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The real reals
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Virgil
Nov 27, 2013, 1:43:24 AM11/27/13
to
Note that for standard sets of reals as defined below,
Cantor's first proof of the uncountability of the set of reals
holds true.
From
http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Synthetic_a
pproach
The synthetic approach axiomatically defines the real number system as
a complete ordered field. Precisely, this means the following. A model
for the real number system consists of a set R, two distinct elements 0
and 1 of R, two binary operations + and * on R (called addition and
multiplication, respectively), a binary relation <= on R, satisfying the
following properties.
1. (R, +, *) forms a field. In other words,
? For all x, y, and z in R, x + (y + z) = (x + y) + z and x * (y * z) = (x * y) * z.
(associativity of addition and multiplication)
? For all x and y in R, x + y = y + x and x * y = y * x.
(commutativity of addition and multiplication)
? For all x, y, and z in R, x * (y + z) = (x * y) + (x * z).
(distributivity of multiplication over addition)
? For all x in R, x + 0 = x. (existence of additive identity)
? 0 is not equal to 1, and for all x in R, x * 1 = x.
(existence of multiplicative identity)
? For every x in R, there exists an element -x in R, such that x + (-x) = 0.
(existence of additive inverses)
? For every x != 0 in R, there exists an element 1/x in R,
such that x * 1/x = 1. (existence of multiplicative inverses)
2. (R, <=) forms a totally ordered set. In other words,
? For all x in R, x <= x. (reflexivity)
? For all x and y in R, if x <= y and y <= x, then x = y. (antisymmetry)
? For all x, y, and z in R, if x <= y and y <= z, then x <= z. (transitivity)
? For all x and y in R, x <= y or y <= x. (totalness)
3. The field operations + and * on R are compatible with the order <=. In other words,
? For all x, y and z in R, if x <= y, then x + z <= y + z.
(preservation of order under addition)
? For all x and y in R, if 0 <= x and 0 <= y, then 0 <= x ? y
(preservation of order under multiplication)
4. The order <= is complete in the following sense:
every non-empty subset of R bounded above has a least upper bound.
In other words,
? If A is a non-empty subset of R, and if A has an upper bound, then A
has a least upper bound u, such that for every upper bound v of A, u <= v.
That final axiom, defining the order as Dedekind-complete, is most
crucial. Without this axiom, we simply have the axioms which define a
totally ordered field, and there are many non-isomorphic models which
satisfy these axioms. This axiom implies that the Archimedean property
applies for this field. Therefore, when the completeness axiom is
added, it can be proved that any two models must be isomorphic, and so
in this sense, there is only one complete ordered Archimedean field.
When we say that any two models of the above axioms are isomorphic, we
mean that for any two models (R, 0R, 1R, +R, *R, <=R) and (S, 0S, 1S,
+S, *S, <=S), there is a bijection f : R -> S preserving both the field
operations and the order. Explicitly,
? f is both injective and surjective.
? f(0R) = 0S and f(1R) = 1S.
? For all x and y in R, f(x +R y) = f(x) +S f(y) and f(x *R y) = f(x) *S f(y)
? For all x and y in R, x <=R y if and only if f(x) <=S f(y).
? For all x and y in R, x <=R y if and only if f(x) <=S f(y).
Wm cannot prove that a real field satisfying the above properties is
countable, because it is provably not countable.
--
fom
Nov 27, 2013, 2:30:54 AM11/27/13
to
http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
It is a nice discussion and adds to what WM
cannot prove.
Ross A. Finlayson
Nov 27, 2013, 3:18:57 AM11/27/13
to
On Tuesday, November 26, 2013 11:30:54 PM UTC-8, fom wrote:
> On 11/27/2013 12:43 AM, Virgil wrote:
>
> > Note that for standard sets of reals as defined below,
>
> > ...
> On 11/27/2013 12:43 AM, Virgil wrote:
>
> > Note that for standard sets of reals as defined below,
>
>
>
> Elsewhere, FredJefferies provided this link:
>
>
>
> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
>
>
>
> It is a nice discussion and adds to what WM
>
> cannot prove.
Heh,
>
> Elsewhere, FredJefferies provided this link:
>
>
>
> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
>
>
>
> It is a nice discussion and adds to what WM
>
> cannot prove.
