[FOM] A logical device of Mario Pieri

[JAMES T SMITH]

In his 1906 paper, “On an Arithmetical Definition of the Irrationals”, Pieri proposed an addition to the logical framework of Peano’s Formulary:  a new logical operator I and some new logical postulates.  Letting “individual” mean “non-class”, he postulated (1) there is an individual;  (2) for every class  C  of individuals,  IC  is an individual but not a member of  C;  (3) for all classes  C,D  of individuals,  IC = ID  iff  C = D.

Pieri’s paper attracted almost no explicit attention.  My question:  did anyone ever use that device, presumably without acknowledging Pieri?

In 1882, Pasch had defined  “S  is a positive real number” as  “S  is an initial segment of the positive rational numbers”, thus simplifying Dedekind’s famous definition.  In 1899 Peano complained that incorporating that into the Formulary would cause inconsistent overloading of operators that had already been defined for classes of rationals.  In 1903 Russell called major attention to Pasch’s definition and dismissed Peano’s objection.  Pieri responded to that.  He suggested changing the definition to  “S = ISʹ  for some initial segment  Sʹ ...”,  and, for real numbers  S,T  defining   “S < T”  as   “S = ISʹ   and  T = ITʹ for some initial segments   Sʹ,Tʹ  such that  Sʹ  is a proper subset of  Tʹ.”  And so on.  This amounts to information-hiding, and does avoid some overloading conflicts.  (My parallel with object-oriented programming terminology is intentional.)

Any suggestions are awaited eagerly!

PS:  I notice a namesake on the FOM list.  I’m the retiree from SF State, not in the UK.
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James T. Smith

Professor Emeritus of Mathematics

San Francisco State University

mailto:smith at sfsu.edu<mailto:smith at math.sfsu.edu>

Sent from Roxane

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