Infinitary Logic¶

An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.

Version: v1 · 2026-09-28 · History
Domain-specific #: 10041
Domain group: Formal Sciences
Origin domain: Mathematics
Subdomains: Mathematical Logic, Model Theory → Mathematics

Core Idea¶

Infinitary Logic is treated here as the recurring mathematics, logic, and statistics identity summarized by this source-grounded definition: An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The concept was introduced by Zermelo in the 1930s. 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.

Scope of Application¶

A word on notation and the axiom of choice. To get around this problem a number of notational conveniences, which, strictly speaking, are not part of the formal language, are used. \cdots is used to point out an expression that.
A word on notation and the axiom of choice. Where this notation becomes ambiguous or confusing, suffixes such as \bigvee{\gamma are used to indicate an infinite disjunction over a set of formulae of cardinality \delta .
Formal languages. The language may also have function, relation, and predicate symbols of finite arity.
Definition of Hilbert-type infinitary logics. As before, all rules of inference in finitary logic can be used, together with an additional one.
Concepts expressible in infinitary logic. The concept of well-foundedness can only be expressed in a logic that allows infinitely many quantifiers in an individual statement.

Clarity¶

A clear use of Infinitary Logic names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. The strongest recognition evidence in the frozen account is: An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.

Manages Complexity¶

Infinitary Logic compresses multiple mathematics, logic, and statistics details into a stable diagnostic relation. The source shows both the central mechanism—the concept of well-foundedness can only be expressed in a logic that allows infinitely many quantifiers in an individual statement.—and the practical consequence—as a language with infinitely long formulae is being presented, it is not possible to write such formulae down explicitly.

Abstract Reasoning¶

Type the carrier. Identify the mathematics, logic, and statistics entities to which the claim applies.
State the relation. Use the source-grounded identity: An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs.
Check operation and conditions. The former is standard finitary first-order logic and the latter is an infinitary logic that only allows statements of countable size.
Demand recognition evidence. An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. 5.

Knowledge Transfer¶

Within the home domain. Knowledge about Infinitary Logic transfers literally when a new case preserves the same carrier type, relation, and recognition test. To get around this problem a number of notational conveniences, which, strictly speaking, are not part of the formal language, are used. \cdots is used to point out an expression that is infinitely long. Where this notation becomes ambiguous or confusing, suffixes such as \bigvee{\gamma are used to indicate an infinite disjunction over a set of formulae.

Relationships to Other Abstractions¶

Current abstraction Infinitary Logic Domain-specific

Parents (1) — more general patterns this builds on

Infinitary Logic is a kind of Formal System Prime

An infinitary logic is a formal system whose formulas or proofs permit specified infinite constructions; it is broader than, not a child of, Omega-logic.

Hierarchy paths (2) — routes to 2 parentless roots

Infinitary Logic → Formal System → Formalization → Representation → Abstraction

Show alternative path (1)

Neighborhood in Abstraction Space¶

Infinitary Logic sits in a moderately populated region (56^th percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Formal Logic & Semantic Systems (18 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08