Question about the sum of Dedekind cuts

[iceblits]

The addition of Dedekind cuts is defined as the sum of the elements (r+s) with r in A and s in B. However, the sets are closed downwards and there is no largest element in the set...so how do you know what to add?
For example, the cut for 2 added to the cut of 3 should equal the cut of 5. so we should obtain all the rationals to the left of 5. However, what element in the cut for 3 is added to say 1 in the cut of 2 to obtain a corresponding element in the cut of 5? I mean I guess all you have to do is add the 3 and the 2 and just say that you have all the elements to the left of 5..but i don't think this method would work for adding the Dedekind cuts of irrationals without explicitly defining irrationals before hand which would seem to defeat the purpose...like the cut for √2 added to the cut for √5 is the cut of √2+√5 ...but I feel like that assumes that the reals have already been constructed.


Also, does anyone know if Dedekind cuts can be used to prove that pi is irrational...for that matter does anyone know the Dedekind cut representation of pi?

-Thanks for your time

 

[mathman]

To get the elements of A + B, add every element of A to every element of B.

pi was proven to be transcendental in the 19th century. The Dedekind cut for pi is like the Dedekind cut for any irrational, i.e. all rationals < pi.

 

[iceblits]

So its really like adding 1+2,1+1,1+0,1+(-1)...and all the combinations possible? I guess that would make sense because the repeats wouldn't be included in the resulting set...
and can you really say all rationals less than pi? pi is irrational and therefore assuming its existence seems to go against the logic of constructing it with a Dedekind cut

 

[mathman]

The existence of irrationals was discovered long before Dedekind cuts. The idea of using the cuts was developed as an attempt to use set theory to get a rigorous basis for real numbers.

http://planetmath.org/encyclopedia/Schnitt.html

http://www-history.mcs.st-and.ac.uk/~john/analysis/Lectures/A3.html

 

[Petek]

I think that iceblits is saying that to define $\pi$ as the set of rationals less than $\pi$ is invalid. I agree. After all, we don't define √2 as the set of rationals less than √2. Instead, we define it as the set {a $\in \mathbb{Q}$ : a² < 2 or a $\leq$ 0}. We already know what "2" is, so we can determine whether the square of any rational number is less than 2. We can't define a Dedekind cut for $\pi$ by assuming that we already know its value.

Textbooks that cover Dedekind cuts almost always cite the definition for √2 given above. I think it would be interesting to describe a Dedekind cut for a transcendental number such $\pi$ or e.

 

[mathman]

For e it is fairly easy. e = 1 + 1 + 1/2! + 1/3! + ...
Let e_k be the set of rationals < sum of k terms. Then e = ∪e_k.

For π, a similar approach can be used, using the terms of a series expansion where all the terms are positive.

 

[Vargo]

Iceblits asked if Dedekind cuts can be used to prove that pi is irrational. Not really. I'll explain.

Start with the axioms for the real numbers. You can look them up. They are a list of basic properties. Every fact about the real numbers (in turn functions and calculus) rests on those axioms. This includes the irrationality of pi.

So what do Dedekind cuts have to do with anything? Well, if you make up a list of "axioms" for something, then a logician will insist that you check that those axioms are consistent with each other. Conceivably, there could be a contradiction hidden deep within the axioms for the real numbers. Math has many surprising results, so maybe we haven't found the contradiction yet because we haven't looked long enough. So how do we check that the axioms are mutually consistent with each other? By exhibiting a model, that's how. We need to "construct" the real numbers from something more basic--something we believe is consistent. That's what Dedekind cuts do. When you prove that Dedekind cuts satisfy the real number axioms, you have demonstrated that those axioms are consistent. After that you can forget about Dedekind cuts and go back to the number line.

 

[iceblits]

Petek: yes that's what I meant to ask
Mathman: thanks for that..it makes sense I suppose
Vargo: that's very interesting...so I guess the construction of real numbers through dedekind cuts doesn't actually give you anything mathematically interesting..just something logically pleasing.

So by mathman's definition..can we perhaps use Ramanujan's formula of pi or 4*Sum((-1)^k/(2k+1) to define the Dedekind cut?

 

[Vargo]

Ramanujan's formula is for 1/pi. Its terms are all positive, so the partial sums will all be less than 1/pi. So if you take their reciprocals, you will end up with numbers c_k just a bit bigger than pi. Let C_k be the rationals less than c_k. The intersection of all such C_k will give us the Dedekind cut for pi, but we only know that because we already know that pi is irrational and therefore does not itself belong to any of the sets C_k.

The formula that uses the expansion of arctan is an alternating series whose partial sums oscillate above and below pi. So you would want to just take the odd partial sums so that all your c_k are below pi. Then take the union of all corresponding C_k. That would work.

 

[Skrew]

I have nothing to add other then Dedekind cuts are the most obtuse way of constructing real numbers.

More reasonable ways are cauchy sequences and infinite decimal expansions.

 

[mathman]

I have nothing to add other then Dedekind cuts are the most obtuse way of constructing real numbers.

More reasonable ways are cauchy sequences and infinite decimal expansions.

The point of Dedekind cuts was not to construct the real numbers, but to give a rigorous apporach for their definition.

 

[Skrew]

The point of Dedekind cuts was not to construct the real numbers, but to give a rigorous apporach for their definition.

I'm not following.

You can either 1. Use the axioms of the real numbers and move on or 2. construct them.

As far as I know Dedekind wanted to construct the real numbers using more basic ideas and that's how dedekind cuts came to be.

There are other ways to do this which I feel are better.

 

[Vargo]

"define", "construct"... same thing I think. Personally I think the axioms better reflect our understanding of what the real numbers are. Dedekind cuts validate the axioms by showing them to be free of contradiction. Another cool property of the axioms is that they are complete in the sense that all models are "isomorphic". So, like Skrew says, you could replace Dedekind cuts with Cauchy sequences of rationals, infinite sequences of digits (decimal or binary expansions) or w/e else someone can think of.

This contrasts nicely with Euclidean geometry. It is possible to define the Euclidean plane purely through axioms. But the list of properties is so long that most people don't even know what the axioms are and instead just use the standard model R^2 with the Euclidean metric. But is that the "definition" of the Euclidean plane? No, because there are a million other models that are equally deserving of the title. For example, the Euclidean plane is the set of linear combinations of (2/pi)sin(x), (2/pi)cos(x) over [0,pi] with the L2 metric.

 

[HallsofIvy]

The crucial point about the Dedekind cut definition is that it makes it easy to show that "if a set of real numbers has an upper bound, it has a least upper bound".

Similarly, you can define the real numbers in terms of equivalence classes of increasing sequences of rational number having an upper bound which makes it easy to prove "in the real number system all increasing sequences of real numbers converge".

Or you can define the real numbers in terms of equivalence classes of Cauchy sequences of rational numbers which makes it easy to prove "in the real number system, all Cauchy sequences converge".