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Foundations of real numbers
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Paul
Oct 10, 2012, 1:30:09 PM10/10/12
to
It seems ridiculous to me to define the real numbers by using the least-upper-bound property as an axiom, although this seems to be the most common way.
The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too. I don't at all understand how it makes sense to regard the least-upper-bound property of the reals as an axiom. The least-upper-bound property should be regarded as a theorem, not an axiom.
Why did the least-upper-bound-property-as-an-axiom approach become so prevalent? If you define real numbers that way, the correspondence between our intuitive sense of real numbers and the formalisation is so much less clear than with either Dedekind cuts or Cauchy sequences.
Paul Epstein
The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too. I don't at all understand how it makes sense to regard the least-upper-bound property of the reals as an axiom. The least-upper-bound property should be regarded as a theorem, not an axiom.
Why did the least-upper-bound-property-as-an-axiom approach become so prevalent? If you define real numbers that way, the correspondence between our intuitive sense of real numbers and the formalisation is so much less clear than with either Dedekind cuts or Cauchy sequences.
Paul Epstein
Ken Pledger
Oct 10, 2012, 4:17:14 PM10/10/12
to
In article <9082b607-540d-4e9d...@googlegroups.com>,
This is about syntax vs semantics.
If you set up a list of axioms for the reals as a complete ordered
field, then you might well include the least-upper-bound axiom. The
formal theory based on all the axioms is essentially just
symbol-manipulation.
To give the symbols some sort of meaning you need a model for the
theory. Dedekind cuts or Cauchy sequences can be used as the last stage
of the process in setting up such a model.
Ken Pledger.
If you set up a list of axioms for the reals as a complete ordered
field, then you might well include the least-upper-bound axiom. The
formal theory based on all the axioms is essentially just
symbol-manipulation.
To give the symbols some sort of meaning you need a model for the
theory. Dedekind cuts or Cauchy sequences can be used as the last stage
of the process in setting up such a model.
Ken Pledger.
Peter Webb
Oct 11, 2012, 1:56:52 AM10/11/12
to
"Paul" wrote in message
news:9082b607-540d-4e9d...@googlegroups.com...
The problem (as I see it) with Cauchy sequences is that they are sequences,
and hence require a mapping from N to Rationals. The LUB property is more
general, in that it applies to unordered sets. So whilst Cauchy is a very
intuitive definition for Reals, the LUB property is of more use in proofs.
Given that you can prove LUB from Cauchy and vice versa it doesn't really
matter whether you define Reals as limits of Cauchy sequences or in terms of
LUB; as I said Cauchy is more intuitive but LUB more useful. IMHO.
Robin Chapman
Oct 11, 2012, 3:58:29 AM10/11/12
to
On 10/10/2012 18:30, Paul wrote:
> It seems ridiculous to me to define the real numbers by using the
> least-upper-bound property as an axiom,
> It seems ridiculous to me to define the real numbers by using the
> least-upper-bound property as an axiom,
> So the Cauchy sequence definition is much better, and I like the
> Dedekind cuts definition too.
The first approach is a way to *characterize* R.
> Dedekind cuts definition too.
The other approaches are ways to *construct* R.
Here's a nice paper:
http://uk.arxiv.org/abs/1204.4483
Frederick Williams
Oct 11, 2012, 1:15:01 PM10/11/12
to
Paul wrote:
>
> It seems ridiculous to me to define the real numbers by using the least-upper-bound property as an axiom, although this seems to be the most common way.
> The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
>
> So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too.
One nice thing about Cauchy sequences is they generalize to metric
>
> It seems ridiculous to me to define the real numbers by using the least-upper-bound property as an axiom, although this seems to be the most common way.
> The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
>
> So the Cauchy sequence definition is much better, and I like the Dedekind cuts definition too.
spaces. Meanwhile, if you mash up von Neumann ordinals and Dedekind
cuts you get Conway's surreal numbers.
[I have heard young people use the phrase 'mash up'. I don't know what
it means, but I think it's a prerequisite for posting to sci.math that
one uses terminology one doesn't understand.]
Meray's nested intervals are also nice. And one can use decimal
expansions, which looks like a bad idea but can be made to work quite
nicely.
--
Where are the songs of Summer?--With the sun,
Oping the dusky eyelids of the south,
Till shade and silence waken up as one,
And morning sings with a warm odorous mouth.
