Ancestral relations make their first appearance in Frege's Begriffsschrift. Frege later employed them in his Grundgesetze as part of his definition of the finite cardinals. Hence the ancestral was a key part of his search for a logicist foundation of arithmetic.
The numbered propositions below are taken from his Begriffsschrift and recast in contemporary notation.
A property P is called R-hereditary if, whenever x is P and xRy holds, then y is also P:
Frege defined b to be an R-ancestor of a, written aR*b, if b has every R-hereditary property that all objects x such that aRx have:
The ancestral is a transitive relation:
Let the notation I(R) denote that R is functional (Frege calls such relations "many-one"):
Relationship to transitive closureEdit
The Ancestral relation is equal to the transitive closure of . Indeed, is transitive (see 98 above), contains (indeed, if aRb then, of course, b has every R-hereditary property that all objects x such that aRx have, because b is one of them), and finally, is contained in (indeed, assume ; take the property to be ; then the two premises, and , are obviously satisfied; therefore, , which means , by our choice of ). See also Boolos's book below, page 8.
However, it is worth noting that the ancestral relation cannot be defined in first-order logic. It is controversial whether second-order logic with standard semantics is really "logic" at all. Quine famously claimed that it was really 'set theory in logic's clothing.' In his books setting out formal systems related to PM and capable of modelling significant portions of Mathematics, namely - and in order of publication - 'A System of Logistic', 'Mathematical Logic' and 'Set Theory and its Logic', Quine's ultimate view as to the proper cleavage between logical and extralogical systems appears to be that once axioms that allow incompleteness phenomena to arise are added to a system, the system is no longer purely logical.