[FOM] First-order arithmetical truth
Stephen Pollard
spollard at truman.edu
Sat Oct 14 15:03:26 EDT 2006
I wrote:
> The first-order number theoretic truths are exactly the first-order
> sentences in the language of arithmetic that follow from the axioms
> of Peano Arithmetic supplemented by the following version of the
> least number principle: "Among any numbers there is always a
> least." (This principle is not firstorderizable; but that doesn't
> make it unintelligible.)
Francis Davey replied:
> I'm not sure that really answers my problem (though I would be
> interested to know if I am right about that). The LNP sounds like
> something you would need to formalise using a concept of "set",
> which is even harder to understand than that of "natural number".
Two responses: (1) Someone who understands the LNP is, indeed, in a
position to understand what sets of natural numbers there are. Say
that some numbers "form a set" if and only if there is a set whose
members are exactly those numbers. (The primes, for example, form a
set if and only if there is a set whose members are exactly the
primes.) The unrestricted set formation principle for natural numbers
is this: "Any numbers will form a set." The point is: a person who
understands the phrase "any numbers" can easily be taught what sets
of numbers there are. That doesn't mean, however, that our
understanding of "any numbers" depends on our understanding of "set
of numbers." (I can report that, in my own case, the converse seems
to hold.) (2) If some people really didn't understand the LNP, I
would wonder whether they really understood the order-type of the
naturals.
Stephen Pollard
Professor of Philosophy
Division of Social Science
Truman State University
spollard at truman.edu
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