Michael Stemper
Oct 11, 2012, 1:28:39 PM10/11/12
to
In article <k55n3g$pdu$1...@news.albasani.net>, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> writes:
>The problem (as I see it) with Cauchy sequences is that they are sequences,
>and hence require a mapping from N to Rationals.
Why would that be a problem?
>The problem (as I see it) with Cauchy sequences is that they are sequences,
>and hence require a mapping from N to Rationals.
> The LUB property is more
>general, in that it applies to unordered sets.
define any upper (or lower) bounds.
--
Michael F. Stemper
#include <Standard_Disclaimer>
Life's too important to take seriously.
Frederick Williams
Oct 11, 2012, 2:38:01 PM10/11/12
to
Frederick Williams wrote:
>
> Meray's nested intervals are also nice.
Not Meray, rather Paul Gustav Heinrich Bachmann. My apologies to both
>
> Meray's nested intervals are also nice.
of them.
Herman Rubin
Oct 11, 2012, 4:31:21 PM10/11/12
to
On 2012-10-10, Paul <peps...@gmail.com> wrote:
> It seems ridiculous to me to define the real numbers by using the
least-upper-bound property as an axiom, although this seems to be the
most common way.
The mistake are making is that you are looking for a DEFINITION
> It seems ridiculous to me to define the real numbers by using the
least-upper-bound property as an axiom, although this seems to be the
most common way.
of the real numbers rather than a CHARACTERIZATION.
If it looks like the real numbers and acts like the real
numbers, it is a version. All of the approaches yu give
are ways of showing that it is possible to construct something
with those properties. That those are existence theorems is
of some small importance, but the lub property is the one which
is most universally useful.
> The definition of the reals should formalise the way we approximate
pi by a decimal expansion: 3, 3.1, 3.14 etc.
> So the Cauchy sequence definition is much better, and I like the
Dedekind cuts definition too. I don't at all understand how it makes
sense to regard the least-upper-bound property of the reals as an axiom.
The least-upper-bound property should be regarded as a theorem, not
an axiom.
> Why did the least-upper-bound-property-as-an-axiom approach become
so prevalent? If you define real numbers that way, the correspondence
between our intuitive sense of real numbers and the formalisation is so
much less clear than with either Dedekind cuts or Cauchy sequences.
in that manner. All of the constructions run into some difficulties,
which can be taken care of with technical proofs. Personally, I like
the Cauchy sequence construction best, but one must be careful to
avoid the need for the axiom of choice in using it.
> Paul Epstein
--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hru...@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558
Michael Stemper
Oct 11, 2012, 5:38:21 PM10/11/12
to
In article <slrnk7eb4t...@skew.stat.purdue.edu>, Herman Rubin <hru...@skew.stat.purdue.edu> writes:
>On 2012-10-10, Paul <peps...@gmail.com> wrote:
>> It seems ridiculous to me to define the real numbers by using the
>least-upper-bound property as an axiom, although this seems to be the
>most common way.
>
>The mistake are making is that you are looking for a DEFINITION
>of the real numbers rather than a CHARACTERIZATION.
>On 2012-10-10, Paul <peps...@gmail.com> wrote:
>> It seems ridiculous to me to define the real numbers by using the
>least-upper-bound property as an axiom, although this seems to be the
>most common way.
>
>The mistake are making is that you are looking for a DEFINITION
>of the real numbers rather than a CHARACTERIZATION.
>Again, we do not define real numbers that way, we characterize them
>in that manner. All of the constructions run into some difficulties,
>which can be taken care of with technical proofs. Personally, I like
>the Cauchy sequence construction best, but one must be careful to
>avoid the need for the axiom of choice in using it.
I (as a non-mathematician) prefer the Cauchy sequence construction,
>in that manner. All of the constructions run into some difficulties,
>which can be taken care of with technical proofs. Personally, I like
>the Cauchy sequence construction best, but one must be careful to
>avoid the need for the axiom of choice in using it.
probably because it matches my intuitive understanding of real numbers
(built on years of using the decimal system). But, I'm suprised to
discover that AC rears its ugly head here.
What I've seen of the Cauchy construction uses things like "show
sequences x_i and y_i converge to the same point" or "show that
sequence z_i exists with the necessary properties." Could you
give an example of where Choice might be invoked? Preferably,
one that could be understood by a freshman or sophomore?
