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Thread: Dedekind cuts

  1. #1 Dedekind cuts 
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    Normally, I avoid number theory, but here's a nifty one I discovered recently.

    Take the real number line, R, and cut it any any point. Let's say at 10.
    Then we have two subsets of R, let's call them L and U (for upper and lower cuts). Then, obviously, 10 must either be the upper bound of L or the lower bound of U. Let's say the latter. Thus L is open on the right, and U is closed on the left.
    This means that, having made the cut, we can never glue it back together again, for the upper bound of L can only ever approach abitrarily close to 10.

    I think that's pretty cool. Apparently it says something profound about the rational and irrationals. If I can bear more number theory, I'll dig a bit more.


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  3. #2  
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    Interesting. It's kinda saying that you can't split infinity... or the infinitessimal.

    The problem is, of course, present at the splitting rather than the regluing. But isn't made apparent until one considers rejoining the two lines once more.

    Somehow, in the splitting, an arbitrary length of the L line is lost. And goes unnoticed until you try to reattach it to U.

    The problem is one of perception. How fine can you split a hair and how close is close enough? We can spend an infinite amount of time getting closer and closer to 10 without ever reaching it. And in a formal system, such as number theory, we have to.

    But. Luckily, humans can intuitively understand that despite the formal discrepancy involved, we can reach a state of nearness to 10 that is 'good enough' and leave it at that.

    I suppose a machine could do the same, but it would need to be programmed how far to go before saying good enough. Humans can do so with the greatest of ease.

    (Apologies for breaking in with implications of Goedel's Incompleteness Theorum and Turing's Halting Problem, but these are the issues that have been on my mind of late and your thread links to them inexorably. These "Dedekind Cuts" are another example of the limitations of formality.)


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  4. #3  
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    Quote Originally Posted by invert_nexus
    Apologies for breaking in with implications of Goedel's Incompleteness Theorum and Turing's Halting Problem, but these are the issues that have been on my mind of late and your thread links to them inexorably.
    OK, so let's do it. This place is dead as Heaven on a Saturday night (it's a quote - know it?)
    So, I've seen Goedel and Turing on unsolvability. Take me through it, because I didn't understand what I read.
    Perhaps it might wake up some of our members.
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  5. #4  
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    "I think that's pretty cool. Apparently it says something profound about the rational and irrationals. If I can bear more number theory, I'll dig a bit more."

    Well, it turns not be quite the jaw-dropper Id hoped for.

    Make Dedekind cuts of the ordered set of rationals, and note that, as one of the resulting subsets is open at the cut, then that set will contain the irrationals at that point. Which motivates the definition of the reals as the set of Dedekind cuts on the rationals.
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  6. #5  
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    Someone care to provide a definition of Dedekind cut? Or does this whole area require a grounding in topology theory?
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    Quote Originally Posted by Silas
    Someone care to provide a definition of Dedekind cut? Or does this whole area require a grounding in topology theory?
    Well, it was defined in my opening post of this thread. Topology may come into it, but let's just stay with the number line (which is of course a topolgy), let's say the rationals.

    Pick any point, and cut the line, say at 5. Then you have 2 choices - either 5 is included in the upper cut or in the lower cut, it can't be in both.
    Let's say 5 is in the upper cut. Then we can say that the upper cut is closed on the left (i.e. its lower bound) and the lower cut is open on the right.
    Why? Because it is possible to approach abitrarily close to 5 from "below", but never quite reach 5.

    That's your Dedekind cut. Now if your number line is the rationals, all I said in my previous post (seems) to follow.
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  8. #7  
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    They can still be joined back together.

    negative infinity to <= 10, >10 to infinity.
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  9. #8  
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    Quote Originally Posted by (In)Sanity
    They can still be joined back together.

    negative infinity to <= 10, >10 to infinity.
    No!! You're not getting it. Cut at 10, and 10 can only be in the upper or lower subset, not both.
    How do you propose to join a subset open at the upper bound, say, with one that is closed at the lower?
    There is an infinity of points between 9.99999999 and 10. You can't re-join them
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  10. #9  
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    Quote Originally Posted by Guitarist
    Quote Originally Posted by (In)Sanity
    They can still be joined back together.

    negative infinity to <= 10, >10 to infinity.
    No!! You're not getting it. Cut at 10, and 10 can only be in the upper or lower subset, not both.
    How do you propose to join a subset open at the upper bound, say, with one that is closed at the lower?
    There is an infinity of points between 9.99999999 and 10. You can't re-join them
    No I get it, that's why I said

    infinity to <= 10, >10 to infinity

    or infinity to <= 10 join >10 to infinity

    <= 10 includes all the infinite posibilities less then or equal to 10. > 10 is all those greater then 10. So my logic holds out. You can't express it as a number but you can express it with greater then or less then or equal to.
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  11. #10  
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    Insanity's right.
    If you cut it down to any specific number, then you lose the ability to join. But if you leave it at the set {(-infinity, 10),[10, infinity)} then it links back together just fine.

