Infinitary logic
Encyclopedia
An infinitary logic is a logic that allows infinitely long statements
and/or infinitely long proofs
. Some infinitary logics may have different properties from those of standard first-order logic
. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logic. So for infinitary logics the notions of strong compactness and strong completeness are defined. In this article we shall be concerned with Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied.
Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis
.
is used to point out an expression that is infinitely long. Where it is unclear, the length of the sequence is noted afterwards. Where this notation becomes ambiguous or confusing, suffixes such as 
are used to indicate an infinite disjunction over a set of formulae of cardinality 
. The same notation may be applied to quantifiers for example 
. This is meant to represent an infinite sequence of quantifiers for each 
where 
.
All usage of suffixes and
are not part of formal infinitary languages. The axiom of choice is assumed (as is often done when discussing infinitary logic) as this is necessary to have sensible distributivity laws.
, β = 0 or ω ≤ β ≤ α, has the same set of symbols as a finitary logic and may use all the rules for formation of formulae of a finitary logic together with some additional ones:
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 finitary logic, a formula all of whose variables are bound is referred to as a sentence
.
A theory T in infinitary logic
is a set of statements in the logic. A proof in infinitary logic from a theory T is a sequence of statements of length 
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 finitary logic can be used, together with an additional one:
We give only those logical axiom schemata specific to infinitary logic. For each
and 
such that 
we have the following logical axioms:
The last two axiom schemata 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.
A logic
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 infinitary logic can be complete without being strongly complete.
A logic is compact if for every theory T of cardinality
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
, have models then T has a model. If a logic is strongly compact, and complete, then it is strongly complete.
The cardinal
is weakly compact
if
is compact and 
is strongly compact
if
is strongly compact.
:
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 finitary logic, can be in a suitable infinitary logic. Other examples include the theories of non-archimedean fields and torsion-free groups. These three theories can be defined without the use of infinite quantification; only infinite junctions are needed.
and 
. The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size.
is also strongly complete, compact and strongly compact.
Statement (logic)
In logic a statement is either a meaningful declarative sentence that is either true or false, or what is asserted or made by the use of a declarative sentence...
and/or infinitely long proofs
Mathematical proof
In mathematics, a proof is a convincing demonstration that some mathematical statement is necessarily true. Proofs are obtained from deductive reasoning, rather than from inductive or empirical arguments. That is, a proof must demonstrate that a statement is true in all cases, without a single...
. Some infinitary logics may have different properties from those of standard first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logic. So for infinitary logics the notions of strong compactness and strong completeness are defined. In this article we shall be concerned with Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied.
Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis
Continuum hypothesis
In mathematics, the continuum hypothesis is a hypothesis, advanced by Georg Cantor in 1874, about the possible sizes of infinite sets. It states:Establishing the truth or falsehood of the continuum hypothesis is the first of Hilbert's 23 problems presented in the year 1900...
.
A word on notation and the axiom of choice
As a language with infinitely long formulae is being presented, it is not possible to write expressions down as they should be written. To get around this problem a number of notational conveniences, which, strictly speaking, are not part of the formal language, are used. is used to point out an expression that is infinitely long. Where it is unclear, the length of the sequence is noted afterwards. Where this notation becomes ambiguous or confusing, suffixes such as are used to indicate an infinite disjunction over a set of formulae of cardinality . The same notation may be applied to quantifiers for example . This is meant to represent an infinite sequence of quantifiers for each where .All usage of suffixes and
are not part of formal infinitary languages. The axiom of choice is assumed (as is often done when discussing infinitary logic) as this is necessary to have sensible distributivity laws.Definition of Hilbert-type infinitary logics
A first-order infinitary logic Lα,β, α regularRegular cardinal
In set theory, a regular cardinal is a cardinal number that is equal to its own cofinality. So, crudely speaking, a regular cardinal is one which cannot be broken into a smaller collection of smaller parts....
, β = 0 or ω ≤ β ≤ α, has the same set of symbols as a finitary logic and may use all the rules for formation of formulae of a finitary logic together with some additional ones:
- If we have a set of variables
and a formulae 
then 
and 
are formulae (In each case the sequence of quantifiers has length 
). - If we have a set of formulae
then 
and 
are formulae (In each case the sequence has length 
).
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 finitary logic, a formula all of whose variables are bound is referred to as a sentence
Sentence (mathematical logic)
In mathematical logic, a sentence of a predicate logic is a boolean-valued well formed formula with no free variables. A sentence can be viewed as expressing a proposition, something that may be true or false...
.
A theory T in infinitary logic
is a set of statements in the logic. A proof in infinitary logic from a theory T is a sequence of statements of length 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 finitary logic can be used, together with an additional one:- If we have a set of statements
which have occurred previously in the proof then the statement 
can be inferred.
We give only those logical axiom schemata specific to infinitary logic. For each
and such that we have the following logical axioms:
- For each
we have 
- Chang's distributivity laws (for each
): 
where 
and 
- For
we have 
where 
is a well ordering of 
The last two axiom schemata 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 finitary 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
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 infinitary logic can be complete without being strongly complete.A logic is compact if for every theory T of cardinality
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, have models then T has a model. If a logic is strongly compact, and complete, then it is strongly complete.The cardinal
is weakly compactWeakly compact cardinal
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence can not be proven from the standard axioms of set theory....
if
is compact and is strongly compactStrongly compact cardinal
In mathematical set theory, a strongly compact cardinal is a certain kind of large cardinal number; their existence can neither be proven nor disproven from the standard axioms of set theory....
if
is strongly compact.Concepts expressible in infinitary logic
In the language of set theory the following statement expresses foundationAxiom of regularity
In mathematics, the axiom of regularity is one of the axioms of Zermelo-Fraenkel set theory and was introduced by...
:
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 finitary logic, can be in a suitable infinitary logic. Other examples include the theories of non-archimedean fields and torsion-free groups. These three theories can be defined without the use of infinite quantification; only infinite junctions are needed.
Complete infinitary logics
Two infinitary logics stand out in their completeness. These are and . The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size. is also strongly complete, compact and strongly compact.