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Topic: Why are the real numbers well-ordered?
Replies: 32   Last Post: Sep 11, 2001 9:08 AM

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 Dave Seaman Posts: 2,446 Registered: 12/6/04
Re: Why are the real numbers well-ordered?
Posted: Sep 11, 2001 9:08 AM

In article <9nj6jk\$of0\$1@tron.sci.fi>, Tapio <hurmecom@dlc.fi> wrote:

>"Dave Seaman" <ags@seaman.cc.purdue.edu> wrote in message
>news://9niqte\$fol@seaman.cc.purdue.edu...

>OK, the confusing notation has been perhaps a reason why we had so much
>problems and missunderstanding. I assume you mean now that the structure of
>the number system would be something like this:

>...[w+2 sequences][w+1 sequences][w sequences= integers N(inf) and
>N][decimal sequences][w+1 decimal sequences][w+2 decimal sequences] ...

Each of those is certainly a well-defined set of mappings. Is there some
connection here to your claim that the reals are well-ordered? The two
problems I see are that (a) your digit strings are not the real numbers,
and (b) your digit strings are not well-ordered.

>> >the result is 1(something) as the carry goes over aleph length of string.
>> >1(something) cannot be after addition smaller than [9]. The digit 1 must

>be
>> >"behind the infinity" and therefore w(inf) = 1,000... = [0)1][0].

>> Then w(inf) is the (w+1)-sequence a: (w+1) -> D given by

>> a_k = 0, if k < w
>> a_w = 1.

>Yes!! I assume now you have got the basic idea correctly.

If you have access to any of our discussions from several years ago, you
may find that I was suggesting way back then that you were using
(w+1)-sequences of digits when talking about things like 0.999...8, but
you have been resisting that characterization up until now. You do
realize, I take it, that (w+1)-strings do not map to real numbers or to
integers in any natural way?

>> >> Remember, you can't use N(inf) in the definition of N(inf), because
>that
>> >> would make it a circular definition. Look at how other number systems
>> >> are defined. The reals are defined in terms of the rationals, the
>> >> rationals in terms of the integers, and the integers in terms of the
>> >> natural numbers. In each case we define a new and more complex number
>> >> system out of a simpler one. So it is with N(inf). If you want to
>> >> describe what your numbers are in terms that people can understand, you
>> >> need to describe them in terms of simpler kinds of numbers so as to

>avoid
>> >> circularity.

>Yes. I do understand your point. Therefore I set a question for myself. If
>we define that the string is infinite instead of finite, then why should I
>use in the sum (a_k)10^k k-values in N as I have N(inf) in use? I think we
>are in "half way", if we do not use the opportunity of infinite sequences
>"infinite integers" for k in N(inf). Just like if we have k in N as soon as
>N is available. I assume this need further consideration. Do you have any
>constructive idea?

Once you have defined N(inf), you can use it to define something else,
say N(hyperinf), that would have the kind of extended sequences you want.
Or, you could define N(inf) directly to use ordinal sequences, which you
are now doing already in the case of (w+1)-sequences, thereby avoiding
circularity.

But where is all this leading? You still have not answered these
questions:

(1) What is the smallest member of Z?
(2) What is the smallest member of N(inf)\N?

(2) should be an w-sequence. You cannot answer these questions, and this
shows that R and N(inf) are not well-ordered.

>> >I think I asked once about the circularity to define N as sum. But you
>did
>> >not answer, perhaps because you saw the analogy.

>> I have no idea what you are talking about.

>See above.

I still have no idea what you are talking about.

>>N can be defined from the
>> Peano axioms or from the axioms of ZF. Neither approach makes any
>> mention of sums until well after the natural numbers are already defined.
>> There is no circularity. The notion that sums come first and numbers
>> afterwards is entirely your own aberration.

>Not exactly, the key point is that the strings are first defined to be
>infinite.

Huh? Thre is nothing in the Peano axioms and nothing in ZF about
"strings".

>> This part of the discussion originated when I asked you for the smallest
>> member of N(inf)\N.

>Yes.

>> 1. It is not a member of N(inf), since [1(0] is an
>> (w+1)-sequence and the members of N(inf) are all w-sequences.

>No. I think you consider that the indexing of placeholders depends on N, but
>not in N(inf)??? I cannot exactly follow your logic although above you
>seemingly and finaly understood the structure of the system. Everything
>within [] belongs to w-sequence area.

Does your [1(0] represent a mapping a: w -> D? If so, what are the
values? For which k do you get a_k = 1? For which k do you get a_k = 0?
It appears to me that all but one of the digits are 0, which makes it a
member of N and therefore not a member of N(inf)\N. If you are not
talking about an w-sequence, then you do not have a member of N(inf).

>Actually we should talk about w_inf sequence areas w_inf +1, w_inf +2 etc.
>Numbers are now infinite in N(inf) finite numbers (integers) are special
>cases.

>> 2. You keep claiming that [1(0] = [9] + 1.

>No, no, there is missunderstanding or typo.

I was taking [1(0] to be an (w+1)-sequence, but you seem to be saying
that it is an w-sequence. I don't understand how that can be, since it
appears to have a leftmost (highest-numbered) digit and there is no
largest member of w.

>[9]+1= [9]+[0)1]= [0)1][0] = 1,000.... where "," is the w+1 point of

I don't understand your notation, but I think you are talking about the
(w+1)-sequence given by a_w = 1 and a_k = 0 for k < w.

--
Dave Seaman dseaman@purdue.edu
Amnesty International calls for new trial for Mumia Abu-Jamal

Date Subject Author
8/31/01 Dave Seaman
9/7/01 Tapio Hurme
9/7/01 Dave Seaman
9/7/01 Tapio Hurme
9/8/01 Dave Seaman
9/8/01 Tapio Hurme
9/8/01 Dave Seaman
9/8/01 Tapio Hurme
9/8/01 Dave Seaman
9/8/01 Tapio Hurme
9/8/01 Lee Rudolph
9/8/01 Tapio Hurme
9/8/01 Dik T. Winter
9/8/01 Denis Feldmann
9/8/01 Dave Seaman
9/8/01 Tapio Hurme
9/8/01 Dave Seaman
9/9/01 Tapio Hurme
9/9/01 Dave Seaman
9/9/01 Tapio Hurme
9/9/01 Dave Seaman
9/9/01 Tapio Hurme
9/10/01 Dave Seaman
9/10/01 Tapio Hurme
9/10/01 Dave Seaman
9/8/01 Dave Seaman
9/9/01 Tapio Hurme
9/9/01 Dave Seaman
9/10/01 Tapio Hurme
9/10/01 Dave Seaman
9/10/01 Tapio Hurme
9/11/01 Dave Seaman