My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Thanks Tree1Thanks
Reply
 
LinkBack Thread Tools Display Modes
August 22nd, 2017, 04:23 AM   #11
Math Team
 
Joined: Jan 2015
From: Alabama

Posts: 2,740
Thanks: 709

The Dedekind cut is one way to construct the reals.

Here is another: Consider the set of all "Cauchy sequences" of rational numbers. That is, the set of all infinite sequences of rational numbers, $\displaystyle a_0, a_1, a_2, \cdot\cdot\cdot, a_n, \cdot\cdot\cdot$ such that $\displaystyle |a_n- a_m|$ goes to 0 as m and n go to infinity independently. We say that two such sequences $\displaystyle \{a_n\}$ and $\displaystyle \{b_n\}$ are equivalent if and only if the sequence $\displaystyle (a_0- b_0), (a_1- b_1), \cdot\cdot\cdot, (a_n- b_n), \cdot\cdot\cdot$ goes to 0.

It is easy to show that this is an equivalence relation and then we define the set of all real numbers to be the set of all equivalence classes.

Last edited by greg1313; August 24th, 2017 at 01:41 AM.
Country Boy is offline  
 
August 22nd, 2017, 05:55 AM   #12
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,973
Thanks: 2296

Math Focus: Mainly analysis and algebra
Most numbers are not describable/computable$*$. Doesn't that mean that we can't describe an appropriate Cut without referring to the number itself.

The only way out of this that I can see is to use the limit of a Cauchy sequence to define the Cut. But this approach effectively uses Cauchy sequences to define the reals, making the Cuts redundant except for the way they impose an ordering relation on the reals.

$*$ I'm aware that "describability" is a slippery concept, but the flavour of the idea is what I wish to invoke.
v8archie is offline  
August 22nd, 2017, 01:15 PM   #13
Senior Member
 
Joined: Aug 2012

Posts: 1,574
Thanks: 380

Quote:
Originally Posted by v8archie View Post
Most numbers are not describable/computable$*$. Doesn't that mean that we can't describe an appropriate Cut without referring to the number itself.

The only way out of this that I can see is to use the limit of a Cauchy sequence to define the Cut. But this approach effectively uses Cauchy sequences to define the reals, making the Cuts redundant except for the way they impose an ordering relation on the reals.

$*$ I'm aware that "describability" is a slippery concept, but the flavour of the idea is what I wish to invoke.
I don't see how either the Dedekind or the Cauchy constructions relate to computability. In the former case we're considering the set of all pairs of Lower/Upper-type partitions of the rationals. That's a valid set that we can build given that we believe in the rationals.

Likewise the set of all Cauchy sequences of rationals is also a set by the rules of set theory, once we believe in the rationals. (Then we have the set of all functions from the naturals to the rationals that satisfy the definition of a Cauchy sequence).

Definability/computability never come into this discussion in any way that I can see. Can you explain why you see a connection?

Last edited by Maschke; August 22nd, 2017 at 01:25 PM.
Maschke is online now  
August 22nd, 2017, 03:28 PM   #14
Math Team
 
Joined: Dec 2013
From: Colombia

Posts: 6,973
Thanks: 2296

Math Focus: Mainly analysis and algebra
Quote:
Originally Posted by Maschke View Post
In the former case we're considering the set of all pairs of Lower/Upper-type partitions of the rationals.
That's not a visualisation I've seen before. I'd always seen a cut defined with an explicit condition such as
$$L = \{x \in \mathbb Q: x^2 \lt 2\} \quad R = \{x \in \mathbb Q: x^2 \ge 2\}$$
I'd already spotted a possible inconsistency in considering this to be less well defined than the set of all Cauchy sequences, although I haven't worked it all through in my mind.

But while I know that we can't describe a condition for every real without using an infinite number of characters, the set of binary sequences (not ending in an infinite string of 1s) is well defined and can be easily used to define a Cauchy sequence, giving us the reals in the half-open interval $[0,1)$ and thus all reals.

I'm aware that I'm using infinite binary sequences while at the same time not being comfortable with infinite sequences of symbols in the set definition, but haven't thought the conclusion of that dichotomy through. But I also have a feeling that the Cauchy route uses a sequence that I know makes sense to generate a number. The Dedekind Cut seems to require more from its infinite sequence of symbols unless it relies on the Cauchy sequence itself.

