My Math Forum Question on Dedekind Cuts

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January 21st, 2019, 09:48 AM   #31
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Quote:
 Originally Posted by v8archie

This is the funny part :

Quote:
 Originally Posted by zylo EDIT By the way, it follows from my decimal definition of real numbers that they are complete, ie, there is no infinite decimal that is not in the definition. Obviously, there are other numbers beside the rationals, which are either repeating or finite decimals.
I knew somehow we'd get back to the above with zylo…

I tell you what zylo... I'll agree with your definition of the reals in $(0,1)$ based on their decimal expansions if you clean up your error involving reals of the form $\frac{n}{10^m}$, where $n,m \in \mathbb{N}$.

January 21st, 2019, 10:50 AM   #32
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Quote:
 Originally Posted by AplanisTophet I tell you what zylo... I'll agree with your definition of the reals in $(0,1)$ based on their decimal expansions if you clean up your error involving reals of the form $\frac{n}{10^m}$, where $n,m \in \mathbb{N}$.
This is my definition of the reals. It says nothing about expansions.

Definition of real numbers in [0,1): List the natural numbers, put a period after them, and read them in reverse.
1. -> .1
2. -> .2
.
10. -> .01
.
95141. -> .14159

And I don't have the slightest idea what you mean by "clean-up"

What is your definition of the real numbers, or what was the point of bringing up Dedekind cuts?

Edit:
Actually, I did answer the OP, specifically and succinctly:

Quote:
 Originally Posted by AplanisTophet What happens for noncomputable reals where there is no formula with which to apply the axiom schema of specification when partitioning $\mathbb{Q}$? /ls.
I now realize it is a very good question. Thanks.

Last edited by zylo; January 21st, 2019 at 11:34 AM.

January 21st, 2019, 12:26 PM   #33
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Quote:
 Originally Posted by zylo This is my definition of the reals. It says nothing about expansions. Definition of real numbers in [0,1): List the natural numbers, put a period after them, and read them in reverse. 1. -> .1 2. -> .2 . 10. -> .01 . 95141. -> .14159 And I don't have the slightest idea what you mean by "clean-up"
Ok, I no longer accept your definition if this is what it is. You know this doesn't even hit all the rationals. I'm not going to keep doing this with you.

If you instead define the set $A = \{0,1,2,3,4,5,6,7,8,9\}$ and then define each real number in $(0,1]$ as an infinite string of the form:

$$0.x_1x_2x_3\dots, \text{ where each } x_i \in A \text{ such that no string contains a } j \in \mathbb{N} \text{ where all } x_{p \geq j} = 0$$

then we'll be much closer. It's sloppy still and no real has a finite expansion, but hey. Maybe you could clean up the above a little bit.

Last edited by AplanisTophet; January 21st, 2019 at 12:37 PM.

January 21st, 2019, 12:49 PM   #34
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Quote:
 Originally Posted by zylo . . . it follows from my decimal definition of real numbers
You've consistently been unable to provide that definition without being unable to account for 1/9, for example. When I've asked about that, you've never done anything better than repeat the so-called definition (sometimes with subtle unexplained changes) without answering my question. Often, you don't reply to my specific questions at all.

January 22nd, 2019, 08:12 AM   #35
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Once again, the decimal representation of 1/9 is .11... where the sequence of 1's is endless. It is a number in the sequence:

Quote:
 Originally Posted by zylo Definition of real numbers in [0,1): List the natural numbers, put a period after them, and read them in reverse. 1. -> .1 2. -> .2 . 10. -> .01 . 95141. -> .14159 .
which is also endless.

EDIT Back to OP

A cut for an irrational (endless) decimal is an infinite sequence of cuts: Define a cut for $\displaystyle S_{n}=\sum_{i=1}^{n}a_{i}, \quad a_{i}=0,1,...,9$ for all n. Of course each step in the process is a rational number but the result is cut of an irrational number.

A cut is really a meaningless figment of the imagination, but as an isolated concept interesting. It assumes you can divide all the rational numbers (endless) into two sets A and B st every member of A is less than every member of B and A has no largest member. You can of course assume it can be done. It only becomes insidious when it intrudes on analysis and casts an impenetrable fog of obfuscation and cripples the ability to think and understand.

A rational version of decimals and the real numbers is given here:
Decimals and the Continuum

Last edited by zylo; January 22nd, 2019 at 08:48 AM.

 January 22nd, 2019, 09:21 AM #36 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,675 Thanks: 2654 Math Focus: Mainly analysis and algebra
January 22nd, 2019, 09:42 AM   #37
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Quote:
 Originally Posted by v8archie
+1

As long as there is a moderator to clean up every time zylo piddles on the floor, it should be ok.

January 22nd, 2019, 09:46 AM   #38
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Quote:
 Originally Posted by zylo Once again, the decimal representation of 1/9 is .11... where the sequence of 1's is endless.
I didn't ask for the decimal representation of 1/9, zylo. I asked how you get it by using your definition. You've shown how some other numbers are produced, but not 1/9.

January 22nd, 2019, 12:37 PM   #39
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Quote:
 Originally Posted by skipjack I didn't ask for the decimal representation of 1/9, zylo. I asked how you get it by using your definition. You've shown how some other numbers are produced, but not 1/9.
What is 1/9?
1/9 is the infinite (unending) decimal .111......., by standard (High School) division. This is an example of how one arrives at decimal representation.

What is the infinite (unending) decimal .111.....?
An infinite decimal whose sum evaluated to n places gets arbitrarily close to 1/9 as n increases indefinitely. The comparison can be made because .1111 .. . evaluated to n places is a rational number.

What is 1/9 in radix 9 notation? .1: Divide a unit line into nine intervals. 1/9 is end of first interval.

That's the way a ruler works:
Divide the standard inch in half: 1/2
Divide each interval in half: 1/4
Divide each interval in half again: 1/8
................................................
A carpenter doesn't deal in decimals. His ruler (a tape) is divided in 1/32's. He works to 1/32's or maybe 1/16's.
A cabinet maker would work to 1/64th's. His ruler is probably a yard stick graduated to 1/64th's.
A machinist deals in thousandths. His ruler is a micrometer or vernier caliper which reads to three (.001) or four (.0001) decimal places.
There is constant conversion in practice: 1/16 = .0625 for example

To convert Pi to a decimal representation: Take a circle of diameter 1. Then unwind the circle onto a line with same unit 1. The decimal representation of the point where the line ends can be found as in:

Decimals and the Continuum

January 22nd, 2019, 12:41 PM   #40
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Quote:
 Originally Posted by zylo What is 1/9? 1/9 is the infinite (unending) decimal .111......., by standard (High School) division. This is an example of how one arrives at decimal representation.
Aren't you claiming that every decimal expansion is derived from reversing the digits of a positive integer? How does .111... get generated in your system?

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