Shmuel Metz
Oct 11, 2012, 9:05:35 AM10/11/12
to
In <k55n3g$pdu$1...@news.albasani.net>, on 10/11/2012
at 04:56 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>The problem (as I see it) with Cauchy sequences is that they are
>sequences, and hence require a mapping from N to Rationals. The LUB
>property is more general, in that it applies to unordered sets. So
>whilst Cauchy is a very intuitive definition for Reals, the LUB
>property is of more use in proofs. Given that you can prove LUB from
>Cauchy and vice versa it doesn't really matter whether you define
>Reals as limits of Cauchy sequences or in terms of LUB; as I said
>Cauchy is more intuitive but LUB more useful. IMHO.
For proving properties of the Reals, a small set of axioms is easiest;
you don't need a construction. However, both equivalence classes of
Cauchy sequences and Dedekind cuts are useful for demonstrating that
there is a system satisfying the axioms.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
at 04:56 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>The problem (as I see it) with Cauchy sequences is that they are
>sequences, and hence require a mapping from N to Rationals. The LUB
>property is more general, in that it applies to unordered sets. So
>whilst Cauchy is a very intuitive definition for Reals, the LUB
>property is of more use in proofs. Given that you can prove LUB from
>Cauchy and vice versa it doesn't really matter whether you define
>Reals as limits of Cauchy sequences or in terms of LUB; as I said
>Cauchy is more intuitive but LUB more useful. IMHO.
you don't need a construction. However, both equivalence classes of
Cauchy sequences and Dedekind cuts are useful for demonstrating that
there is a system satisfying the axioms.
--
Shmuel (Seymour J.) Metz, SysProg and JOAT <http://patriot.net/~shmuel>
Unsolicited bulk E-mail subject to legal action. I reserve the
right to publicly post or ridicule any abusive E-mail. Reply to
domain Patriot dot net user shmuel+news to contact me. Do not
reply to spam...@library.lspace.org
1treePetrifiedForestLane
Oct 11, 2012, 10:31:01 PM10/11/12
to
cannonical digital representation of base-one accouting,
is?
is?
Peter Webb
Oct 12, 2012, 12:19:25 AM10/12/12
to
"Michael Stemper" wrote in message news:k56vk7$lja$2...@dont-email.me...
In article <k55n3g$pdu$1...@news.albasani.net>, "Peter Webb"
<webbfamily@DIE_SPAMoptusnet.com.au> writes:
>The problem (as I see it) with Cauchy sequences is that they are sequences,
>and hence require a mapping from N to Rationals.
Why would that be a problem?
Because in order to define a Real number you have define a mapping from N to
Rationals. Lots of situations arise where there is no obvious mapping. The
LUB property makes it easy to say that the sequence pi, pi-3, pi-3.14,
pi-3.141, ... defines a unique Real; try doing that with Cauchy (you can, of
course, but it involves work). These sorts of sets occur frequently in
analysis.
> The LUB property is more
>general, in that it applies to unordered sets.
It does? I always thought that order was necessary before you could
define any upper (or lower) bounds.
Wrong. You need an order relation to define "least", but the set itself need
not be ordered. The set {x|x>1 & x<2} clearly has a LUB even though there is
no mapping from N to the set.
Virgil
Oct 12, 2012, 12:56:16 AM10/12/12
to
In article <k585os$h34$1...@news.albasani.net>,
Actually, there are lots of mappings from N to the set {x|x>1 & x<2}, or
to any real interval of positive length. It is true, but irrelevant,
that there are no surjections.
--
to any real interval of positive length. It is true, but irrelevant,
that there are no surjections.
--
LudovicoVan
Oct 12, 2012, 6:04:26 AM10/12/12
to
"Virgil" <vir...@ligriv.com> wrote in message
news:virgil-81315D....@bignews.usenetmonster.com...
>> Wrong. You need an order relation to define "least", but the set itself
>> need
>> not be ordered. The set {x|x>1 & x<2} clearly has a LUB even though there
>> is
>> no mapping from N to the set.
>
> Actually, there are lots of mappings from N to the set {x|x>1 & x<2}, or
> to any real interval of positive length. It is true, but irrelevant,
> that there are no surjections.
That must be why you are mentioning it.