"It may happen that the reals are in 1-1 correspondence with a subset
of the natural numbers, while at the same time they form an uncountable set."
-Andrej Bauer, visiting the Institute for Advanced Study
Heh: the Institute for Advanced Study.
"It may happen that the reals contain nilpotent infinitesimals, which validate
the 17th century calculations that physicists still use because, luckily, they
did not subscribe entirely to the ϵδ-dogma of analysis."
That may not be exactly so, where, physicists do subscribe to delta-epsilonics, and, they do subscribe to Leibniz' notation.
https://groups.google.com/forum/#!msg/sci.math/4RBNLj-Q4Mo/hsgK3usvIAcJ
Message has been deleted
George Greene
Nov 28, 2013, 7:01:57 PM11/28/13
to
So allow me to actually provide relevant context.
Ross should have done this himself:
http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/
Bauer's point is ridiculous.
He decides to flout a clear known definition and then to claim
that people defending the definition have succumbed to "dogma".
He acts like he does NOT even KNOW what a definition IS.
This is NOT forgivable.
His example involves "the reals":
[quoting hereafter]
Let us consider an example. The real numbers are a mathematical object of fundamental importance, and have many aspects:
The reals as a set are uncountable and in bijection with the powerset of natural numbers.
The reals as an algebraic structure form a linearly ordered field.
The reals as a space are locally compact, Hausdorff, and connected.
The reals are a measurable space on which measure theory rests.
The reals of non-standard analysis contain infinitesimals.
The reals as understood by Leibniz contain nilpotent infinitesimals.
The reals as Brouwerian continuum cannot be decomposed into two disjoint inhabited subsets.
The reals are overt.
We can have some of these properties but not all at once. History has chosen for us a combination that is taught today as a dogma. Any attempt to deviate from it is met with opposition. Thus you probably consider 1, 2, 3, and 4 as true, 5 as something exotic you heard of, 6 as Leibniz’s biggest mistake, 7 as intuitionistic hallucination (because obviously the reals can be decomposed into the non-negative and negative numbers), and 8 as something you never heard of (but you should have because it is the concept dual to compactness and you have been using it all your life).
Once we break free from Cantor’s paradise that Hilbert threw us in we discover unsuspected possibilities:
[close quote]
This is just a bunch of horse-shit.
It really is important for terms to have DEFINITIONS.
It is important to know WHAT WE MEAN by "the reals".
It is NOT necessary to "break free from Cantor's paradise" in order
to know that.
It IS necessary to know what THE DEFINITION of "the reals" is.
Once you have picked a definition, some of the above are true and some
are false, and that is just all there is to it. Nobody is being "dogmatic"
by insisting that terms have definitions.
Conspicuous by its absence from the list above is
"The reals are the unique (up to isomorphism)
complete ordered Archimedean field."
That is NOT a forgivable omission.
Ross should have done this himself:
http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/
Bauer's point is ridiculous.
He decides to flout a clear known definition and then to claim
that people defending the definition have succumbed to "dogma".
He acts like he does NOT even KNOW what a definition IS.
This is NOT forgivable.
His example involves "the reals":
[quoting hereafter]
Let us consider an example. The real numbers are a mathematical object of fundamental importance, and have many aspects:
The reals as a set are uncountable and in bijection with the powerset of natural numbers.
The reals as an algebraic structure form a linearly ordered field.
The reals as a space are locally compact, Hausdorff, and connected.
The reals are a measurable space on which measure theory rests.
The reals of non-standard analysis contain infinitesimals.
The reals as understood by Leibniz contain nilpotent infinitesimals.
The reals as Brouwerian continuum cannot be decomposed into two disjoint inhabited subsets.
The reals are overt.
We can have some of these properties but not all at once. History has chosen for us a combination that is taught today as a dogma. Any attempt to deviate from it is met with opposition. Thus you probably consider 1, 2, 3, and 4 as true, 5 as something exotic you heard of, 6 as Leibniz’s biggest mistake, 7 as intuitionistic hallucination (because obviously the reals can be decomposed into the non-negative and negative numbers), and 8 as something you never heard of (but you should have because it is the concept dual to compactness and you have been using it all your life).
Once we break free from Cantor’s paradise that Hilbert threw us in we discover unsuspected possibilities:
[close quote]
This is just a bunch of horse-shit.
It really is important for terms to have DEFINITIONS.
