Lexicographical Order Definition

[Gauss]

From the ‘order of axioms’ in real number analysis, for every [math]\alpha , \beta \in \mathbb{R}[/math], exactly one and only one of the following holds.

 

a) [math] \alpha < \beta[/math]

b) [math] \alpha = \beta[/math]

c) [math] \alpha > \beta[/math]

 

Then an order for the real numbers can be laid down lexicographically.

 

Let [math]\alpha \in \mathbb{R}[/math], and let [math]\alpha [/math] be expressed in the form:

 

[math]\alpha = a_0.a_1a_2a_3 . . . a_k[/math]

 

Let [math]\beta \in \mathbb{R}[/math], and let [math]\beta [/math] be expressed in the form:

 

[math]\beta = b_0.b_1b_2b_3 . . . b_k[/math]

 

When the first [math]a_k[/math] that differs from [math]b_k[/math] and [math]a_k[/math] < [math]b_k[/math] then [math]\alpha < \beta[/math], if [math]a_k[/math] = [math]b_k[/math] [math]\forall _k[/math] then [math]\alpha = \beta[/math] and if the first [math]a_k[/math] that differs from [math]b_k[/math] and [math]a_k[/math] > [math]b_k[/math] then [math]\alpha > \beta[/math].

 

Where [math]a_k[/math] are integers and [math]0 \leqslant a_k \leqslant 9[/math] and [math]b_k[/math] are integers and [math]0 \leqslant b_k \leqslant 9[/math] and [math]k = \mbox { }\{0, 1, 2, 3, . . .\mbox { }\}[/math]

 

_____________________________________________________________

 

I would appreciate your feedback on improving (mathematically, more rigorous) the last four lines of the lexicographical definition.

[jordan]

I really wish I knew what half of that was saying. I think I can follow a bit of it but I'm not familiar with all the signs. If you happen to get the chance or feel inspired, would you mind writting some of that out in words?

[Gauss]

A friend comes along and gives you two numbers one is called [math]\alpha[/math] (alpha) and the other is called [math]\beta[/math] (beta). The friend also tells you that they (the numbers given to you) belong to a certain group of numbers called the real numbers and these numbers are represented by [math]\mathbb{R}[/math].

 

So you have these two numbers [math]\alpha[/math] and [math]\beta[/math]. Since you are a mathematical genius you want to arrange these two numbers in order. Then from the defintion of 'order of axioms' you can arrange these two numbers by looking at them and applying the following rules.

 

a) [math] \alpha < \beta[/math] (This means the first number alpha is less than beta)

b) [math] \alpha = \beta[/math](This means the first number alpha is equal to beta)

c) [math] \alpha > \beta[/math](This means the first number alpha is greater than beta)

 

Suppose your friend gave you these two numbers [math]\alpha = 3.78634 . . .[/math] and [math]\beta = 3.78629 . . .[/math]. Then from the rest of the lexicographical definition you can compare each single digit in the numbers [math]\alpha[/math] and [math]\beta[/math] and find out if one is less than, equal to or greater than the other.

 

Of course you are very smart and you put the second number down first ([math]\beta[/math]) and then to the right and next to it you put down the first number ([math]\alpha[/math]). Because 3.78629 . . . is less than 3.78634 . . .

 

I hope this helps you in understanding what is going on with all the mathematical terms.

[jordan]

Upon reading it a second time, it makes much more sense to me. I understand exactly what you are saying, though I don't think I can offer much help with the more rigorous proof you requested.

 

if [math]a_k[/math] = [math]b_k[/math] [math]\forall _k[/math]

 

[math]0 \leqslant a_k \leqslant 9[/math]

Above are the symbols I didn't recognize. From the text in the math brackets I read after quoting you' date=' it seems that [math']\forall[/math] would mean that it holds true "for all values of k". And the other symbols appears to be a "less than or equal to" sign, though I've never seen it with the "equal to" line slanted and didn't know if it meant something different. Would that be right?

 

Thanks for taking time to explain it for me.

[Gauss]

You are correct the symbol [math]\forall[/math] means "for all values of".

 

The symbol [math] < [/math] means "less than" of course and adding a slant line to the less than symbol gives you the symbol [math]\leqslant[/math] which means "less than or equal to". The symbol [math]\geqslant[/math] means "greater than or equal to".

 

I hope this helps you.

[e(ho0n3]

The lexicographical order definition you give does not order the reals. Here is a simple counter-example: Let [math]\textstyle \alpha[/math] = 1.000... and [math]\textstyle \beta[/math] = 0.999...

[jordan]

Those are supposed to be repeating infinitely right? If so, that's a good point. Perhaps restricting the defintion to terminating decimals would help? I can't come up with a counter-example for that.

[e(ho0n3]

Yes they do repeat infinitely. If the numbers you use contain a finite number of digits, then you are not dealing with the reals, but rather a subset of them.

[jordan]

Yes, you're right. So I reconmended either using that subset or changing the defintion. At the time, without much time to think about it, I wasn't sure how easy changing the definition would be so I suggested restricting non-terminating decimals until they could be worked in somehow. If I come up with anything as I am cutting the grass, I'll post and see what you think. It should keep my mind occupied, but I wouldn't count on me coming up with anything.