-LV
news:virgil-81315D....@bignews.usenetmonster.com...
> In article <k585os$h34$1...@news.albasani.net>,
> "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> wrote:
<snip>
> "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> wrote:
>> Wrong. You need an order relation to define "least", but the set itself
>> need
>> not be ordered. The set {x|x>1 & x<2} clearly has a LUB even though there
>> is
>> no mapping from N to the set.
>
> Actually, there are lots of mappings from N to the set {x|x>1 & x<2}, or
> to any real interval of positive length. It is true, but irrelevant,
> that there are no surjections.
-LV
Michael Stemper
Oct 12, 2012, 8:40:53 AM10/12/12
to
In article <k585os$h34$1...@news.albasani.net>, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> writes:
>"Michael Stemper" wrote in message news:k56vk7$lja$2...@dont-email.me...
>In article <k55n3g$pdu$1...@news.albasani.net>, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> writes:
>>>The problem (as I see it) with Cauchy sequences is that they are sequences,
>>>and hence require a mapping from N to Rationals.
>>
>>Why would that be a problem?
>>
>>
>"Michael Stemper" wrote in message news:k56vk7$lja$2...@dont-email.me...
>In article <k55n3g$pdu$1...@news.albasani.net>, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au> writes:
>>>The problem (as I see it) with Cauchy sequences is that they are sequences,
>>>and hence require a mapping from N to Rationals.
>>
>>Why would that be a problem?
>>
>>
>>Because in order to define a Real number you have define a mapping from N to
>>Rationals.
Repeating the statement doesn't explain it.
>>Rationals.
>> Lots of situations arise where there is no obvious mapping. The
>>LUB property makes it easy to say that the sequence pi, pi-3, pi-3.14,
>>pi-3.141, ... defines a unique Real; try doing that with Cauchy (you can, of
>>course, but it involves work).
sequence you just wrote implies another one: 0, 3, 3.14, 3.141, ...
This is the beginning of a perfectly cromulent Cauchy sequence for pi.
>>> The LUB property is more
>>>general, in that it applies to unordered sets.
>>
>>It does? I always thought that order was necessary before you could
>>define any upper (or lower) bounds.
>>
>>Wrong. You need an order relation to define "least",
As far as I can tell, you also need that to define "upper bound" or
"lower bound", as well. If that's not the case, maybe you could give
an example, perhaps on C.
>> The set {x|x>1 & x<2} clearly has a LUB even though there is
>>no mapping from N to the set.
But, what of it? That set isn't a Cauchy sequence, and doesn't represent
a real.
G. A. Edgar
Oct 12, 2012, 9:40:52 AM10/12/12
to
For the beginning of a subject, sometimes the choices made are limited.
For example: I can imagine* a textbook that begins by postulating a
complete ordered field R. (Complete in the order sense.) Then
constructing, within R, the set N or natural numbers. Sometime later
defining "sequence" as a function with domain N. And only still later
defining "Cauchy" sequence. So, for this arrangement the least upper
bound axiom makes perfect sense.
* Perhaps called CALCULUS, written by M. Spivak ...
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
For example: I can imagine* a textbook that begins by postulating a
complete ordered field R. (Complete in the order sense.) Then
constructing, within R, the set N or natural numbers. Sometime later
defining "sequence" as a function with domain N. And only still later
defining "Cauchy" sequence. So, for this arrangement the least upper
bound axiom makes perfect sense.
* Perhaps called CALCULUS, written by M. Spivak ...
--
G. A. Edgar http://www.math.ohio-state.edu/~edgar/
Shmuel Metz
Oct 12, 2012, 9:51:06 AM10/12/12
to
In <k585os$h34$1...@news.albasani.net>, on 10/12/2012
at 03:19 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because in order to define a Real number you have define a mapping
>from N to Rationals.
You're restating the phrase instead of explaining why it is a problem.
>Lots of situations arise where there is no obvious mapping.
1. You haven't explained why you need an obvious mapping
2. You haven't given an example where it it is a problem.
>The LUB property makes it easy to say that the sequence pi, pi-3,
>pi-3.14, pi-3.141, ... defines a unique Real;
Since each of those terms is irrational, it is not easy. Since the
sequence is a Cauchy sequence, you have the obvious mapping that you
say you need.
at 03:19 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because in order to define a Real number you have define a mapping
>from N to Rationals.
