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Math Help - Thoughts on the empty set and logic

  1. #1
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    Thoughts on the empty set and logic

    Hello all,

    I was just thinking about this today and was wondering, if anyone else had any input on the matter.

    As we know the empty set \{\} = \theta
    Now by definition \theta is subset of all sets, because it has no elements or by contradiction because there are not any elements of \theta that are not a member of a set.

    so we have
      \theta \sqsubseteq A for any A

    thus   \theta \sqsubseteq A is a tautology.

    Yet by using Axiom of extensionality and scheme of comprehension. We can define the empty set \theta = \{x \in A | x \not= x \} which is a contradiction.

    So the definition of an empty set is a contradiction which allows for it to exist, yet its a subset of all sets by tautology.

    Is my reasoning here correct, or have i made a logical mistake?
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  2. #2
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    Quote Originally Posted by drgonzo View Post
    Hello all,

    I was just thinking about this today and was wondering, if anyone else had any input on the matter.

    As we know the empty set \{\} = \theta
    Now by definition \theta is subset of all sets, because it has no elements or by contradiction because there are not any elements of \theta that are not a member of a set.

    so we have
      \theta \sqsubseteq A for any A

    thus   \theta \sqsubseteq A is a tautology.

    Yet by using Axiom of extensionality and scheme of comprehension. We can define the empty set \theta = \{x \in A | x \not= x \} which is a contradiction.
    What is the contradiction here?

    CB
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  3. #3
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    Quote Originally Posted by CaptainBlack View Post
    What is the contradiction here?

    CB
    well that could also be defined as
    \forall x (\ x \in A \wedge x \not= x) = \ x \in A \wedge x \iff \~{x}
    Which is always false.
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  4. #4
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    Quote Originally Posted by drgonzo View Post
    well that could also be defined as
    \forall x (\ x \in A \wedge x \not= x) = \ x \in A \wedge x \iff \~{x}
    This is not a well-formed formula. First, it is not clear what the notation \~{x} means. Second, = usually connects terms, not formulas (see the syntax of first-order logic). Conversely, \iff or \leftrightarrow usually connects formulas, not terms.
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    Quote Originally Posted by emakarov View Post
    This is not a well-formed formula. First, it is not clear what the notation \~{x} means. Second, = usually connects terms, not formulas (see the syntax of first-order logic). Conversely, \iff or \leftrightarrow usually connects formulas, not terms.
    Yeah I should go over first order logic a bit thanks for that.

    This is what i was trying to say
     \forall x( x \in A \wedge x \not = x) \leftrightarrow (x \in A \wedge \lnot(x = x)) which is alway false if my logic is correct.
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  6. #6
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    Quote Originally Posted by drgonzo View Post
     \forall x( x \in A \wedge x \not = x) \leftrightarrow (x \in A \wedge \lnot(x = x)) which is alway false if my logic is correct.
    Usually x\ne y is a contraction for \neg (x=y), so your formula is  \forall x\,(( x \in A \wedge \neg (x=x)) \leftrightarrow (x \in A \wedge \lnot(x = x))) , i.e., it has the shape \forall x\,(B(x)\leftrightarrow B(x)) for some formula B(x). All formulas of this shape are tautologies.
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  7. #7
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    Quote Originally Posted by drgonzo View Post
    Yeah I should go over first order logic a bit thanks for that.

    This is what i was trying to say
     \forall x( x \in A \wedge x \not = x) \leftrightarrow (x \in A \wedge \lnot(x = x)) which is alway false if my logic is correct.
    But still not a contradiction, so where is this contradiction (you would have to show that some such x existed).

    You appear to be trying to set up a definition of the null set of the form

    \phi=\{x: "\text{some property of {\it{x}} that is contradictary}"\}

    The point is that in the system you are setting this up a contractictary property for $$x implies there is no such $$x, which is the whole point of trying to define the null set this way.

    CB
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  8. #8
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    Isn't something that is always a false a contradiction ?

    Another example
     \{ x  | A \oplus A \} = \theta

    Thus any property that is always false and defines a set leads to an empty set?

    Quote Originally Posted by CaptainBlack View Post
    The point is that in the system you are setting this up a contractictary property for $$x implies there is no such $$x, which is the whole point of trying to define the null set this way.

    CB
    Yes this was my point. I think i just poorly explained myself. I find it interesting that the empty set is defined for this reason. Yet the empty is a subset of all sets which is a tautology.