I guess this is a question about what it means to construct the reals. Do we just create an amorphous blob of objects that must contain them all several times over, or do we create each one separately.

Last edited by skipjack; August 22nd, 2017 at 10:56 PM.
v8archie is offline  
August 22nd, 2017, 06:05 PM   #15
Senior Member
 
Joined: Aug 2012

Posts: 1,574
Thanks: 380

Quote:
Originally Posted by v8archie View Post
That's not a visualisation I've seen before. I'd always seen a cut defined with an explicit condition such as
$$L = \{x \in \mathbb Q: x^2 \lt 2\} \quad R = \{x \in \mathbb Q: x^2 \ge 2\}$$
Oh I see. That couldn't possibly work. There aren't enough explicit expressions (no matter how you define them) to construct a complete ordered field, which the reals are required to be. (Defined to be, in fact).

We see expressions like that as examples of cuts; but that's not how cuts are defined.

Rather than go into all the details I'll just refer readers to Rudin, PMA, 3rd edition, page 17ff. Earlier he states without proof theorem 1.19, that there exists a complete ordered field.

Now on page 17 he proves that theorem. He defines a cut as any subset of the rationals that's nonempty, not all the rationals, that's downward closed (if x is in the cut and y < x then y is in the cut); and that has no largest element.

From that definition, he shows that the set of cuts, with <, +, and * defined appropriately, is a complete ordered field that satisfies all of the axioms for the real numbers given earlier in the chapter.

There is no reference to any specific cut. Any concrete example we give is of course of a very special form, namely describable by a formula, or computable, or some such characterization. Of course there aren't enough of those to make up all the reals and the construction never discusses any particular cuts.

Note that the set of cuts exists. First, the power set of the rationals exists by the powerset axiom. Among all those subsets, some are cuts. Namely they are nonempty, downward closed, etc.; and each condition is a predicate that can be applied to the power set via the axiom of specification. So the set of cuts exists.

Last edited by Maschke; August 22nd, 2017 at 06:11 PM.
Maschke is online now  
August 23rd, 2017, 10:29 PM   #16
Math Team
 
agentredlum's Avatar
 
Joined: Jul 2011
From: North America, 42nd parallel

Posts: 3,372
Thanks: 233

Quote:
Originally Posted by v8archie View Post
That's not a visualisation I've seen before. I'd always seen a cut defined with an explicit condition such as
$$L = \{x \in \mathbb Q: x^2 \lt 2\} \quad R = \{x \in \mathbb Q: x^2 \ge 2\}$$
For this to work properly as Dedekind intended , you have to add an extra condition otherwise the partition to L and R sets gets jumbled up

$L = \{x \in \mathbb Q: x^2 \lt 2 \ \ or \ \ x \lt 0 \}$

$R = \{x \in \mathbb Q: x^2 \ge 2 \ \ or \ \ x \gt 0 \}$

With this extra condition a number like $ \ \ -5 \ \ $ surely belongs to $ \ \ L $


Last edited by agentredlum; August 23rd, 2017 at 10:33 PM.
agentredlum is offline  
August 24th, 2017, 01:45 AM   #17
Global Moderator
 
greg1313's Avatar
 
Joined: Oct 2008
From: London, Ontario, Canada - The Forest City

Posts: 7,599
Thanks: 941

Math Focus: Elementary mathematics and beyond
Country Boy, [tex]...[/tex] and [latex]...[/latex] no longer work here. Please use [math]...[/math]. Thanks.
greg1313 is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
dedekind, irrationals, theory



Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Dedekind cut of √2 ActSci Real Analysis 16 September 1st, 2015 10:51 AM
All Fox are irrationals complicatemodulus Number Theory 113 August 20th, 2015 04:08 AM
Find four irrationals shunya Elementary Math 1 March 20th, 2014 12:46 PM
irrationals nikkor180 Real Analysis 1 August 15th, 2011 01:12 PM
dedekind cut tinynerdi Real Analysis 0 October 10th, 2010 02:40 PM





Copyright © 2017 My Math Forum. All rights reserved.