Results 1 to 2 of 2

Math Help - first-order theory

  1. #1
    Newbie
    Joined
    Apr 2010
    Posts
    10

    first-order theory

    Recall that the ordering on Q resp. on R is archimedean, i.e. for every x in Q resp. x in R there is some n in N with -n<x<n. Use the compactness theorem to prove that archimedeanity is not a first-order property.

    Could anyone please give me some hints how to handle this problem? Any help is appreciated!
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2008
    From
    Paris
    Posts
    354
    Hi

    Basically, to show that archimedeanity is not a first order property, you want to obtain two equivalent structures, one archimedean and the other not.

    Let's work with \mathbb{R}, we want to find for instance A\equiv\mathbb{R} in the language \{+,\times ,0,1,\leq\} with an element a\in A such that for all n\in\mathbb{N}, n<a.

    So consider the theory T=Th(\mathbb{R})\cup\{c\ \text{is strictly greater than any natural integer}\} where c is a new constant symbol ( T is therefore a \{+,\times ,0,1,\leq,c\}-theory )


    1. Write \{c\ \text{is strictly greater than any natural integer}\} as a set of first-order formulae.

    2. Use compactness to prove that T is consistent.

    3.Conclude.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Group Theory and Order of Elements
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: July 31st 2013, 09:17 PM
  2. First Order Perturbation Theory help
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: October 23rd 2011, 12:17 PM
  3. Group Theory - order of element
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: December 2nd 2009, 04:50 AM
  4. Number theory, order of an integer
    Posted in the Number Theory Forum
    Replies: 1
    Last Post: November 21st 2006, 05:53 AM
  5. need help, order theory
    Posted in the Discrete Math Forum
    Replies: 7
    Last Post: November 8th 2006, 09:22 AM

Search Tags


/mathhelpforum @mathhelpforum