> Is this thing for real? Looks good to me , but I'm no expert .
As I read it, he says: It is known that if a theory has a model, then the theory is consistent. But in fact PA has the natural numbers as a model, therefore PA is consistent.
>> Is this thing for real? Looks good to me , but I'm no expert .
> As I read it, he says: It is known that if a theory has a model, then > the theory is consistent. But in fact PA has the natural numbers as a > model, therefore PA is consistent.
Actually, it mostly use a confusion between the formal notion of model (for which the result of consistency is well-known since Godel) and the idea of "the natural numbers", which is *not* a model, being not formalised (and in some theories, the naïve integers are not even a set...) if what he asserts was true (i.e. proven), it would refutates Church thesis, and a lot is known about that, especially as he pretend to get a proof of Consis PA by syntaxic means (and of course do nothing of the sort)
On Dec 1, 1:29 pm, Denis Feldmann <feldmann.denis.asuppri...@neuf.fr> wrote:
> the formal notion of model > (for which the result of consistency is well-known since Godel)
Just to be clear, Godel completeness may be stated in this form: If a set of formulas G (such as a theory) is consistent, then G has a model M (M satisfies all the members of G).
On the other hand, the principle that "if a set of formulas G (such as a theory) has a model M (M satisfeis all the formulas of G), then G is consistent" is a very old notion (well before Godel) though not necessarily in the same terminology I just used.
> Is this thing for real? Looks good to me , but I'm no expert .
McCall is right in some sense. We can certainly convince ourselves that PA is consistent through semantics. In fact we can define the set of the naturals from a handful of intuitive concepts such as 0, 1, addition and finiteness, as follows:
1. The number 0 (the number of objects which applies when there is no object present) is a natural number.
2. Whatever number can be reached from 0 by a finite number of additions of 1 is a natural number.
3. Nothing else is a natural number.
There is no corresponding formal definition of N. In first order we cannot express the third clause and that's why we cannot exclude nonstandard models with 'supernatural' numbers. But remember that even second order Peano arithmetic, which under classical semantics has all progressions and only them as models (hence not only the progression of the naturals), has in addition nonstandard models under Henkin semantics; those nonstandard models do not contain the set N (otherwise induction would prove that all objects in the domain are natural numbers).
Now we can define addition and multiplication in N and see that PA's axioms are true in N. We know as well that first order inference rules preserve truth. So we can intuitively know beyond any reasonable doubt that PA is consistent and that the Gödel sentence G(S, g), as standardly interpreted, is true for each formal system S of Peano arithmetic and for each gödelization g of S.
So certainly we know better than PA in that sense.
This does not imply that we are not Turing machines; it only implies that we are not Turing machines equivalent to PA.
Nevertheless, the role of semantics and intuition in the whole issue suggests the following: while we as semantic creatures can get a clear concept of what natural numbers are, no purely syntactic device can; so we are not purely syntactical devices; since Turing machines are purely syntactical devices, we are not Turing machines.
I must say that McCall's argument for the soundness of induction is obscure to me.
On Dec 2, 3:34 pm, LauLuna <laureanol...@yahoo.es> wrote:
> We can certainly convince ourselves > that PA is consistent through semantics.
In Z set theory (and even in weaker systems) we prove PA is consistent. So there's no reason that the consistency of PA should be any more dubious than many another mathematical theorem whose proof is formalized in set theory.