It is important to know WHAT WE MEAN by "the reals".
It is NOT necessary to "break free from Cantor's paradise" in order
to know that.
It IS necessary to know what THE DEFINITION of "the reals" is.
Once you have picked a definition, some of the above are true and some
are false, and that is just all there is to it. Nobody is being "dogmatic"
by insisting that terms have definitions.
Conspicuous by its absence from the list above is
"The reals are the unique (up to isomorphism)
complete ordered Archimedean field."
That is NOT a forgivable omission.
George Greene
Nov 28, 2013, 7:06:56 PM11/28/13
to
"At the moment I am visiting the Institute for Advanced Study as a member of the Univalent Foundations group. We are building a new foundation of mathematics whose language is type theory rather than set theory, and whose primary objects are homotopy types and not just bare sets."
And this IS RIDICULOUS ANYhow since real numbers, e.g. zero and one,
whatever ELSE they may be, ARE NOT "just bare sets" and ARE CERTAINLY NOT
"homotopy types".
The language RIGHT NOW isn't set theory EITHER.
Set theory IS NOT "the language for the foundation".
Set theory is the MODEL CONSTRUCTION language.
The ACTUAL current paradigm IS AXIOMATIC.
Set theory is just one way of constructing models for the axioms
and obviously others are welcome.
But whatEVER "reals" may come out of trying-to-define-reals-in-terms-of-type-
theory, they are NOT going TO BE reals unless they satisfy THE AXIOMS DEFINING
reals.
Nobody is "dogmatic" for saying that. We will just mean (and so will you)
that you are talking about DIFFERENT things FROM "the reals" if the things
you are talking about don't satisfy the axioms.
George Greene
Nov 28, 2013, 7:25:53 PM11/28/13
to
On Wednesday, November 27, 2013 1:43:24 AM UTC-5, Virgil wrote:
> Note that for standard sets of reals as defined below,
> http://en.wikipedia.org/wiki/Construction_of_the_real_numbers#Synthetic_approach
> Note that for standard sets of reals as defined below,
> The synthetic approach axiomatically defines the real number system as
>
> a complete ordered field.
treated below, it mixes in a lot of irrelevant set theory.
> Precisely, this means the following.
> A model for the real number system consists of a set R,
Set theories ARE NOT the ONLY model-construction languages!
George Greene
Nov 28, 2013, 7:31:38 PM11/28/13
to
On Wednesday, November 27, 2013 1:43:24 AM UTC-5, Virgil wrote:
> That final axiom, defining the order as Dedekind-complete, is most
> crucial. Without this axiom, we simply have the axioms which define a
> totally ordered field, and there are many non-isomorphic models which
> satisfy these axioms. This axiom implies that the Archimedean property
> applies for this field. Therefore, when the completeness axiom is
> added,
When the completeness axiom is added, you are CHEATING.
> crucial. Without this axiom, we simply have the axioms which define a
> totally ordered field, and there are many non-isomorphic models which
> satisfy these axioms. This axiom implies that the Archimedean property
> applies for this field. Therefore, when the completeness axiom is
> added,
If you have to do that VIA an axiom, you are CHEATING.
You don't stipulate a completeness axiom and then get the
Archimedean property as a theorem.
You DEFINE the field as Archimedean.
You probably also define it as "complete" in the relevant sense,
but you don't necessarily have to do that with set theory.
If you are going to was foundational and quote Andrej Bauer then it is
IMPORTANT NOT to presume set-theoretic treatments.
George Greene
Nov 28, 2013, 8:14:37 PM11/28/13
to
On Wednesday, November 27, 2013 1:43:24 AM UTC-5, Virgil wrote:
> Note that for standard sets of reals as defined below,
> Cantor's first proof of the uncountability of the set of reals
> holds true.
That is not even the point.
> Cantor's first proof of the uncountability of the set of reals
> holds true.
The relevant question is whether the ZFC proof that the
powerset P(S) of a set S is not bijectible with S somehow becomes
suspect when S is infinite.
Defining the reals as the elements/numbers in the [UNIQUE!] ordered Archimedean field-that-is-"complete"-in-the-relevant-sense is not going to interact AT ALL with the issues about transfinite cardinalities FROM SET theory.