    I wasn't trying to disprove anything just making an observation.
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  9. #9
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    the null set has lots of "bad" properties. for example, every element of the null set is in point of fact, a russell set. mappings from the null set are very poorly behaved, their image sets are also russell sets. in fact, the null set is a russell set itself, and being self-contradictory, cannot exist. by the same token, it also is NOT a russell set, and therefore its existence is perfectly legitimate. the null set is the ultimate law-abiding revolutionary. personally, i feel the null set should be locked up in the closet, and only be taken out for special occasions.
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  10. #10
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    Quote Originally Posted by Deveno View Post
    the null set has lots of "bad" properties. for example, every element of the null set is in point of fact, a russell set. mappings from the null set are very poorly behaved, their image sets are also russell sets. in fact, the null set is a russell set itself, and being self-contradictory, cannot exist. by the same token, it also is NOT a russell set, and therefore its existence is perfectly legitimate. the null set is the ultimate law-abiding revolutionary. personally, i feel the null set should be locked up in the closet, and only be taken out for special occasions.
    By a Russell set I take it you mean a set that contains itself as an element? If not please provide your definition.

    Then as:

    \phi \not\in \phi

    the null set is not a Russell set (as you say). Now please explain in what sense $$\phi is an element of $$\phi (as you also maintain).

    CB
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  11. #11
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    no, the definition is rather the opposite. a russell set is the set of all sets not containing themselves as a member, and as such is self-contradictory.

    without getting too pedantic about it, note the date of my post.

    on a more serious side, if a property isn't satisfied by any set at all, the null set HAS that property. the null set is a very curious set indeed, and it's good not to think about it too much. for example think about this set: S = {A : A is a set}. it's obviously ill-defined, there can be no such set (this is the trouble that basing category theory on set theory runs into. one would like to think of the category of all categories, since for any two categories C and D, one can define morphisms between them to be functors, and functors are associatively composable. the trouble is, what do you call the collection of all categories? even worse, how do you define the category of Set, seeing as how the collection of all sets is not a set? one can take the route of some authors who only consider "small categories" (ones in which the objects and morphisms form sets), or "locally small" categories (where the collection of all morphisms between two objects is a set, even if the collection of all objects is not a set), or one can introduce a kind of type theory, where you have sets inside classes inside...and so forth, which is fine. but then you're no longer using set theory as a basis for category theory, but instead some other, larger theory, which stubbornly resists "encapsulation").

    since there is no "set of all sets", such a set is the empty set. it is logically consistent to ascribe to every member of the empty set, any property you like, even contradictory ones. every element of the empty set is blue, and also green, and red besides. in fact, if two properties are mutually exclusive, the empty set is the only set that has BOTH.

    for example, the intersection of the even integers and the odd integers is the null set. so we can safely say that every element of the null set is even AND odd, and... also an integer! that's what i mean by the empty set shouldn't be allowed out in public, it's very poorly behaved.
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  12. #12
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    Quote Originally Posted by Deveno View Post
    no, the definition is rather the opposite. a russell set is the set of all sets not containing themselves as a member, and as such is self-contradictory.

    without getting too pedantic about it, note the date of my post.
    I am looking at the date and time and it says 04:29 2nd of April, so am I still supposed to find it funny?

    Also <pedantic> it would then be "the Russell set" not "a Russell set".</pedantic>

    CB
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  13. #13
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    not THAT post, the OTHER one. geez.

    well, yes there is the classic russell set R = {A: A ∉ A}, but generally any set of similar construction has come to be called "a russell set". one of my favorites is:

    {the set of all heterologic words in English} (a word is autologic if its self-descriptive, such as: "short", "english", "seventeenlettered". a word is heterologic if it isn't self-descriptive, like "long", "french" or "monosyllabic"). is "heterologic" in that set? what i like about that example is that it shows you don't even need a set of all sets to run into trouble, even with a finite universe you can assign properties that corrupt self-referentiality and run with it.

    furthermore, some versions of set theory take place entirely within a well-defined "universe of discourse", meaning each universe has its own unique version of the russell set (i hasten to add that in many of these theories, certain axioms are designed to preclude the "classical russell set" but it is an open question as to whether or not the theories powerful enough to serve as a suitable foundation for all of mathematics nevertheless may admit some other self-contradictory construction).

    as far as our dear old friend the empty set is concerned, he (for as i remarked, the empty set certainly has the property of belonging to the set of all male sets, as it is a member of every set) at least has the commendable quality of remaining silent throughout this ponderous conversation.
    Last edited by Deveno; April 3rd 2011 at 03:20 AM.
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  14. #14
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    Quote Originally Posted by Deveno View Post
    not THAT post, the OTHER one. geez.
    I know which post I an referring to, perhaps you should try googling "time zones" (which incidentally makes attempted April Fools jokes not containing the time stamp in originators time look pretty foolish).

    Here we also have a traditional rule that April Fools played locally have to take place before mid-day otherwise the perpetrator is the fool. Now I see that your local time for your post was between late evening and night...

    CB



    Thoughts on the empty set and logic-timezone.png
    Last edited by CaptainBlack; April 3rd 2011 at 07:37 PM.
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