to become

• (SE)   

•           “Any sets $x$ and $y$ that are not urelemente are identical if they contain the same elements.”

One axiom is added:

• (SU)   

•           “The urelemente form a set.”

This determines ZU. The language of ST contains an additional predicate constant $\not \mid$. The elementary formula $v\not \mid _{x}w$ is read “$w$ intersects $v$ at $x$.” or “$w$ intersects $<v,x>$”. In the following, the proper axioms of ST are given, along with several (nominal) definitions.

• (P1)   )

•        “$w$ intersects $v$ at $x$ implies that $v$ and $w$ are worldlines and $x$ is a real number.”

•        “$w$ intersects $<v, x>$ implies that $v$ and $w$ are worldlines and $x$ is a real number.”

• (P2)   

•        “Intersection of worldlines is irreflexive.”

We write as an abbreviation

•          

•          “$v$ and $w$ mutually intersect at $x$ and $y$.”

• (P3)   

•       “For any worldlines $v, w$, and any real number $x$, if $w$ intersects $v$ at $x$, then one and only one real number $y$ exists, such that $v$ and $w$ mutually intersect at $x$ and $y$.”

We write

$\mathsf{Lea}_{vw}~xy$ for

      $\mathsf{V}uab~((\mathsf{Isn}_vu xa \wedge \mathsf{Isn}_{wu} yb~\wedge a<b)\vee ~(\mathsf{Isn}_{vw} by\wedge x<b)\vee ~(\mathsf{Isn}_{vw}xa~\wedge a<y))$

•       “A worldline leads from $v$ at $x$ to $w$ at $y$.”

•       “A worldline leads from $x$ on $v$ to $y$ on $w$.”

• (P4)   

•       “For any two worldlines $v$, $w$ and any real number $x$, there is a real number $y$, such that a worldline exists that leads from $y$ on $w$ to $x$ on $v$.”

• (P5)   

•       “For any two worldlines $v$, $w$ and any real number $x$, with $w$ not intersecting $v$ at $x$, two sets $a$, $b$ exist, such that no worldline intersecting $w$ at $a$ or $b$ intersects $v$ at $x$.”

• (P6)   

•       “For any two worldlines $v$, $w$ and all real numbers $x$, $y$, such that $v$ and $w$ mutually intersect at $x$ and $y$, and all real numbers $a$ that are smaller than $x$ and all real numbers $b$ that are greater than $y$, a worldline exists which leads from $a$ on $v$ to $b$ on $w$.”

• (P7)   

•       “For any two worldlines $v$, $w$ and all real numbers $x$, $y$, such that $v$ and $w$ mutually intersect at $x$ and $y$, and all real numbers $b$ that are greater than $y$, there exist a real number $a$ greater than $x$ and a worldline which leads from $a$ on $v$ to $b$ on $w$.”

• (P8)   

•       “For any two worldlines $v$, $w$ and all real numbers $x$, $y$, such that $v$ and $w$ mutually intersect at $x$ and $y$, and all real numbers $a$ that are smaller than $x$, there exist a real number $b$ smaller than $y$ and a worldline which leads from $a$ on $v$ to $b$ on $w$.”

We write

•       $\mathsf{sou}_{wv}~x$ for  $\{y~{\vert }~\mathsf{Lea}_{wv}~ yx)\}$

•       “the source on $w$ of $x$ on $v$”;

•       tar $_{wv}x$ for$~\{y~\vert ~\mathsf{Lea}_{vw}~xy)\}$

•       “the target on $w$ of $x$ on $v$”;

•       $\mathsf{ina}_{wv}~x~$ for$~\{y~{\vert }~y \in {\mathbb {R}}\wedge y\notin ~\mathsf{sou}_{wv}~x\wedge y\notin \mathsf{tar}_{wv}x)\}$

•       “the inaccessible section on $w$ from $x$ on $v$”;

•       

•       “$f$ smoothly and monotonously maps any $x$ on $v$ to the infimum of the inaccessible section on $w$ of $x$ on $v$.”

•       dpb $_{vwxs}~f$ for $\iota y~(\mathsf{Smm}~f~\wedge ~\mathsf{Mia}~_{wv}~f~\wedge$

•       $\wedge ~\mathsf{V}! g~(\mathsf{V}! h ~(\mathsf{Mia}~_{sv}~g~\wedge \mathsf{Mia}~_{sw}~h~\wedge ~y~=~\partial (h\circ f\circ g))^\prime (g^{-1\prime } x)))$

•       “The derivative of the backward coordinate function $f$ from $v$ to $w$ at $x$ that is parametrized by $s$.”

• (P9)   

•             

•             

•             $\wedge \, \mathsf{Mia}~_{tv}~f_{2} \rightarrow ~\mathsf{dpb}~_{wvxs}~f~\ne ~\mathsf{dpb}~_{wuys}~f_{1}~\wedge ~\mathsf{dpb}~_{wvxs}~f~\ne ~\mathsf{dpb}~_{tvxs}~f_{2}))$

•       “For any two worldlines $v$, $w$, if the source on $w$ of $x$ on $v$ is nonempty, any backward coordinate function from $v$ to $w$ is a smooth and monotonous and, for all real numbers $x$, $y$, all backward coordinate functions $f$ from $v$ to $w$, all worldlines $u$ with which $v$ mutually intersects at $x$ and $y$ and all worldlines $t$ intersecting $w$ at the value of $x$ under $f$, the first derivative of $f$ at $x$ parametrized by $s$ differs both from the first derivative of the backward coordinate function from $u$ to $w$ at $y$ parametrized by $s$ and the first derivative of the backward coordinate function from $v$ to $t$ at $x$ parametrized by $s$.”

