On 12/06/2013 4:16 AM, Alan Smaill wrote: > Peter Percival <email@example.com> writes: > >> Nam Nguyen wrote: >> >>> Yes. Something unclear or vague you did have: what _are_ the >>> "natural numbers" ? >> >> The natural numbers are the elements of the intersection of all sets M >> such that: >> (1) 0 in M >> (2) if x in M then x+1 in M. >> It's strange that you yourself have come close to this definition >> (though you've never got it quite right). So what is your objection >> to the[*] right version? >> >> [* Or 'a right version', there is more than one way to do it.] > > we can of course ask what "0" and "+" refer to ... > this is (very roughly) one of Nam's points, I think.
That's right. My main theme here is the natural numbers aren't just individuals of the universe symbolized as U but are also part of a _language structure_ symbolized as N(U).
But as with any language structure, all relevant predicates have to be constructed, enough to the point that if one is given a sentence (formula) F that must be either true or false in N(U), then F must must necessarily be verifiable (in principle at least), as true or as false in N(U).
That's because the whole definition of a language structure is dedicated to this single purpose: _to ASSERT in a VERIFIABLE manner a sentence as_ _true or as false_ .
> He was sort of persuaded that the denotation of "0" is recognisable > in any given particular "language structure" for PA. > > The notion of standard model makes sense, even if "0" does not > correspond to the empty set.
Exactly. The natural numbers collectively is just a language structure constructed to assert, in a verifiable manner, (sentence) formulas as true or false, by definition of a structure.
Therefore if there's a sentence that it's impossible to verifiable as true or false, then - by definition - we don't have a structure, however intended.
The key "strategy" of my argument in this respect is that, to the extend that we have to construct an infinite (language) structure, as N(U) is supposed to be, generalized-inductive-definition constructions of predicate-sets are _NOT_ sufficient to assert the truth value of certain formulas.
We don't have the concept _one_ modulo arithmetic: for each modulo arithmetic, there's a fixed non-zero number m such that the sentence formula m+m=0 is true, but is also false in _another_ modulo arithmetic, all modulo arithmetic structure being finite of course.
Then, _the concept of the natural numbers_ is the concept of _AN_ _infinite modulo arithmetic_ where the the modulo-formula isn't m+m=0 but cGC, ~cGC, or any of infinitely many such "impossible" formulas.
Iow, there's indeed a class of infinitely many concepts each of which can be named (known) as "the natural numbers".
From this class, mathematical relativity can be established.
-- ---------------------------------------------------- There is no remainder in the mathematics of infinity.