[FOM] Re: numbers and sets

Aatu Koskensilta aatu.koskensilta at xortec.fi
Wed Sep 24 03:00:02 EDT 2003

Eric Steinhart wrote:
> The old Benacerraf arguments to the effect that numbers can't be sets 
> are obsolete. It's remarkable that anybody would think this is still an 
> issue.
> Balaguer nicely argues that there's lots more to being the natural 
> numbers than merely being a model of the Dedekind-Peano axioms. (And 
> since there are lots of models of those axioms, the property of being 
> such a model is vague ­ so vague that it certainly can't serve as a 
> definition of anything.) Balaguer's logic is general and applies to all 
> sorts of alleged "non-uniqueness" problems in mathematics (and 
> elsewhere). His article:
> Balaguer, M. (1998) Non-uniqueness as a non-problem. Philosophia 
> Mathematica (3) Vol. 6, 63 - 84.
> Steinhart (that's me) shows that Benacerraf's own reasoning is unsound. 
> The article:
> Steinhart, E. (2003) Why numbers are sets. Synthese 133, 343 - 361.
> Given the Dedekind-Peano axioms and Benacerraf's own definition of 
> cardinality, Steinhart proves that the natural numbers are the finite 
> von Neumann ordinals. To put it very quickly: (1) say the natural 
> numbers are 0, 1, 2, and so on; (2) Benacerraf's definition of 
> cardinality requires the formation of the sets {}, {0}, {0, 1}, {0, 1, 
> 2} and so on; (3) it's obvious that {}, {0}, {0, 1}, (0, 1, 2} and so on 
> is a model of the Dedekind-Peano axioms; (4) Benacerraf's own 
> non-uniquness argument forces these two series have to be identified -- 
> WHATEVER the natural numbers might be. The identification is 
> unavoidable. (5) Hence: 0 = {}, 1 = {0}, 2 = {0, 1}, and so on; (6) 
> Therefore: the natural numbers are the finite von Neumann ordinals.

I won't comment on this, not having read the papers. To me the main 
force of Benacerraf's argument has always been in his noting that if we 
identify natural numbers with any particular set theoretic 
representation (satisfying the Peano-Dedekind axioms) then suddenly 
natural numbers have all sorts of properties that seem to have nothing 
to do with their being the natural numbers, such as 1 \in 3. But I'll 
have to read the papers you mention in order to have anything sensible 
to contribute here.

> It's astonishing that anybody would think that any set theoretic model 
> of the Dedekind-Peano axioms is as good as any other. The advantages of 
> the von Neumann ordinals are elaborated in almost every introductory 
> text. Here's just a few quick and dirty examples of advantages of the 
> von Neumann's: (1) the VNs identify less-than with membership; (2) the 
> VNs identify less-than-or-equal-to with subset-inclusion; (3) each VN 
> ordinal is internally well-ordered; (4) the relation that internally 
> well-orders each VN ordinal is the relation that well-orders the whole 
> series of VN ordinals; (4) the VNs are uniformly extendible to the 
> transfinite; (5) the n-th VN ordinal has n members.

These are indeed very convenient properties, and they do show that there 
is a very good reason to prefer the von Neumann ordinals in set 
theoretic settings. However, I fail to see how they could support an 
argument to the effect that natural numbers *are* the finite von Neumann 
ordinals, any more than the fact that the Hindu-Arabic numerals are 
immensely more convenient than the Roman numerals gives reason to 
believe that the Hindu-Arabic numerals *are* the natural numbers.

Aatu Koskensilta (aatu.koskensilta at xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
  - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

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