>Lots of situations arise where there is no obvious mapping.
2. You haven't given an example where it it is a problem.
>The LUB property makes it easy to say that the sequence pi, pi-3,
>pi-3.14, pi-3.141, ... defines a unique Real;
sequence is a Cauchy sequence, you have the obvious mapping that you
say you need.
Peter Webb
Oct 12, 2012, 10:39:33 PM10/12/12
to
"Michael Stemper" wrote in message news:k5934l$2o9$1...@dont-email.me...
In article <k585os$h34$1...@news.albasani.net>, "Peter Webb"
<webbfamily@DIE_SPAMoptusnet.com.au> writes:
>"Michael Stemper" wrote in message news:k56vk7$lja$2...@dont-email.me...
>In article <k55n3g$pdu$1...@news.albasani.net>, "Peter Webb"
><webbfamily@DIE_SPAMoptusnet.com.au> writes:
>>>The problem (as I see it) with Cauchy sequences is that they are
>>>sequences,
>>>and hence require a mapping from N to Rationals.
>>
>>Why would that be a problem?
>>
>>
>>Because in order to define a Real number you have define a mapping from N
>>to
>>Rationals.
Repeating the statement doesn't explain it.
Cauchy sequences are mappings from N to Rationals. So to define a Real
number using Cauchy sequences you need a mapping from N to Q. How can I make
this any simpler?
>> Lots of situations arise where there is no obvious mapping. The
>>LUB property makes it easy to say that the sequence pi, pi-3, pi-3.14,
>>pi-3.141, ... defines a unique Real; try doing that with Cauchy (you can,
>>of
>>course, but it involves work).
I don't see how writing "pi-a_n" is any easier than writing "a_n". The
sequence you just wrote implies another one: 0, 3, 3.14, 3.141, ...
This is the beginning of a perfectly cromulent Cauchy sequence for pi.
You miss the point. How are you going to prove the sequence pi, pi-3,
pi-3.1, pi-3.14 ... converges to a Real number using Cauchy sequences? Its
easy using LUB.
>>> The LUB property is more
>>>general, in that it applies to unordered sets.
>>
>>It does? I always thought that order was necessary before you could
>>define any upper (or lower) bounds.
>>
>>Wrong. You need an order relation to define "least",
As far as I can tell, you also need that to define "upper bound" or
"lower bound", as well. If that's not the case, maybe you could give
an example, perhaps on C.
>> The set {x|x>1 & x<2} clearly has a LUB even though there is
>>no mapping from N to the set.
Apparently, when you say "to the set" you really mean "onto the set".
Yes. Dropped two letters. Sorry.
But, what of it? That set isn't a Cauchy sequence, and doesn't represent
a real.
Incorrect. The sequence pi, pi-3, pi-3.1, pi-3.14 does converge to
(represent) a Real, but this cannot be easily shown using Cauchy (as its not
a Cauchy sequence). It is trivially easy to prove using LUB. It is therefore
an example of where the LUB property is more useful than Cauchy, which is
what the OP wanted.
Peter Webb
Oct 12, 2012, 10:46:53 PM10/12/12
to
"Shmuel (Seymour J.)Metz" wrote in message
news:5078204a$24$fuzhry+tra$mr2...@news.patriot.net...
In <k585os$h34$1...@news.albasani.net>, on 10/12/2012
at 03:19 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because in order to define a Real number you have define a mapping
>from N to Rationals.
You're restating the phrase instead of explaining why it is a problem.
>Lots of situations arise where there is no obvious mapping.
1. You haven't explained why you need an obvious mapping
2. You haven't given an example where it it is a problem.
>The LUB property makes it easy to say that the sequence pi, pi-3,
>pi-3.14, pi-3.141, ... defines a unique Real;
Since each of those terms is irrational, it is not easy.
Yes it is. It is a set of Reals with an upper bound of 4. It must therefore
have a LUB. How hard was that?
Since the
sequence is a Cauchy sequence, you have the obvious mapping that you
say you need.
To use the Cauchy sequence to construct R, the terms must all be rational or
else the construction is circular. "The real numbers are complete under the
metric induced by the usual absolute value, and one of the standard
constructions of the real numbers involves Cauchy sequences of rational
numbers.", from http://en.wikipedia.org/wiki/Cauchy_sequence . You will note
the terms in pi, pi-3, pi-3.14 ... are not rational. So we do not have a
mapping from N to Q as you claim.
