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A question about the empty set

  1. Feb 2, 2004 #1
    It is very important to define a fundamental concept like set before we use it.

    The set concept can be useful iff there is clear separation between two basic concepts, which are: container and content.

    Before we are going to check the ratio between these concepts, we first have to understand that set’s concept in general is only a framework, which we use to explore our ideas.

    To get clearer picture of it we first have to distinguish between two basic states of the set’s concept, which are: unused state, used state.

    The unused state is like a mobile stage before its parts are connected.

    Let us notate this state by }{.

    The first used state is like an empty stage (before playing), and notated as {}.

    The first used state is also the first ratio between container and content concepts.

    The container concept notated by ‘{‘ and ‘}’ and its content notated by using the letter ‘x’ .

    The name of any used set depends on the property of its content.

    Content x can be at least two basic states, which are: something XOR nothing.

    Let us examine the ZF axiom of the empty set from this point of view.


    The axiom of the empty set:

    If A is a set such that for any x , x not in A , then A is {}.


    As we can see, we have here a hidden assumption, which is:

    Content x is something.

    If x is nothing then A cannot be an empty set by ZF axiom of the empty set.

    This result clearly shows that the definition of the set’s concept, which used by ZF is not fundamental enough.


    A question:

    Is there another set theory where this hidden assumption does not exist?
     
    Last edited: Feb 2, 2004
  2. jcsd
  3. Feb 2, 2004 #2

    matt grime

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    Have you considered posting this in the set theory forum?

    Let me explain it with this analogy:

    there is the axiomatized theory of probability.
    One of the axioms is that the probabilty measure of the entire space is 1. The axioms do not define what 1 is. Do you think there is an issue there?
     
    Last edited: Feb 2, 2004
  4. Feb 2, 2004 #3
    Matt,

    Can you be more specific about what I wrote in my first post?


    Thank you,


    Organic
     
  5. Feb 2, 2004 #4

    matt grime

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    My analogy wasn't very good; it was taking the wrong emphasis from your post.
    How about this?

    A set theory satisifies ZF if .. and then there's the list of axioms. Now, this must be a non-empty universe. If it weren't, it wouldn't satisfy ZF. IF there were nothing in it, the empty set wouldn't be in it.

    If there are NO SETS and NO ELEMENTS it doesn't make sense to attempt to make this object, whatever it is, satisfy the axioms of ZF. And it doesn't make sense to call it a set theory if there are no sets and no elements in the sets.

    So there is no presupposition about anything here. You have an object, you compare it to ZF to see if it satisfies those axioms. If it does, anything deducible in ZF is true in your system.

    Just making a list of axioms doesn't make things exist. They exist already, we check if they satisfy the axioms.
     
  6. Feb 2, 2004 #5
    "It is very important to define a fundamental concept like set before we use it."

    why?
     
  7. Feb 2, 2004 #6
    Because if we have some hidden assumption in the basis of a fundamentel idea, it means that our idea is not fundamental.
     
    Last edited: Feb 2, 2004
  8. Feb 2, 2004 #7
    why not have an undefined concept without hidden assumptions?
     
  9. Feb 2, 2004 #8
    My idea is to look on the set's concept as a general framework, which we use to explore any content, where the content can be at least nothing XOR something.

    ZF does not have this fundamental point of view about the set's concept.


    Therefore x cannot be nothing by ZF.
     
    Last edited: Feb 2, 2004
  10. Feb 2, 2004 #9

    matt grime

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    but there's nothing hidden. it is an axiomatized set theory, a theory about sets. the sets exist, as do the elements of the sets. if they don't exist then it's not a set theory.
     
  11. Feb 2, 2004 #10
    Please read again my first post, where I show how "{" and "}" are only
    a general framework to examine ideas, concepts and so on, including the basic idea of no input(=nothing), an input(=something).

    From this point of view, x can be nothing XOR something, or in other words: the container {x} is the framework of x where x can be at least nothing XOR smoething.

    By ZF x is something because the set concept is the object of itself.
     
    Last edited: Feb 2, 2004
  12. Feb 2, 2004 #11

    matt grime

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    In very badly phrased English and Mathematics you seem to make the point that it is important to distinguish between set and element. (container and contained.) true. and?

    you then use the term ratio in a sense i don't understand.

    you alse tell us to consider used and unused states, but don't define them: you say what they are like, and what symbol you are going to use, but don't define what they are.

    nor do you explain what is meant by 'play'.


    you then state that the contents of a set may be something, or nothing. ok.


    there is no hidden assumption in the ZF axiom though, I just don't see that you have a point here. what is wrong with the ZF def of empty set?
     
  13. Feb 2, 2004 #12
    ZF axiom of the empty set:

    If A is a set such that for any x , x not in A , then A is {}.

