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What's wrong with my logic ?

  1. Aug 23, 2013 #1
    Hi,

    Consider the following two logics:

    Logic 1:
    Given an infinite countable set of real number A = {a1, a2, a3 ...... }
    If there exist a real number M such that: a1 + a2 + a3 + ... + an < M for all integer n,
    then the infinite sum a1 + a2 + a3 ... < M

    Logic 2:
    Given an infinite collection of open set U = {A1, A2, A3 ...... }
    If A1 [itex]\cap[/itex] A2 [itex]\cap[/itex] ... [itex]\cap[/itex] An is an open set for all integers n, then the infinite intersection A1 [itex]\cap[/itex] A2 [itex]\cap[/itex] ... is also an open set


    I am pretty sure logic 1 is legit while logic 2 is fallacious
    My question is, what is wrong with the logic of the form:

    If Predicate P(n) is true for all integer n, then [itex]\\lim_{n\rightarrow +\infty}P(n)[/itex] is also true.

    Is there a general rule as to when such logic holds, and when not ?
    Or, it is totally irrelevant and different problem yields different result ?

    Thanks and you folks are great !
     
  2. jcsd
  3. Aug 23, 2013 #2

    mfb

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    Staff: Mentor

    Correct, if you change "<M" to "≤M" in the first example.

    That is not true for all P.

    As another example, consider the set A = {a1, a2, a3 ...... } with ##a_i=\frac{1}{i^2}##. The sum of the first n elements is always a rational number, but the limit ##\frac{\pi^2}{6}## is not.

    Right.
     
  4. Aug 23, 2013 #3
    Hi mfb,

    Thanks for the cool explanation. So, do you think it makes sense if I say:

    +++++

    In general, "Predicate P(n) is true for all integer n" has nothing to do with the truth value of [itex]\\lim_{n\rightarrow +\infty}P(n)[/itex].

    We need to apply different tricks specific to the problem itself to determine if [itex]\\lim_{n\rightarrow +\infty}P(n)[/itex] is true, and even if it is, it probably has nothing to do with "Predicate P(n) is true for all integer n"


    Am I right ?

    +++++

    Sorry for the mumble mumble but my teacher grill me real hard on this.

    Thanks
     
  5. Aug 23, 2013 #4

    rubi

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    Science Advisor

    The general case of this is transfinite induction. If you want to prove something for a bigger class of ordinal numbers than just the natural numbers, it's not enough to just prove [itex]P(\alpha)\rightarrow P(\alpha+1)[/itex], but you also need to prove [itex]P(\alpha)[/itex] for all limit ordinals. In your case, you'd have to prove the case [itex]\alpha=\omega[/itex] (the ordinal corresponding to the case [itex]n\rightarrow\infty[/itex]) separately.

    See http://en.wikipedia.org/wiki/Transfinite_induction for more information.

    One doesn't need to use induction for such proofs, but i thought, this might enlight you.
     
    Last edited: Aug 23, 2013
  6. Aug 23, 2013 #5

    D H

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    Staff Emeritus
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    Your "logic 1" is also fallacious. Consider 1/2+1/4+1/8+...+1/2n. This is less than 1 for all n, yet the infinite sum is exactly 1. As mfb already noted, you have to replace < with ≤.

    What's wrong is that (ordinary) mathematical induction is a statement about finite numbers. A limit to infinity (or to negative infinity) is a horse of a different color.
     
  7. Aug 23, 2013 #6
    Given a sequence [itex]\{P(n)\}_{n=1}^\infty[/itex] of predicates, it's not in general clear what one means by [itex]\lim_{n\to\infty}P(n).[/itex] Without a general meaning for it, we don't have a general meaning for the statement [tex]\text{``}P(n)\text{ true for all }n\in\mathbb N\implies \lim_{n\to\infty}P(n) \text{ true."}[/tex] So if the above statement doesn't have a precise meaning, it's hard to make precise statements about its general validity.
     
  8. Aug 23, 2013 #7
    Let's say I have a countable set of real number X = {a1, a2, a3 ...... }
    Define
    A1 = {a1}
    A2 = {a1,a2 }
    .....
    An = {a1, a2, ... , an}

    Is it legit to write that

    [itex]\displaystyle\lim_{n\rightarrow +\infty} {A_n} = X [/itex] ?


    Is there an ε-δ notation for set operation, equivalent to what we have for sequence and function ?
    Thx!!
     
  9. Aug 23, 2013 #8

    mfb

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    I think that is (at best) a very strange notation. You can write X as union of all those A_n.
     
  10. Aug 23, 2013 #9
    There is such a thing as the set-theoretic limit. And yes, in your example, X is the limit.

    Just like ordinary limits, the set-theoretic limit does not always exist.
     
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