Archive for September 27th, 2010

Hellman’s Second Definability Example

September 27, 2010

Tarski’s theorem shows that the set $\mathsf{Th} (\Omega)^\#$ of code numbers of sentences true in $\Omega$ is not definable.  This has negative consequences for the prospects of aritmetically defining a truth predicate for arithmetic. Nevertheless, Tarski showed how to define truth in terms of satisfaction and by giving an inductive definition of satisfaction beginning with atomic sentences and on up with sentences of higher and higher complexity in terms of the satisfaction of their parts. While a stronger set theory or higher-order logic is still required to convert the inductive definition into an explicit one, Hellman investigates how Det-T and Det-R measure up against this weaker type of definability.

Addison’s theorem establishes that the class of arithmetically definable sets of numbers is itself not an arithmetically definable class of sets.  This means that in the language, $\textup{L}$, of arithmetic extended by a one place predicate $\textup{G}(x)$, no formula $\textup{S}$ is true in $\Omega$ when $\textup{G}(x)$ is assigned a set $\textup{A}$ of numbers such that $\textup{A}$ is definable over $\Omega$.  The proof is involved and is clinched by contradiction on the existence of a generic arithmetical set (I may, or may not, get around to explaining what this is in the next post or so, since it involves explanation of the technique of forcing).

The example turns on this: Addison’s theorem shows that the predicate $\textup{DF}arith$ = ‘set of numbers definable in arithmetic’ is not inductively definable in  arithmetic. Nevertheless, $\textup{DF}arith$ is determined by the primitive predicates of $\textup{L}$.  Set out the following: $\alpha$ is the set of standard $\omega$-models of arithmetic.  Now extend each model $m \in \alpha$ by adding the class $\textup{C}$ of all sets $\textup{X}$ of natural numbers from the domain of $m$ such that $\textup{X}$ is in the extension in $m$ of a formula $\textup{B}(x)$ of $\textup{L}$ with one free variable.  Let $\alpha^{\textup{C}}$ be the class of $\alpha$-structures extended in this way.

So, $\alpha^{\textup{C}}$ contains all the standard models of arithmetic that have standard interpretations of $\textup{DF}arith$. This means that in $\alpha^{\textup{C}}$-structures, $\textup{L}$ reference determines $\textup{DF}arith$ reference but within this same class of structures, $\textup{L} + \textup{G}(x)$ does not inductively define $\textup{DF}arith$.  In spite of this lack of definiability,  Det-R still holds since (and this is all up to isomorphism) any two structures that assign the same interpretations to the primitives of $\textup{L}$ must also assign the same extension to the well-formed-formulas of $\textup{L}$ with only one free variable.  So, up to isomorphism, the same sets of natural numbers are assigned to the distinguished elements of $\textup{C}$

The next example from Hellman is not mathematical, but from classical particle mechanics.  After that I will go into Hellman’s clarification on the difference between the ontological and ideological status of attributes, properties and relations before moving into constructive work on the mental.