Infinitary logic
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Those unfamiliar with mathematical logic or the concept of ordinals are advised to consult those articles first.
An infinite logic is a logic that allows infinitely long statements and infinitely long proofs. Infinite logics have different properties to standard first order logics. In particular infinite logics often fail to be compact or complete. Notions of compactness and completeness that are equivalent in finite logic sometimes are not so in infinite logic. So for infinite logics the notions of strong compactness and strong completeness are defined. In this article we shall be concerned with Hilbert type infinite logics as these have been extensively studied and constitute the most straightforward extensions of finite logic. These are not, however, the only infinite logics around.
Considering whether a certain infinite logic named <math>\Omega</math> logic is complete promises to throw light on the continuum hypothesis.
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A word on notation and the axiom of choice
As we are presenting a language with infinitely long formulae it is not possible to write expressions down as they should be written. To get around this problem we use a number of notational conveniences which strictly speaking are not part of the formal language we are defining. We use <math>\cdots</math> to indicate an expression that is infinitely long. Where it is not clear the length of the sequence is noted afterwards. Where this notation becomes ambiguous or confusing we use suffixes such as <math>\lor_{\gamma < \delta}{A_{\gamma}}</math> to indicate an infinite disjunction over a set of formulae of cardinality <math>\delta</math>. The same notation may be applied to predicates for example <math>\forall_{\gamma < \delta}{V_{\gamma}:}</math>. This is meant to represent an infinite sequence of predicates for each <math>V_{\gamma}</math> where <math>\gamma < \delta</math>.
All usage of suffices and <math>\cdots</math> are not part of formal infinite languages. We assume the axiom of choice (as is often done when discussing infinite logic) as this is necessary to have sensible distributivity laws.
Definition of Hilbert type infinite logics
A first order infinite logic <math>L_{\alpha ,\beta}\ ,\alpha \in regular ,\beta = 0\ or\ \omega < \beta < \alpha </math> has the same set of symbols as a finite logic and may use all the rules for formation of formulae of a finite logic together with some additional ones:
- If we have a set of variables <math>V=\{V_\gamma | \gamma< \delta < \beta \}</math> and a formulae <math>A_0</math> then <math>\forall V_0 :\forall V_1 \cdots (A_0)</math> and <math>\exists V_0 :\exists V_1 \cdots (A_0)</math> are formulae (In each case the sequence of quantifiers has length <math>\delta</math>).
- If we have a set of formulae <math>A=\{A_\gamma | \gamma < \delta <\alpha \}</math> then <math>(A_0 \lor A_1 \lor \cdots)</math> and <math>(A_0 \and A_1 \and \cdots)</math> are formulae (In each case the sequence has length <math>\delta</math>).
The concepts of bound variables apply in the same manner to infinite sentences. Note that the number of brackets in these formulae is always finite. Just as in finite logic a formulae all of whose variables are bound is referred to as a sentence.
A theory T in infinite logic <math>L_{\alpha , \beta}</math> is a set of statements in the logic. A proof in infinite logic from a theory T is a sequence of statements of length <math>\gamma</math> which obeys the following conditions: Each statement is either a logical axiom, an element of T or is deduced from previous statements using a rule of inference. As before all rules of inference in finite logic can be used together with an additional one:
- If we have a set of statements <math>A=\{A_\gamma | \gamma < \delta <\alpha \}</math> which have occurred previously in the proof then the statement <math>\and_{\gamma < \delta}{A_{\gamma}}</math> can be inferred.
We give only those logical axiom schema specific to infinite logic. For each <math>\delta</math> and <math>\gamma</math> such that <math>0 < \alpha < \delta </math> we have the following logical axioms:
- <math>((\and_{\epsilon < \delta}{(A_{\delta} \implies A_{\epsilon})}) \implies (A_{\delta} \implies \and_{\epsilon < \delta}{A_{\epsilon}}))</math>
- For each <math>\gamma < \delta</math> we have <math>((\and_{\epsilon < \delta}{A_{\epsilon}}) \implies A_{\gamma})</math>
- Chang's distributivity laws (for each <math>\gamma</math>): <math>(\lor_{\mu < \gamma}{(\and_{\delta < \gamma}{A_{\mu , \delta}})})</math> where <math>\forall \mu : \forall \delta : \exists \epsilon < \gamma:A_{\mu , \delta} = A_{\epsilon}</math> and <math>\forall g \in \gamma^{\gamma} : \exists \epsilon < \gamma: \{A_{\epsilon} , \neg A_{\epsilon}\} \subseteq \{A_{\mu , g(\mu)} : \mu < \gamma\}</math>
- For <math>\gamma < \alpha</math> we have <math>((\and_{\mu < \gamma}{(\lor_{\delta < \gamma}{A_{\mu , \delta}})}) \implies (\lor_{\epsilon < \gamma^{\gamma}}{(\and_{\mu < \gamma}{A_{\mu ,\gamma_{\epsilon}})}}))</math> where <math>g_{\epsilon}</math> is a well ordering of <math>\gamma^{\gamma}</math>
The last two axiom schema require the axiom of choice because certain sets must be well orderable. The last axiom schema is strictly speaking unnecessary as Chang's distributivity laws imply it, however it is included as a natural way to allow natural weakenings to the logic.
Completeness, compactness and strong completeness
A theory is any set of statements. The truth of statements in models are defined by recursion and will agree with the definition for finite logic where both are defined. Given a theory T a statement is said to be valid for the theory T if it is true in all models of T.
A logic <math>L_{\alpha , \beta}</math> is complete if for every sentence S valid in every model there exists a proof of S. It is strongly complete if for any theory T for every sentence S valid in T there is a proof of S from T. An infinite logic can be complete without being strongly complete.
A logic is compact if for every theory T of cardinality <math>\alpha</math> if all subsets S of T have models then T has a model. A logic is strongly compact if for every theory T if all subsets S of T, where S has cardinality<math>< \alpha</math>, have models then T has a model.
The cardinal <math>\kappa \neq \omega</math> is weakly compact if <math>L_{\kappa , \kappa}</math> is compact and <math>\kappa</math> is strongly compact if <math>L_{\kappa , \kappa}</math> is strongly compact.
Concepts expressible in infinite logic
In the language of set theory the following statement expresses foundation:
<math>\forall_{\gamma < \omega}{V_{\gamma}:} \neg \and_{\gamma < \omega}{V_{\gamma +} \in V_{\gamma}}</math>.
Unlike the axiom of foundation this statement admits no non-standard interpretations. The concept of well foundedness can only be expressed in a logic which allows infinitely many quantifiers in an individual statement. As a consequence many theories, including peano arithmetic, which cannot be properly axiomatised in finite logic can be in a suitable infinite logic. Other examples include the theories of non archimedean fields and torsion free groups.
Complete infinite logics
Two infinite logics stand out in their completeness. These are <math>L_{\omega , \omega}</math> and <math>L_{\omega_1 , \omega}</math>. The former is standard finite first order logic and the latter is an infinite logic which only allows statements of countable size.
<math>L_{\omega , \omega}</math> is also strongly complete, compact and strongly compact.
References
- Carol Karp, Languages with expressions of infinite length.
- Kenneth Jon Barwise, Admissible sets.