[FOM] historical question about the axiomatisation of identity
ron.rood at planet.nl
Tue Sep 20 10:57:39 EDT 2005
As to question 1: no it seems that, in 1928, Hilbert & Ackermann where
not the first to axiomatise identity as an equivalence obeying
substitutivity. It seems to me that Peano already did some such thing
in 1889, in is "Arithmetices principia", although I am not sure about
substitutivity here. Perhaps Peano himself was not very clear on this
point himself; see p.87 of the van Heijenoort anthology (it seems
you've got that one...). See also p.219 of Russell's The Principles of
Mathematics from 1903. Perhaps it also interesting to check Russell and
Whitehead's Principa; I myself do not have a copy within my reach now.
But perhaps there is a little problem here: to *axiomatise* a relation
(such as identity), and to define it accordingly, is typical for
Hilbert's axiomatic method; it is not clear–to me, at least–whether and
to what extent both Peano and Russell accepted this method, whether
they both where aware of it. Both your questions presuppose the
As to question 2: no, in 1930, Gödel was not the first to axiomatise
identity as a reflexive relation obeying substitutivity. The earliest
reference I have found is Hilbert 1904, "Über die Grundlagen der Logik
und der Arithmetik"; see p.132 of van Heijenoort's anthology. In this
paper, Hilbert does not de facto deduce symmetry and transitivity. See
also Hilbert's 1927 paper "Die Grundlagen der Mathematik"; cf. p.467 of
the van Heijenoort anthology. Again, perhaps it is also of interest to
check Principia here.
Muller F.A. heeft op dinsdag, 20 sep 2005 om 00:43 (Europe/Amsterdam)
het volgende geschreven:
> Dear all,
> In *Grundzuge der theoretischen Logik* (1928),
> Hilbert & Ackermann axiomatised identity as
> an equivalence-relation that obeys substitutivity
> (page 86).
> In his completeness paper of 1930, Godel axiomatised
> identity as a reflexive relation that obeys substitutivity
> (Van Heijenoort's source book, page 589) --- symmetry
> and transitivity can be deduced.
> Two historical questions.
> Q1: Were Hilbert & Ackermann the first to do what
> they did (see first paragraph)?
> Q2: Was Godel the first to see that reflexivity
> and substitutivity suffice?
> --> F.A. Muller
> Utrecht University
> Erasmus University Rotterdam
> FOM mailing list
> FOM at cs.nyu.edu
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