We write

Cwl $_{vw}~ux$ for $Wv~\wedge ~ Ww~\wedge v~\ne ~w$

     $\wedge ~\mathsf{V} z~(\mathsf{Lea}_{vu}~zx)~\wedge ~\mathsf{V} z~(\mathsf{Lea}_{uv}~xz)~\wedge ~\mathsf{V} z~(\mathsf{Lea}_{wu}~ zx)~\wedge ~\mathsf{V} z~(\mathsf{Lea}_{uw}~xz)$

     $\wedge \mathsf{V}~z_{1}z_{2}~(z_{1}~\ne ~z_{2}~\wedge ~\lnot \mathsf{Lea}_{vu}~z_{1}x~\wedge ~\lnot \mathsf{Lea}_{vu}~z_{2}x~\wedge ~\lnot \mathsf{Lea}_{uv}~xz_{1}~\wedge ~\lnot \mathsf{Lea}_{uv}~xz_{2})$

     $\wedge ~\mathsf{V} z_{1}z_{2}~(z_{1}~\ne ~z_{2}~\wedge ~\lnot \mathsf{Lea}_{wu}~z_{1}x~\wedge ~\lnot \mathsf{Lea}_{wu}~z_{2}x~\wedge ~\lnot \mathsf{Lea}_{uw}~ xz_{1}~\wedge ~\lnot \mathsf{Lea}_{uw}~xz_{2})$

     “$v$ and $w$ are coordinate worldlines of $x$ on $u$.”

• (P10)    $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \,\,\wedge ~\mathsf{ina}_{wu}~x~=~\mathsf{ina}_{wt}~y)))$ $\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \,\,\wedge ~\mathsf{ina}_{wu}~x~=~\mathsf{ina}_{wt}~y))$))) $\qquad \qquad \wedge ~t~\ne ~u~\wedge ~y~\ne ~x~\rightarrow ~\mathsf{ina}_{vu}~x~\ne ~\mathsf{ina}_{vt}~y~\wedge ~\mathsf{ina}_{wu}~x~\ne ~\mathsf{ina}_{wt}~y))$)

•       “For all worldlines $u$ and real numbers $x$, there are worldlines $v$, $w$ and real numbers $a$, $b$, with $v$ and $w$ being coordinate worldlines of $x$ on $u$ and $a$ being smaller than $b$, such that, for all $x_{1}$ in the inaccessible section on $v$ of $x$ on $u$, there are $y_{1}, y_{2}$ not contained in the inaccessible section on $v$ of $x$ on $u$, with $y_{1}$ being smaller than $x_{1}$ and $x_{1}$ being smaller than $y_{2}$, such that the set of real numbers formed by augmenting the inaccessible section on $v$ of $x$ on $u$ by the interval from any real number $z_{2}$ between $y_{1}$ and $x_{1}$ to $x_{1}$ and diminishing it by the interval from $y_{1}$ to $z_{2}$ is an inaccessible section on $v$ of some real number $y$ on some worldline $t$, to which a worldline leads from $a$ on $u$ and from which a worldline leads to $b$ on $u$, where the inaccessible section on $w$ of $y$ on $t$ is identical with the inaccessible section on $w$ of $x$ on $u$, and such that the set of real numbers formed by augmenting the inaccessible section on $v$ of $x$ on $u$ by the interval from $x_{1}$ to any real number $z_{2}$ between $x_{1}$ and $y_{2}$ and diminishing it by the interval from $z_{2}$ to $y_{2}$ is an inaccessible section on $v$ of some real number $y$ on some worldline $t$, to which a worldline leads from $a$ on $u$ and from which a worldline leads to $b$ on $u$, where the inaccessible section on $w$ of $y$ on $t$ is identical with the inaccessible section on $w$ of $x$ on $u$ and, furthermore, such that the inaccessible sections on $v$ and $w$ of any real number $y$ on any worldline $t$ to which a worldline leads from $a$ on $u$ and from which a worldline leads to $b$ on $u$ differ from the inaccessible sections on $v$ and $w$ of $x$ on $u$, respectively.”

• (P11)                                                   

•       “For all worldlines $u$ and all real numbers $x$, there are two real numbers $a$, $b$, with $a$ being smaller than $b$, such that, for all worldlines $v$ intersecting $u$ at $x$ or being identical with $u$, there is precisely one set of worldlines $z$ containing $v$ whose elements all intersect some worldline $w$ at some real number $y$ to which a worldline leads from $u$ at $a$ and from which a worldline leads to $u$ at $b$ and where, for every worldline $w$ and real number $y$, such that a worldline leads from $u$ at $a$ to $w$ at $y$ and a worldline leads from $w$ at $y$ to $u$ at $b$, there is precise one element of $z$, which intersects $w$ at $y$ or is identical with $w$.”