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Re: Can L(<) be the language of the naturals?
Posted:
Sep 2, 2013 10:55 AM


On 9/1/2013 10:07 PM, Virgil wrote: > In article > <0.bae957e5d0142561b752.20130901195027BST.87wqn0wf4c.fsf@bsb.me.uk>, > Ben Bacarisse <ben.usenet@bsb.me.uk> wrote: > >> Virgil <virgil@ligriv.com> writes: >> >>> In article <kvvu1c$b1j$2@news.albasani.net>, >>> Peter Percival <peterxpercival@hotmail.com> wrote: >>> >>>> David Hartley wrote: >>>>> In message <52236CD3.1030800@osu.edu>, Jim Burns <burns.87@osu.edu> >>>>> writes >>>>>> If I say that I have a set with a semiinfinite, >>>>>> discrete, linear order, (N, <), is that enough to >>>>>> define the naturals? >>>>> >>>>> I'm afraid not. Thee are many other orderings satisfying your axioms. >>>>> E.g. N + Z  i.e. a copy of N followed by a copy of Z. >>>> >>>> Also, there is no recursive set of first order axioms that will capture >>>> just the natural numbers. >>> >>> What's wrong with the von Neumann model? >> >> Nothing, but it's only one model. There will always be others that >> satisfy the axioms but which have nonstandard numbers. > > No others that can be defined purely in terms of an empty set, and such > elementary set operations. >
You have a number of views which are nonstandard. Alan Smaill pointed that out to you recently.
Peter made a statement in his reply concerning how "secondorder is a different story". The remarks in the link,
http://plato.stanford.edu/entries/paradoxskolem/#3
should convey some of how "the different story" relates to Peter's response.
It is the mathematics of universal algebra rather than the principles of logic which dominate model theory.
You might find the latter part of Section 3.2 interesting.
It discusses the fact that there are skeptics of Skolem's ideas. Among the remarks there you will find:
"any sufficiently general criticism of realism would apply to the Skolemite's own model theory as much as it does to classical set theory"



