[FOM] First-order arithmetical truth

[Stephen Pollard]

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