Ancestral relation

From Wikipedia, the free encyclopedia

In mathematical logic, the ancestral relation (often shortened to ancestral) of an arbitrary binary relation R is defined below.

The ancestral makes its first appearance in Frege's Begriffsschrift. Frege later employed it in hisGrundgesetze as part of his definition of the natural numbers (actually the finite cardinals). Hence the ancestral was a key part of his search for a logicist foundation of arithmetic.

Contents 1 Definition 2 Discussion 3 See also 4 References 5 External links

[edit] Definition

The numbered propositions below are taken from his Begriffsschrift and recast in contemporary notation.

The property F is "R-hereditary" if, whenever x is F and xRy, y is also F:

Frege then defined b to be an R-ancestor of a, written aR*b, iff b has every R-hereditary property that all objects x such that aRx have:

76: .

The ancestral is transitive:

98:

Let the notation I(R) denote that R is functional (Frege calls such relations "many-one"):

115:

If R is functional, we say nowadays that the ancestral of R is connected:

133:

[edit] Discussion

Principia Mathematica made repeated use of the ancestral, as does Quine's (1951) Mathematical Logic.

[edit] See also

• Begriffsschrift
• Gottlob Frege

[edit] References

• George Boolos, 1998. Logic, Logic, and Logic. Harvard Univ. Press.
• Ivor Grattan-Guinness, 2000. In Search of Mathematical Roots. Princeton Univ. Press.

[edit] External links

• Stanford Encyclopedia of Philosophy: "Frege's Logic, Theorem, and Foundations for Arithmetic" -- by Edward N. Zalta. Section 4.2.