Results 1 to 9 of 9

Math Help - Bounded Comprehension Principle

  1. #1
    Junior Member
    Joined
    Nov 2010
    Posts
    40

    Bounded Comprehension Principle

    Using the Bounded Comprehension Principle, show that if x and y are sets then
    so are:

    a. \{z:\forall w( w\epsilon z \rightarrow w\epsilon x) \}

    b. \{w:\exists z( z\epsilon x \wedge w= y\cap z) \}

    c. \{w:\exists z( z\epsilon x \wedge w= y\cup z) \}

    I'm just not sure the Bounded Comprehension Principle applies here... could someone explain this?
    Last edited by gutnedawg; January 19th 2011 at 05:44 PM.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    Let me try the first one:

    \{z:\forall w( w\epsilon z \rightarrow w\epsilon x) \}=\{ z\in P(x):\forall w( w\epsilon z \rightarrow w\epsilon x) \}=\{ z\in P(x):z=z\}=P(x)

    (to answer the question you only need the first equation)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    I guess I wasn't clear in my description

    You have to demonstrate that for all of the sets
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    Quote Originally Posted by DrSteve View Post
    Let me try the first one:

    \{z:\forall w( w\epsilon z \rightarrow w\epsilon x) \}=\{ z\in P(x):\forall w( w\epsilon z \rightarrow w\epsilon x) \}=\{ z\in P(x):z=z\}=P(x)

    (to answer the question you only need the first equation)
    What is P(x)? My definition of Bounded Comprehension is for every property phi(x) and every ordinal a, the set {x: rk(x) <a and phi(x)} exists at time a
    Last edited by gutnedawg; January 19th 2011 at 08:26 PM.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    P(x) is the Power Set of x, that is P(x)=\{ A: A\subseteq X \}.

    Since x is a set, so is P(x) (by the power set axiom).

    I realize that you need to do all of them. I did the first one to start you off. If you understand this one, then you should at least be able to attempt the other two. Give them a try, show your thoughts, and I'll help out if you get stuck.
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    I don't understand how this is using the bounded comprehension principle
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    The Comprehension schema \{ x:\phi (x)\}

    Bounded Comprehension is \{ x\in a:\phi (x)\} where a is a set.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    Junior Member
    Joined
    Nov 2010
    Posts
    40
    Quote Originally Posted by DrSteve View Post
    The Comprehension schema \{ x:\phi (x)\}

    Bounded Comprehension is \{ x\in a:\phi (x)\} where a is a set.
    Bounded Comprehension is for every property \phi(x) and every ordinal \alpha, the set \{x: rk(x) < a \wedge \phi(x)\} exists at time \alpha

    this is my definition and I'm not sure how it relates to yours
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Senior Member
    Joined
    Nov 2010
    From
    Staten Island, NY
    Posts
    451
    Thanks
    2
    I've never heard that definition, but it is equivalent to the one I have given you. For example, if x has rank \alpha, the rank of P(x) can be expressed in terms of \alpha (I think it has rank \alpha +1). Conversely you can find a bounding set for a collection of sets of rank less than a fixed ordinal.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Set Term Comprehension
    Posted in the Discrete Math Forum
    Replies: 1
    Last Post: October 19th 2011, 01:09 PM
  2. comprehension -/sinx-cosx/=√1-sin2x
    Posted in the Trigonometry Forum
    Replies: 9
    Last Post: July 17th 2011, 05:08 AM
  3. [SOLVED] Correct use of Axiom Schema of Comprehension
    Posted in the Discrete Math Forum
    Replies: 2
    Last Post: March 20th 2011, 03:51 PM
  4. Well ordering principle and the maximum principle
    Posted in the Advanced Algebra Forum
    Replies: 6
    Last Post: August 3rd 2010, 09:31 AM
  5. Momentum Principle and Energy Principle Problem
    Posted in the Math Topics Forum
    Replies: 3
    Last Post: October 4th 2009, 02:42 AM

Search Tags


/mathhelpforum @mathhelpforum