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Mickey1
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Dedekind cut
« on: Feb 10th, 2012, 12:51pm » |
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Dedekind’s cut is exemplified the following way:
All rational numbers whose squares are less than 2 form a set of number that can be associated with sqrt(2), thereby extending the rational numbers into real numbers.
In Goedel’s inconsistency proof he shows that one can order and number all statements, such as the one about q*q< 2. It seems therefore that if the example of Dedekind’s cuts where meant to be general i.e. refer to all real numbers, there would exist a one-to-one relationship between the rational and real numbers, which is a contradiction.
So where is the "real" example valid for all real numbers?
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| « Last Edit: Feb 11th, 2012, 1:02pm by Mickey1 » |
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Grimbal
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Re: Dedekind cut
« Reply #1 on: Feb 13th, 2012, 6:42am » |
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The "real" examples are the sets S such that
if S contains a and b<a then S contains b.
Or, starting from an infinite sequence of rationals ri, build the sets
Si = {q|q<ri}
and
S = union of all Si = {q | q in Si for some i}
But it is true that there are more reals than can be expressed in a finite expression. Whether a real can exist if it can not be expressed explicitly is a philosophical quesion. But reals defined like that are still quite useful.
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Mickey1
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Re: Dedekind cut
« Reply #2 on: Feb 13th, 2012, 8:18am » |
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I got a similar answer from Prof Karlis Podnieks,
University of Latvia:
"Dear colleague,
Any Goedel-style enumeration can cover only those real numbers that are definable by formulas (in some fixed language). Thus, if in one's set theory, there are uncountably many real numbers, then some of these numbers must be undefinable (by formulas)".
I thought that classical mathematical teaching viewed the real numbers to constitute an enlargement of the rational number, in much the same way as the rational numbers are an enlargement of the natural numbers and I saw a contradiction given by the enumeration characteristics. After having spent some - modest - additional time on the internet I now see that the introduction of real numbers usually relies on the introduction of a new idea, the geometrical concept of the number line.
I still object to the wording “enlargement” or “extension”. The terminology is misleading given the earlier (looking at this as a journey from N to I to Q to R) and stricter enlargement of the natural to the rational numbers.
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| « Last Edit: Feb 14th, 2012, 12:36am by Mickey1 » |
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Mickey1
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Re: Dedekind cut
« Reply #4 on: Feb 14th, 2012, 12:37am » |
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I see.
It seems there is nothing new under the sun.
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Mickey1
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Re: Dedekind cut
« Reply #5 on: Feb 19th, 2012, 2:07am » |
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I believe, after some thought, that a stricter formulated objection can be made:
Remembering once again the formulation by Prof. Karl Podniek, Univ. of Latvia : “Any Goedel-style enumeration can cover only those real numbers that are definable by formulas (in some fixed language). Thus, if in one's set theory, there are uncountably many real numbers, then some of these numbers must be undefinable (by formulas)”, and I continue:
Given the fact that some real X cannot be defined, the method of the cut is invalidated completely, in as far as it claims to define the real numbers, since the definition of the cut requires a finite number of steps. It looks dramatic, but then again perhaps not when you consider Wikipedia on Dedekind:
“ ...Arguably he was even ahead of much contemporary philosophy of mathematics, especially in terms of his sensitivity to both sides. This is not to say that his position is without problems. Dedekind himself was deeply troubled by the set-theoretic antinomies. And the twentieth century produced additional surprises, such as Gödel's Incompleteness Theorems, that are difficult to accommodate more generally.” (This is a quote from Reck, Erich, "Dedekind's Contributions to the Foundations of Mathematics", The Stanford Encyclopedia of Philosophy, Fall 2011 Edition.)
It seems to me that there is no way around this, philosophically or otherwise. If you are willing to accept infinite steps in the definition of X, why not just define, in a similar circular way, the number X by its own decimal expansion.
Either way, there is no need for the cut.
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| « Last Edit: Feb 19th, 2012, 8:48am by Mickey1 » |
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rmsgrey
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Re: Dedekind cut
« Reply #6 on: Feb 20th, 2012, 8:18am » |
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on Feb 19th, 2012, 2:07am, Mickey1 wrote:It seems to me that there is no way around this, philosophically or otherwise. If you are willing to accept infinite steps in the definition of X, why not just define, in a similar circular way, the number X by its own decimal expansion.