Peter Webb
Oct 13, 2012, 12:28:10 AM10/13/12
to
"Peter Webb" wrote in message news:k5akne$i3l$1...@news.albasani.net...
"Shmuel (Seymour J.)Metz" wrote in message
news:5078204a$24$fuzhry+tra$mr2...@news.patriot.net...
In <k585os$h34$1...@news.albasani.net>, on 10/12/2012
at 03:19 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because in order to define a Real number you have define a mapping
>from N to Rationals.
You're restating the phrase instead of explaining why it is a problem.
>Lots of situations arise where there is no obvious mapping.
1. You haven't explained why you need an obvious mapping
Because the Cauchy derivation of a Real involves an explicit mapping.
2. You haven't given an example where it it is a problem.
Prove that the sequence pi, pi + 1/2, pi+1/2+1/4, ... converges to a Real.
In the Cauchy derivation of Reals, the terms are Rationals or the argument
is circular.
>The LUB property makes it easy to say that the sequence pi, pi-3,
>pi-3.14, pi-3.141, ... defines a unique Real;
Since each of those terms is irrational, it is not easy.
__________________________________________
Yes it is. It is a set of Reals with an upper bound of 4. It must therefore
have a LUB. How hard was that?
Since the
sequence is a Cauchy sequence, you have the obvious mapping that you
say you need.
Well no. In the construction of R using Cauchy sequences, each term is from
Q, or else the argument is circular. The sequence I gave is not a mapping
from N to Q.
________________________________________________
To use the Cauchy sequence to construct R, the terms must all be rational or
else the construction is circular. "The real numbers are complete under the
metric induced by the usual absolute value, and one of the standard
constructions of the real numbers involves Cauchy sequences of rational
numbers.", from http://en.wikipedia.org/wiki/Cauchy_sequence . You will note
the terms in pi, pi-3, pi-3.14 ... are not rational. So we do not have a
mapping from N to Q as you claim.
Gee, you don't seem to object to that statement, even though it is the
argument I have given throughout. I'm guessing that you looked at the web
page I cited and realised I am correct. You should have read to the end
before starting to reply.
Graham Cooper
Oct 13, 2012, 4:07:35 AM10/13/12
to
On Oct 11, 6:17 am, Ken Pledger <ken.pled...@vuw.ac.nz> wrote:
>
> This is about syntax vs semantics.
>
> If you set up a list of axioms for the reals as a complete ordered
> field, then you might well include the least-upper-bound axiom. The
> formal theory based on all the axioms is essentially just
> symbol-manipulation.
>
> To give the symbols some sort of meaning you need a model for the
> theory. Dedekind cuts or Cauchy sequences can be used as the last stage
> of the process in setting up such a model.
>
> Ken Pledger.
Oh MODEL BOBBLE
>
> This is about syntax vs semantics.
>
> If you set up a list of axioms for the reals as a complete ordered
> field, then you might well include the least-upper-bound axiom. The
> formal theory based on all the axioms is essentially just
> symbol-manipulation.
>
> To give the symbols some sort of meaning you need a model for the
> theory. Dedekind cuts or Cauchy sequences can be used as the last stage
> of the process in setting up such a model.
>
> Ken Pledger.
There is no such thing as a model in logic.
It's a convoluted rationale on proof by resolution, a top down macro
explanation on why
FACT-A & not(FACT-A)
cannot both exist in the same theory!
*a theorem t is true if a model in the theory interprets t*
*if a theorem is true in all models it's consistent*
JUST REPLACE MODEL WITH 'DEDUCTION SEQUENCE'.
A FORMULA IS TRUE IF A DEDUCTION SEQUENCE FROM THE AXIOMS PROVES IT.
IF ALL DEDUCTION SEQUENCES DON'T DISPROVE THE FORMULA, THEN IT'S
CONSISTENT.
OTHERWISE, YOU HAVE TO NEGATE THAT FORMULA IF IT WAS AN ASSUMPTION.
The amount of DOUBLE SPEAK here to make a THEOREM PROVER WORK is
INCREDULOUS!
The minute you start to PROVE WHY YOU CANT MAKE IT WORK you should
start asking yourselves if you're still on the right track!??