The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention infininty (OR FINITUDE) AT ALL, so the cranks here who have problems with that
are just cranks. This is not going to interact with ANYthing about the reals AT ALL UNTIL AFTER you draw the analogy between a real number and an infinite subset of the naturals. If you are just considering real numbers directly then the issues are entirely separate. The Cantorian-set-theory proof IS MUCH MORE IMPORTANT THAN ANYthing anyone would want to say about the reals directly because the Cantorian set theory proof correctly understands that INFINITY*IS*NOT* relevant to the problem!
George Greene
Nov 28, 2013, 8:16:14 PM11/28/13
to
On Wednesday, November 27, 2013 2:30:54 AM UTC-5, fom wrote:
> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
> It is a nice discussion and adds to what WM
> cannot prove.
It's a bunch of insulting aspersions upon the character of everybody-at-large,
> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
> It is a nice discussion and adds to what WM
> cannot prove.
which is the EXACT OPPOSITE of nice.
Wanting to construct an alternative foundation using type theory is entirely
laudable. Alleging that everybody else is suffering from "dogma" is entirely horseshit.
Virgil
Nov 28, 2013, 9:42:14 PM11/28/13
to
In article <8b8c5190-5ca7-4a29...@googlegroups.com>,
argue ir out with WIKI, not me.
> If you have to do that VIA an axiom, you are CHEATING.
I do not. Wiki did, though on careful reading of the wiki article, it
was not needed.
--
George Greene <gre...@email.unc.edu> wrote:
> On Wednesday, November 27, 2013 1:43:24 AM UTC-5, Virgil wrote:
> > That final axiom, defining the order as Dedekind-complete, is most
> > crucial. Without this axiom, we simply have the axioms which define a
> > totally ordered field, and there are many non-isomorphic models which
> > satisfy these axioms. This axiom implies that the Archimedean property
> > applies for this field. Therefore, when the completeness axiom is
> > added,
>
> When the completeness axiom is added, you are CHEATING.
Since what I posted was was a quotation from WIKI, and labled so,
> On Wednesday, November 27, 2013 1:43:24 AM UTC-5, Virgil wrote:
> > That final axiom, defining the order as Dedekind-complete, is most
> > crucial. Without this axiom, we simply have the axioms which define a
> > totally ordered field, and there are many non-isomorphic models which
> > satisfy these axioms. This axiom implies that the Archimedean property
> > applies for this field. Therefore, when the completeness axiom is
> > added,
>
> When the completeness axiom is added, you are CHEATING.
argue ir out with WIKI, not me.
> If you have to do that VIA an axiom, you are CHEATING.
was not needed.
--
WM
Nov 29, 2013, 3:27:08 AM11/29/13
to
Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
> The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention infininty (OR FINITUDE) AT ALL,
That means the proof is cheating since ZF needs completed it everywhere.
> The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention infininty (OR FINITUDE) AT ALL,
Without completed infinity we can prove that the cardinalities of the initial segments of the even natural numbers
|{2}| = 1
|{2, 4}| = 2
|{2, 4, 6}| = 3
...
are never larger than all elements of the initial segments. On the contrary, the number which are large than the cardinality grows with the sets. So there is no chance to prove forall n in |N: aleph_0 > n.
Regards, WM
Virgil
Nov 29, 2013, 5:15:33 PM11/29/13
to
In article <66e51715-2199-475b...@googlegroups.com>,
actually infinite, no one expects finite sets to exceed actually
infinite ones in size.
>On the contrary,
> the number which are large than the cardinality grows with the sets. So there
> is no chance to prove forall n in |N: aleph_0 > n.
What is unprovable inside of WM's wild weird world of WMytheology is
often trivially obvious outside of WM's wild weird world of
WMytheology.
Since for every n in |N FISON_n is a finite proper subset of finite
set FISON_(n+1) which is, in turn, a proper subset of |N,
Card(FISON_n) < Card(FISON_(n+1)) <= aleph_0.
At least unless WM has some highly non-standard definition of
CARDINALITY.
The STANDARD definitions say that if there is an injective mapping from
set A to set B then Card(A) <= Card(B).
And further
if if there is an injective mapping from set A to set B but NO injective
mapping from set B to set A then Card(A) < Card(B).
Does WM object in any way to either of those definitions?