Either way, there is no need for the cut. |
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1.000... or 0.999... which expansion do you use to define X when you want the real number which is the non-zero solution to X=X2?
If you want to understand why people might want to use the Dedekind Cut (rather than need to), you should probably start by looking at the contexts in which it is used, and the context in which it was invented. At a minimum, it's another trick in the mathematician's arsenal that may prove useful in future...
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Mickey1
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Re: Dedekind cut
« Reply #7 on: Feb 20th, 2012, 12:57pm » |
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The answer to your question will be revealed when you reach infinity (or Nirvana, whichever comes first). I take the same negative view on Wu’s combination of pi and sqrt(2), in the hard section. I don’t think the question can be posed properly without contradictions.
I do share your historic view, and I was pleased to find Erich Reck’s article.
Personally, I have difficulty with my sense of order. If I turn a page and rewrite three terms, two of them will be wrong. I therefore realized early that any understanding I might develop in math would be restricted to the philosophical part, such as the axioms, and not the trickery of calculation, the engineering part (the importance of which I fully acknowledge both it its own right and as a driving force for the philosophical part).
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peoplepower
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Re: Dedekind cut
« Reply #8 on: Dec 15th, 2012, 6:05am » |
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The underlying question seems to be, can we utilize a collection of all objects which satisfy some property as if it is any mathematical object? Russell's paradox reveals that we need to be careful, and freedom is lost in this approach. However, I believe the utility of such an object (whether or not it exists) is so great that I am motivated to define it and use it before checking it.
To sum up the difference in our opinions, you implicitly deny that we can form the collection of all bounded, strictly increasing sequences of rational numbers. I take such a collection as given and form the full set of real numbers as a result.
The primary reason why I do not worry about this construction being possible is that in specific instances one is only interested in objects which can be explicitly constructed (admittedly, this is not exactly the case in set theory where one includes/excludes existence hypotheses all the time).
Finally, if you look in any mathematical textbook you will see examples of the (instructional!) gains of using these kinds of objects. A group, field, vector space, etc., can be defined by its properties rather than an algorithm for its construction. If the conceptual advantages are not obvious, do a problem set and really monitor your thinking process. You will see that thinking of objects in terms of their distinguishing properties alone significantly clarifies your thoughts involving them.
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rmsgrey
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Re: Dedekind cut
« Reply #9 on: Dec 17th, 2012, 4:12am » |
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on Dec 15th, 2012, 6:05am, peoplepower wrote:| Finally, if you look in any mathematical textbook you will see examples of the (instructional!) gains of using these kinds of objects. A group, field, vector space, etc., can be defined by its properties rather than an algorithm for its construction. If the conceptual advantages are not obvious, do a problem set and really monitor your thinking process. You will see that thinking of objects in terms of their distinguishing properties alone significantly clarifies your thoughts involving them. |
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Purely considering the distinguishing properties of an object lets you draw conclusions like "if there is an object with those properties, then it will also have all these properties". What it can't do is establish that there is an object with those properties in the first place - I could reason for weeks about the properties of the smallest strictly positive rational number - and eventually observe that half that number is both smaller and still a strictly positive rational...
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Grimbal
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Re: Dedekind cut
« Reply #10 on: Dec 17th, 2012, 5:33am » |
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The ultimate property being that it does not exist.
Makes me think that mathematics is superior to physics.
The rules of physics apply to everything that exists. But as you have shown, the rules of mathematics apply even to things that don't exist.
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peoplepower
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Re: Dedekind cut
« Reply #11 on: Dec 17th, 2012, 7:09am » |
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For an example which does not lead to a contradiction, I remember spending a few days determining the behavior of a certain function on weakly inaccessible cardinals and weakly Mahlo cardinals, neither of which need to exist in ZFC though they are both definable under those axioms.
Of course there are lots of examples in the differential ZF versus ZFC. For instance, (a weaker form of) the axiom of choice is necessary for there to be a nonmeasurable set of real numbers.
Reasoning about these things seems to arouse an equal excitement as reasoning about anything else, so why not?
P.S. I think the ability to measure the size of a set using a cardinal without being able to construct all of its elements is an awesome outgrowth of reasoning about things outside the "countable" world.
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| « Last Edit: Dec 17th, 2012, 7:27am by peoplepower » |
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