    By this axiom x must be some input, therefore x cannot hold the idea of no input.

    If x holds for the idea of no input, then A is not empty by the same definition, therefore A can be empty XOR non-empty by the same axiom.

    edit:
    The conceptual mistake in ZF is that it does not distinguish between x-model and x itself, for example: if emptiness is nothing then the notation x does not exist, therefore if x notation exists it cannot be empty.
     
    Last edited: Feb 2, 2004
  14. Feb 2, 2004 #13

    matt grime

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    no it can't. this is you misunderstanding mathematical convention. do you see the 'for all' there?

    it makes no sense to talk of 'x holding'; x *is* some object, what does it mean if 'x holds'?
     
  15. Feb 2, 2004 #14
    The conceptual mistake in ZF is that it does not distinguish between x-model and x itself, for example: if emptiness is nothing then the notation x does not exist, therefore if x notation exists it cannot be empty.
     
  16. Feb 2, 2004 #15

    matt grime

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    and your mistake is to use x for at least two different things. you are off on another tangent again. what's an x model? actually don't bother, because it's bound to be something vastly unedifying.

    How about this. We deal with sets every day. If we want to put them on a rigorous footing we have ZF (there are other theories too), which does what we want, pretty much. You would require the our unverse contains something, which you label 'empty input' that can be in the empty set. you are therefore not in the ZF world. good luck to you, but don't say ZF has a conceptual mistake in it when you clearly don't understand it. it's almost as though you are insisting that there can be a group with a non-invertible element. The mistake is in your misunderstanding of mathematical notation and convention.
     
  17. Feb 2, 2004 #16
    Hmmmm,

    In Euclidean Geometry, basic ideas like point, line, and triangle are taken to be self-evident. These are "hidden assumptions" (if you like).

    Similarly in Set Theory, statements like "a is in..." and "B is a set" are also taken to be self-evident-- so in this sense the phrase:
    "If A is a set such that for any x, x not in A, then A = {}"
    Has several such hidden assumptions.

    If I understand correctly you are asserting that inherent in that expression is the underlying assumption of existant objects-- an input if you will, and you further claim that because the criterion for identifying the empty set involves such objects, then the objects are of a more basic nature then the empty set, and as such the empty set really shouldn't be considered an 'axiom'.

    But by this reasoning, a line in Euclidean geometry would not be considered a primitive object because it is made out of points. But where in the definition of point can one deduce that of line? The problem may be resolved by allowing relationships between primitive objects, and that's how I would answer your objection. I would say the empty set is a primitive, the phrase "a is in..." is a primitive, and "set" is a primitive"... I would further assert that set theory is built not just on ZF but also on the axioms of logic.... so I'm not saying you haven't noticed something interesting... I'm saying that you may want to pursue it down into the axioms of logic-- after all, so much of what's bothering you is coming from the phrase "If A is a set such that for any x, x not in A, then A = {}", and that's using both x and A as variables, and where were variables defined in the ZF axioms?
     
  18. Feb 2, 2004 #17
    No Matt,

    The set concept is too important to say that it belongs to some theory.

    I examine the set concept and simply say that it is only a general framework to explore our concepts and ideas.

    "x" is a general notation for any idea that can be examined by using "{" "}" framework, including the idea that "x" is a model of "no input".

    But by ZF "x" is not a model but x itself, therefore if "x" notation
    exists then it is not empty.

    This approach is a very big conceptual mistake of the set's concept in any set theory.
     
    Last edited: Feb 2, 2004
  19. Feb 2, 2004 #18

    matt grime

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    that is your theory that you are free to explore. I don't think it is a view shared by any mathematicians. it would probably disappoint you to know that the vast majority of mathematicians don't give a damn about set theory because you rarely come up against the boundaries it presents.

    we know intuitively what sets are, we also know that there is russell's paradox, and that we can avoid this with a stronger set of rules for the manipulation of sets (ZF). it would appear that you don't understand the concept of axioms.

    so if it's so important for you to define sets, go ahead, define a set.
     
  20. Feb 2, 2004 #19
  21. Feb 2, 2004 #20

    Hurkyl

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    Well, one thing I think you need to realize, Organic, is that "for all" has a rigorous definition too.

    For instance, one of the 'axiom's related to forall is that, where P and Q are logical predicates and T is any logical variable, we can make the deduction:

    [tex]\forall x: P(x)[/tex]
    -------------
    [tex]P(T)[/tex]

    Another valid deduction is

    [tex]\forall x: P(x)[/tex]
    [tex]\forall x: Q(x)[/tex]
    ---------------------
    [tex]\forall x:( P(x) \wedge Q(x))[/tex]


    So, for example, the axiom of the empty set means (among other things) that if T is a logical variable, then we can conclude [itex]T \notin \varnothing[/itex].
     
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