Drexel dragonThe Math ForumDonate to the Math Forum



Search All of the Math Forum:

Views expressed in these public forums are not endorsed by Drexel University or The Math Forum.


Math Forum » Discussions » sci.math.* » sci.math.independent

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

Advanced Search

Back to Topic List Back to Topic List Jump to Tree View Jump to Tree View   Messages: [ Previous | Next ]
fom

Posts: 1,968
Registered: 12/4/12
Re: Can L(<) be the language of the naturals?
Posted: Sep 2, 2013 10:55 AM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

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 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,

http://plato.stanford.edu/entries/paradox-skolem/#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"







Point your RSS reader here for a feed of the latest messages in this topic.

[Privacy Policy] [Terms of Use]

© Drexel University 1994-2014. All Rights Reserved.
The Math Forum is a research and educational enterprise of the Drexel University School of Education.