Results 1 to 5 of 5

Math Help - cardinality of L inside V

  1. #1
    Member
    Joined
    Aug 2010
    Posts
    130

    cardinality of L inside V

    A quick question, just to make sure: assuming ~(V=L), then L is countable when viewed as a set inside V, correct?
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor FernandoRevilla's Avatar
    Joined
    Nov 2010
    From
    Madrid, Spain
    Posts
    2,162
    Thanks
    45

    Re: cardinality of L inside V

    Quote Originally Posted by nomadreid View Post
    A quick question, just to make sure: assuming ~(V=L), then L is countable when viewed as a set inside V, correct?
    I don't see the sense of the question. Could you reword it?.
    Last edited by FernandoRevilla; July 3rd 2011 at 04:09 AM.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Aug 2010
    Posts
    130

    Re: cardinality of L inside V

    Ah, sorry. I will put it in two ways, hoping that one of them makes sense.

    (1) Is this true or false: Given a sufficiently strong cardinal \kappa so that the axiom of constructibility fails, then < V_\kappa, \epsilon> \models "there exists a set L closed under Def, contains the empty set, and is equinumerable with the set of natural numbers" ?

    (2) The definition of L as a stepwise process in which each step depends on the number of finite first-order formulas; this number would be countable in the above-mentioned model. Or, looked at another way, since no first-order formula can assure that a set is uncountable, it would seem that, in a larger-than-L universe, L would be countable. If one claims uncountability for L from the fact that the process is taken over the ordinals seems like a vicious circle.

    Hm, that's still a bit vague. I am hoping that the direction of my question is clearer, so that someone's answer can point me in the right direction to make it more precise.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Senior Member
    Joined
    Feb 2010
    Posts
    466
    Thanks
    4

    Re: cardinality of L inside V

    L is a proper class, so it doesn't have a cardinality.

    I don't know what you have in mind with "when viewed as a set inside V".

    V is a proper class and L too is a proper class. L is a subclass of V. With the axiom of constructibility, L is not just a subclass of V but L is V. With the negation of the axiom of constructibility, L is a proper subclass of V.

    But in any case, neither V nor L have a cardinality.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Aug 2010
    Posts
    130

    Re: cardinality of L inside V

    Ah, right, I should have thought of that when mentioning the bit about On. Silly of me, so thanks for the correction. I should stick to some L_\kappa to get a countable model.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Area inside cardioid and also inside circle??
    Posted in the Calculus Forum
    Replies: 3
    Last Post: April 19th 2010, 06:57 PM
  2. Cardinality
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: February 8th 2009, 03:23 PM
  3. Cardinality of a Set
    Posted in the Discrete Math Forum
    Replies: 3
    Last Post: January 27th 2009, 04:33 PM
  4. cardinality
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: April 15th 2008, 08:18 PM
  5. cardinality
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: April 9th 2008, 04:37 AM

Search Tags


/mathhelpforum @mathhelpforum