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.
Read more about this topic: Infinitary Logic
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