    The problem comes in when you look at it too closely. When you try to make a measurement. The problem is in the preciseness of measurement. It's kills possibility by looking too close.
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  12. #11  
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    Quote Originally Posted by invert_nexus
    Insanity's right.
    If you cut it down to any specific number, then you lose the ability to join. But if you leave it at the set {(-infinity, 10),[10, infinity)} then it links back together just fine.
    Hey, no boss! Look what you just wrote {(-infinity, 10),[10, infinity)}. You have the number 10 in two disjoint subsets. Illegal move!
    You correctly wrote the intersection as ),[ but I defy you to to prove that 8, 9, 10, 10, 11 makes any sort of sense.

    And anyway - it's not my own pet theory, it seems to be an accepted part of what sad number theorists take for granted. So don't attack me for it.
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  13. #12  
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    But that's not what it means.
    (-infinity, 10) means -infinity>x>10. That is. X is between -infinity and 10. But is equal to neither one.
    [10, infinity) means 10</=x<infinity) That is X can equal 10. But not infinity.

    See?

    (How do you write Less than or equal to anyway?)

    So. When you combine them into a single set you have -infinity>x>infinity.
    Number line rejoined.

    It's only when you get into specifics such as x=9.999999999999999 that you run into problems.


    Edit: Or maybe I just don't understand the rules. It makes perfect sense to me that they could be rejoined in this way, but that doesn't mean it doesn't break a rule of set theory. I really know next to nothing about number theory.

    As to Godel. I know little about that either. I'm reading Godel, Escher, Bach and I've read Lucas Against Determinism. But the actual theorum is beyond my skill to comprehend. I can only understand it through metaphor and anology. Which is imperfect.
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  14. #13  
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    Oops. Kinda screwed the pooch up there, didn't I?

    I got my signs backwards.
    I meant:
    -Infinity<x<10 and 10</=x<Infinity

    Anyway. I'm sure you understood what I meant. The question is, does this violate set theory? You don't end up with a series of numbers with any duplicates. THere are not two 10's as in the list you posted: 7, 8, 9, 10, 10, 11, 12, ....
    Instead it is as I posted (with reversed signs): -Infinity<x<Infinity

    Right?
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  15. #14  
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    Quote Originally Posted by invert_nexus
    Oops. Kinda screwed the pooch up there, didn't I?

    I got my signs backwards.
    I meant:
    -Infinity<x<10 and 10</=x<Infinity
    Nah, you had it right first time. Trouble is with number theory, all objects must have a numerical value. < and > don't cut it (oops!).
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  16. #15  
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    Quote Originally Posted by Guitarist
    Pick any point, and cut the line, say at 5. Then you have 2 choices - either 5 is included in the upper cut or in the lower cut, it can't be in both.
    Let's say 5 is in the upper cut. Then we can say that the upper cut is closed on the left (i.e. its lower bound) and the lower cut is open on the right.
    Why? Because it is possible to approach abitrarily close to 5 from "below", but never quite reach 5.
    You don't have "cut points" when defining Dedekind cuts in general on the rationals (which is where they usually live). The idea is to pick up the "hole" that the rationals have and build a new and improved set from these new objects.

    A dedekind cut can be defined as a partition of the rationals, L and U to stay with your earlier terminology, where l in L and u in U implies l<u and L does not contain a greatest element. You then go on to define the usual structure on this set of dedekind cuts that make it a complete ordered field and call this new set the real numbers. The cuts that corresponded to nice rational "cut points" will then be identified with the rationals, but you get a whole whack of real numbers that don't have a nice rational point defining the cut. In hindsight we could say, yea that cut corresponded to the real number we're calling sqrt(2), but if you're trying to build the reals from the rationals you don't really know what sqrt(2) is yet, only that the rationals don't have one, whatever it is.

    (edit-of course I'm talking above about trying to define your cut as "L is the set of rational x where x<a" where a is some point. You can talk about the cut for sqrt(2) in terms of x^2: "L is the set of rationals where x^2<2 or x<=0", etc. but things aren't as nice for transcendental numbers, say)

    I don't understand what you mean by "gluing" the cut back together.

    ps.Number Theory is the study of the structure of the natural numbers- divisibility, primes, etc. I'd toss this under foundations of analysis if it needed to be anywhere specific, but not number theory.
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  17. #16  
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    Ha! Otherwise I did pretty well, huh?
    Seriously, shmoe, thanks for putting it straight. Like I said I had recently stumbled across them, and wanted to share what I "thought" I learned.
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