Herc
Frederick Williams
Oct 13, 2012, 12:10:11 PM10/13/12
to
Graham Cooper wrote:
>
> The amount of DOUBLE SPEAK here to make a THEOREM PROVER WORK is
> INCREDULOUS!
Incredible.
>
> The amount of DOUBLE SPEAK here to make a THEOREM PROVER WORK is
> INCREDULOUS!
Frederick Williams
Oct 13, 2012, 12:18:47 PM10/13/12
to
Peter Webb wrote:
>
> "Peter Webb" wrote in message news:k5akne$i3l$1...@news.albasani.net...
>
> "Shmuel (Seymour J.)Metz" wrote in message
>
>
> "Peter Webb" wrote in message news:k5akne$i3l$1...@news.albasani.net...
>
> "Shmuel (Seymour J.)Metz" wrote in message
>
> ________________________________________________
> To use the Cauchy sequence to construct R, the terms must all be rational or
> else the construction is circular. "The real numbers are complete under the
> metric induced by the usual absolute value, and one of the standard
> constructions of the real numbers involves Cauchy sequences of rational
> numbers.", from http://en.wikipedia.org/wiki/Cauchy_sequence . You will note
> the terms in pi, pi-3, pi-3.14 ... are not rational. So we do not have a
> mapping from N to Q as you claim.
> ________________________________________________
> Gee, you don't seem to object to that statement, even though it is the
> argument I have given throughout. I'm guessing that you looked at the web
> page I cited and realised I am correct. You should have read to the end
> before starting to reply.
Can you get your newsreader to quote properly? It is very difficult to
> To use the Cauchy sequence to construct R, the terms must all be rational or
> else the construction is circular. "The real numbers are complete under the
> metric induced by the usual absolute value, and one of the standard
> constructions of the real numbers involves Cauchy sequences of rational
> numbers.", from http://en.wikipedia.org/wiki/Cauchy_sequence . You will note
> the terms in pi, pi-3, pi-3.14 ... are not rational. So we do not have a
> mapping from N to Q as you claim.
> ________________________________________________
> Gee, you don't seem to object to that statement, even though it is the
> argument I have given throughout. I'm guessing that you looked at the web
> page I cited and realised I am correct. You should have read to the end
> before starting to reply.
see what comes from you and what comes from the person you're replying
to.
Shmuel Metz
Oct 13, 2012, 8:59:41 PM10/13/12
to
In <k5ak9n$hdp$1...@news.albasani.net>, on 10/13/2012
at 01:39 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Incorrect. The sequence pi, pi-3, pi-3.1, pi-3.14 does converge to
>(represent) a Real, but this cannot be easily shown using Cauchy (as
>its not a Cauchy sequence).
If you mean that it's not a Cauchy sequence in Q, then say so. If not,
then it's incorrect.
In <k5akne$i3l$1...@news.albasani.net>, on 10/13/2012
at 01:46 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Yes it is. It is a set of Reals with an upper bound of 4.
The sequence that you have in mind is a Cauchy sequence, but that does
not mean that it is easy to define the sequence that you have in mind.
An ellipsis is not a definition.
In <k5aqlb$4mv$1...@news.albasani.net>, on 10/13/2012
at 03:28 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because the Cauchy derivation of a Real involves an explicit mapping.
No. There is no requirement to explicitly state a representative
sequence for any particular Real.
>Prove that the sequence pi, pi + 1/2, pi+1/2+1/4, ... converges to a
>Real.
That follows from the Axioms for the Reals, so all that is necessary
is to prove that equivalence classes of Cauchy sequences satisfy those
axioms.
>Yes it is. It is a set of Reals with an upper bound of 4.
No, it is a partial description of a sequence. Provinding a complete
description would be harder than you believe.
>How hard was that?
Harder than proving the axioms of the Reals for either of the standard
models of the Reals.
>Gee, you don't seem to object to that statement, even though it is
>the argument I have given throughout. I'm guessing that you looked
>at the web page I cited and realised I am correct.
What are you smoking? You are wrong on all counts.
at 01:39 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Incorrect. The sequence pi, pi-3, pi-3.1, pi-3.14 does converge to
>(represent) a Real, but this cannot be easily shown using Cauchy (as
>its not a Cauchy sequence).
then it's incorrect.
