Axiom:Axiom of Continuity
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This page needs to be split into two: the "Axiom of Continuity" (relating to sets) and the "Axiom Schema of Continuity" which asserts the first statement. One may transclude the other. I think it is worth making this extra effort to separate out what is an "axiom" with what is an "axiom schema", and it would then be appropriate to do something similar with the ZFC axioms which do something similar which is also an axiom schema. Then we can be rigorous at asserting which axioms are dependent upon non-first-order logic. About time this site went down this route. But not me, not today, as I have my nephew's 18th birthday celebrations to attend.
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Contents |
Axiom Schema
This is more properly an collection of axioms rather than a single axiom.
Let $a,b,x,y,$ be points.
Let $\mathsf{B}$ be the relation of betweenness.
Let $\alpha, \beta$ be first-order formulas.
Let $\alpha$ contain no free occurrences of $a,b,y$.
Let $\beta$ contain no free occurrences of $a,b,x$.
This axiom asserts:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\) | \(\displaystyle \exists a: \forall x,y: \left({ \alpha \land \beta \implies \mathsf{B}axy }\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\implies\) | \(\displaystyle \exists b: \forall x,y: \left({ \alpha \land \beta \implies \mathsf{B}xby }\right)\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Alternative form
Let $\mathsf{B}$ be the relation of betweenness.
This axiom asserts:
- $\left({\exists a : \forall x,y : \left({x \in X \land y \in Y}\right)\implies \mathsf{B}axy}\right) \implies \left({\exists b : \forall x,y : \left({x \in X \land y \in Y}\right)\implies \mathsf{B}xby}\right)$
where $a,b,x,y$ are points and $X, Y$ are point sets.
Note that this alternate form of the Axiom of Continuity assumes the existence of sets.
As such, it cannot be expressed in the framework of first-order logic.
Intuition
Consider an infinite line.
Consider it as consisting of a left side and a right side, where the transition is any place you care to choose.
No matter what place you choose, that place of transition is a point.
If you are fairly certain that there are none, please remove {{proofread}} from the code.
Also see
Sources
- June 1999: Alfred Tarski and Steven Givant: Tarski's System of Geometry (The Bulletin of Symbolic Logic Vol. 5, no. 2: 175 – 214) : Page 185 : Axiom $11$
