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Topic: Can L(<) be the language of the naturals?
Replies: 7   Last Post: Sep 2, 2013 10:55 AM

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Registered: 12/4/12
Re: Can L(<) be the language of the naturals?
Posted: Sep 2, 2013 10:55 AM
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On 9/1/2013 10:07 PM, Virgil wrote:
> In article
> <>,
> Ben Bacarisse <> wrote:

>> Virgil <> writes:

>>> In article <kvvu1c$b1j$>,
>>> Peter Percival <> wrote:

>>>> David Hartley wrote:
>>>>> In message <>, Jim Burns <>
>>>>> writes

>>>>>> If I say that I have a set with a semi-infinite,
>>>>>> 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 non-standard 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 non-standard.
Alan Smaill pointed that out to you recently.

Peter made a statement in his reply concerning how
"second-order is a different story". The remarks in
the link,

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

"any sufficiently general criticism of realism would
apply to the Skolemite's own model theory as much as
it does to classical set theory"

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