In <k5akne$i3l$1...@news.albasani.net>, on 10/13/2012
at 01:46 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Yes it is. It is a set of Reals with an upper bound of 4.
not mean that it is easy to define the sequence that you have in mind.
An ellipsis is not a definition.
In <k5aqlb$4mv$1...@news.albasani.net>, on 10/13/2012
at 03:28 PM, "Peter Webb" <webbfamily@DIE_SPAMoptusnet.com.au>
said:
>Because the Cauchy derivation of a Real involves an explicit mapping.
sequence for any particular Real.
>Prove that the sequence pi, pi + 1/2, pi+1/2+1/4, ... converges to a
>Real.
is to prove that equivalence classes of Cauchy sequences satisfy those
axioms.
>Yes it is. It is a set of Reals with an upper bound of 4.
description would be harder than you believe.
>How hard was that?
Harder than proving the axioms of the Reals for either of the standard
models of the Reals.
>Gee, you don't seem to object to that statement, even though it is
>the argument I have given throughout. I'm guessing that you looked
>at the web page I cited and realised I am correct.
Graham Cooper
Oct 13, 2012, 11:18:56 PM10/13/12
to
On Oct 14, 2:10 am, Frederick Williams <freddywilli...@btinternet.com>
wrote:
Herc
wrote:
> Graham Cooper wrote:
>
> > The amount of DOUBLE SPEAK here to make a THEOREM PROVER WORK is
> > INCREDULOUS!
>
> Incredible.
>
Verb, sentence needs.
>
> > The amount of DOUBLE SPEAK here to make a THEOREM PROVER WORK is
> > INCREDULOUS!
>
> Incredible.
>
Herc
christian.bau
Oct 14, 2012, 12:53:33 PM10/14/12
to
On Oct 10, 6:30 pm, Paul <pepste...@gmail.com> wrote:
> The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
So if you take the set with the elements 3, 3.1, 3.14, 3.141 and so
> The definition of the reals should formalise the way we approximate pi by a decimal expansion: 3, 3.1, 3.14 etc.
on, what would be the least upper bound?
What seems ridiculous to you seems actually very natural to me.
Graham Cooper
Oct 14, 2012, 7:35:50 PM10/14/12
to
On Oct 15, 2:53 am, "christian.bau" <christian....@cbau.wanadoo.co.uk>
wrote:
There is a function that already pin points that value.
You can't USE a least upper bound of an infinite set.
If you have a list of reals:
LIST
f(1) = 0. 314 159 2653 ...
f(2) = 0. 222 222 222 ...
f(3) = 0. 28 18 28 44 ...
f(4) = 0. 55555 55555 ...
..
then you can use them in arguments as STREAMS
using LAZY EVALUATION.
FUNCTION TIMES-2 ( a , b )
TIMES-2 ( 3.141.. , 0.555.. )
= 1.74...
*************
A FEW SECONDS LATER
TIMES-2( 3.141 265.. , 0.555 555.. )
= 1.74 5914..
*************
YOU CAN CALCULATE THE FUNCTION RESULT
AS THE INPUT COMES IN.
SIMULATING AN INFINITE VALUE
*************
IF you had the brains to use PROCESSES
then you would REALISE that
AD(LIST) = 555 4 555 ...
NEVER CALCULATES A UNIQUE DIGIT SEGMENT MISSING FROM THE INFINITE LIST
Herc
wrote:
You can't USE a least upper bound of an infinite set.
If you have a list of reals:
LIST
f(1) = 0. 314 159 2653 ...
f(2) = 0. 222 222 222 ...
f(3) = 0. 28 18 28 44 ...
f(4) = 0. 55555 55555 ...
..
then you can use them in arguments as STREAMS
using LAZY EVALUATION.
FUNCTION TIMES-2 ( a , b )
TIMES-2 ( 3.141.. , 0.555.. )
= 1.74...
*************
A FEW SECONDS LATER
TIMES-2( 3.141 265.. , 0.555 555.. )
= 1.74 5914..
*************
YOU CAN CALCULATE THE FUNCTION RESULT
AS THE INPUT COMES IN.
SIMULATING AN INFINITE VALUE
*************
IF you had the brains to use PROCESSES
then you would REALISE that
AD(LIST) = 555 4 555 ...
NEVER CALCULATES A UNIQUE DIGIT SEGMENT MISSING FROM THE INFINITE LIST
Herc
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