Frege’s Definition on Ancestral of a Relation R
Extra notes for LSG Ch.4.2

Table of Contents

The second-order predicate calculus with which we began has been extended with:

Frege’s analysis of numbers:

\begin{align*} &C_0 = [\lambda x \, x\neq x] \\ &C_1 = [\lambda x \, x= \# C_0] \\ &C_2 = [\lambda x \, x= \# C_0 \lor x= \# C_1] \\ &C_3 = [\lambda x \, x= \# C_0 \lor x= \# C_1 \lor x= \# C_2] \\ &\text{etc.} \end{align*}

Each concept \(C_{k}\) can define a number like:

\begin{align*} &0=\#C_{0}\\ &1=\#C_{1}\\ &2=\#C_{2}\\ &3=\#C_{3}\\ &\text{etc.} \end{align*}

Such a sequence is simply a list, it does not constitute a definition of a concept (e.g., natural number) that applies to all and only the numbers defined in the sequence. Also, this sequence cannot show that every natural number has a successor.

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Frege defined the concept of natural number by first defined a 2-place relation \(R\), the relation is a parent of. And the general concept \(x\) is an ancestor of \(y\) in the R-series, namely the ancestral of the relation \(R\) — \(R^*\) — was first defined in Begriffsschrift (BS (1879), §26, Proposition 76; BS (1884), §79)

Consider \(a\) is an ancestor of \(b\), and \(b\) is an ancestor of \(c\) and so on, the series will be in the form of \(Rab, Rbc, Rcd, \dots\). Then Frege showed how to define the relation \(R^*\) (\(x\) is an ancestor of \(y\) in the \(R\)).

1. The predecessoral relation \(R\)

The concept of \(x\) immediately precedes \(y\) defined by (GL, §76, and GG I, §43):

\(x\) (immediately) precedes \(y\) if and only if there is a concept \(F\) and an object \(w\) such that: (a) \(w\) falls under \(F\), (b) \(y\) is the number of \(Fs\), and (c) \(x\) is the number of the concept object falling under \(F\) other than \(w\).

This can be formally renderred as:

\begin{align*} &\mathit{Precedes}(x,y) \equiv \\ &\quad \exists F\exists w(Fw\& y = \#F \& x = \#[\lambda z\, Fz \& z\neq w]) \end{align*}

(Here the lambda-calculus λ-notation \([\lambda u \: \phi]\) is to name the concept of being an object \(u\) such that \(\phi\))

2. The (strong) ancestral relation \(R^*\)

Frege first defined the ancestral of a relation \(R\) in BS (Part III, Proposition 76).

A preliminary definition required for his definition of the ancestral of \(R\) is

\(F\) is hereditary in the R-series if and only if every pair of R-related objects \(x\) and \(y\) are such that \(y\) falls under \(F\) whenever \(x\) falls under \(F\).

Formally (KS §24),

\[ \text{Her}_{xyz}(F_x,R_{yz})\equiv\forall x\forall y(Rxy \land Fx \to Fy)) \]

Then, Frege’s definition of the strong \(R^*\) (Frege 1967, §26; Frege, 1980, §79):

\[ R^*(a,b) \equiv \forall F[\text{Her}_{xyz}(Fx,Rxy) \land \forall x(Rax \to Fx) \to Fb] \]

His definition of the strong \(R^*\) in GG:

\[ R^*(a,b) \equiv \forall F[\forall x\forall y(Fx\land Rxy \to Fy)\land\forall x(Rax\to Fx)\to Fb] \]

3. The (week) ancestral relation \(R^+\)

The week ancestral relation is the concept y is a member of the R-series beginning with x, (BS, Part III, Proposition 99; GL, §81, and GG I, §46)

\(b\) is a member of the R-series beginning with \(a\) if and only if either \(a\) bears the ancestral of \(R\) to \(b\) or \(a=b\).

Formally,

\[ R^{+}ab \equiv R^* ab \lor a =b \]

Frege’s definitions of ancestral finally allow him to define the natural numbers:

\[ \mathbb{N}a\equiv P^{+}0a \]

4. Abbreviations

Abb. Year Name
BS 1879 Begriffsschrift
GL 1884 Die Grundlagen der Arithmetik
GG 1893/1903 Grundgesetze der Arithmetik, Vol. 1, 1893; Vol. II, 1903