---
tag: PQG7VAZ1R2
kind: concept
name: Infinitary logic
msc: [03C75]
sources:
  - url: https://en.wikipedia.org/wiki/Infinitary_logic
same_as:
  wikidata: Q6029713
  wikipedia: Infinitary logic
---

## Summary

An infinitary logic is an extension of [[first-order logic|086QKPF0A6]] that allows formulas of infinite length.
Examples are conjunctions of infinitely many formulas and infinite blocks of quantifiers.
The logics $L_{\kappa\lambda}$ are the standard family, and $L_{\omega_1\omega}$, with countable conjunctions and finite quantifier blocks, is the most studied one.
These logics can express properties that first-order logic cannot, but they lose the compactness theorem.

## Explanations

### L1 human

In ordinary logic a formula is a finite string of symbols.
Infinitary logic allows a formula to say "$A_1$ or $A_2$ or $A_3$ or ..." without end.
For example, "every element $x$ of the group has some finite order" is the infinite disjunction "$x = e$ or $x^2 = e$ or $x^3 = e$ or ...".
No single first-order sentence says this about all groups.