[FOM] historical question about the axiomatisation of identity

Ron Rood 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 
axiomatic method.

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.

Ron Rood

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
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