# The Philosophy of Mathematics Today

By Matthias Schirn | Go to book overview

is good to 0. If Ϛ is good to n, then Ϛ ⋃ {< n, least member of α not in the range of Ϛo>} is good to sn (the existence of the least member of α not in the range of Ϛ follows from our supposition). By induction, for every n, some o is good to n. If Ϛ is good to n, Ϛ′ is good to n′, and n≤n′, then by induction for every ii) = Ϛ'(i). By comprehension, let ρ = {〈n, k〉: ∃Ϛ (Ϛ is good to sn and Ϛ(n) = k)}. It is sufficiently clear that ρ is a function; domain (ρ) = N; for all i, ρ(i) ∈ α; if ji), and if k ∈ α, k < ρ(i), then for some ji) whenever j

APPENDIX 3. TRANSLATIONS INTO
PRESENT-DAY NOTATION OF §§115, 117, 119 AND
PART OF §§113 OF GRUNDGESETZE

Below we use boldface instead of Fraktur and notation introduced above in place of Frege's own, we omit the signs indicating which rule of inference is applied and reference numbers to axioms of Frege's system, and we utilize certain easy equivalences (e.g. p ∧ q for ¬(p → ¬q)).

§113 (part)

aq*b → ∃e(eqb ∧ aq*=e) (141)

∀e(eqb → ¬q*=e) → ¬q*b (142)

¬aP*=d → d=c → ¬aP*=c

(88): ¬aP*=d → dPb → cPb → ¬aP*=c

¬aP*=d → dPb → ∀e(ePb → ¬aP*=e)

(142): ¬aP*=d → dPb → ¬aP*b

aP*b → dPb → aP*=d (143)

§115
130 bq*=a → ¬bq*a → a=b

¬bq*a → bq*=a → b=a (146)

bq*=a ∧ b≠a → bq*a (147)

147 bP*=a ∧ b≠a → bP*a

(143): dPa → (bP*=a A b≠a) → bP*=d

(bP*=d → (bP*=a ∧ b≠a)) → dPa → ((bP*=a ∧ b≠a) ↔ bP*=d) (β)

134 bP*=d → dPa → bP*a
¬,aP*a → bP*=d → dPa → b≠a

(145): OP*=a → bP*"d → dPa → b≠a

bP*=a → OP*=a → bP*=d → dPa → (bP*=a ∧ b≠a)

(137): dPa → OP*=a → bP*=d → (bP*=a ∧ b≠a)

(β): dPa → OP*=a → ((bP*=a ∧ b≠a) ↔ bP*=d)

(77): dPa → OP*=a → ([x: xP*=a a x≠a]b ↔ bP*=d) (148)

dPa → OP*=a → ∀a([x: xP*=a A x≠a]a ↔ aP*=d)

(96): dPa → OP*=a → #[x: xP*=a ∧ x≠a] = #[x: xP*=d] (149)

-426-

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