Since they are so clear and straightforward, WM will probably reject
them, as they do not allow him his needed wiggle room.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
>
> > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
> > infininty (OR FINITUDE) AT ALL,
>
> That means the proof is cheating since ZF needs completed it everywhere.
What is "completed it" supposed to mean?
> Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
>
> > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
> > infininty (OR FINITUDE) AT ALL,
>
> That means the proof is cheating since ZF needs completed it everywhere.
>
> Without completed infinity we can prove that the cardinalities of the initial
> segments of the even natural numbers
>
> |{2}| = 1
> |{2, 4}| = 2
> |{2, 4, 6}| = 3
> ...
>
> are never larger than all elements of the initial segments.
Being larger than all elements of the initial segments means being
> Without completed infinity we can prove that the cardinalities of the initial
> segments of the even natural numbers
>
> |{2}| = 1
> |{2, 4}| = 2
> |{2, 4, 6}| = 3
> ...
>
> are never larger than all elements of the initial segments.
actually infinite, no one expects finite sets to exceed actually
infinite ones in size.
>On the contrary,
> the number which are large than the cardinality grows with the sets. So there
> is no chance to prove forall n in |N: aleph_0 > n.
often trivially obvious outside of WM's wild weird world of
WMytheology.
Since for every n in |N FISON_n is a finite proper subset of finite
set FISON_(n+1) which is, in turn, a proper subset of |N,
Card(FISON_n) < Card(FISON_(n+1)) <= aleph_0.
At least unless WM has some highly non-standard definition of
CARDINALITY.
The STANDARD definitions say that if there is an injective mapping from
set A to set B then Card(A) <= Card(B).
And further
if if there is an injective mapping from set A to set B but NO injective
mapping from set B to set A then Card(A) < Card(B).
Does WM object in any way to either of those definitions?
Since they are so clear and straightforward, WM will probably reject
them, as they do not allow him his needed wiggle room.
--
Ross A. Finlayson
Nov 30, 2013, 8:26:29 PM11/30/13
to
see issues between models of the reals, as applicable, and their
interpretations, as mutually consistent (with regular/well-founded/ordinary
foundations). It's not out of context that Bauer lists this with the mutually
contradictory, it's in the context that it is.
For me then I just took the integral as the line as it is and then showed that
was consistent with Cantorian arithmetic, which otherwise is not, as simply
uniqueness.
So trust me that was easy.
Regards, Ross Finlayson
andrej...@andrej.com
Dec 1, 2013, 4:52:53 AM12/1/13
to
I was alerted to this discussion, so I will make one post. But I will not participate because the level of civility is far too low for me to lose time on this. I do not appreciate Gorge Green's the ad hominem attacks on me.
I would just like make a couple of clarifications.
The real numbers of course have a well known and fixed definition, such sa the one given by Virgil above. It is technically slightly better to reformulate the usual axioms in a form which is classically equivalent, but also works intuitionistically. This gives more generality at no expense to classical mathematics. So my first point is that of course I am not changing definitions.
But even if I did, there would be nothing wrong with that. Often we start with a "pre-mathematical" understadning of some phenomenon and only later arrive at what counts as a mathematical definition. In fact, Dedekind and Cauchy did exactly this when they treated real numbers: they made imprecise ideas precise. However, there are many ways to make an imprecise idea mathematically precise. For instance, Dedekind and Cauchy found two ways of treating the real numbers. Those turn out to be equivalent, but that is hindsight. When you were taught mathematics you were given definitions as cast in stone, unchanging. But that is far from truth in research mathematics, where old definitions are revisited, criticized and improved upon.
But anyhow, this is not the case here, we can all agree on one precise definition of real numbers. The essential components are: arithmetic, order, and completeness. Completeness is tricikiest, both because there are several ways to express it and because it requires more complex logical language, for instance second-order (or higher order) logic, or some form of set theory.
A definition of reals, however, does not fix all of its properties. This is so because the reals do not exist in a vacuum, but within a larger context of a mathematical universe, be it a topos, a model of ZFC, or some form of type theory. For instance, within the context of set theory, it is possible to construct models of ZFC such that every subset of the reals is measurable.
If we widen the context and allow also non-classical mathematical universes, for instance toposes, then we see an even wider spectrum of possibilities. One and the same definition of reals gives us in different toposes objects of reals which all share the same basic properties (arithmetic, order, completeness), but differ in other aspects. In some toposes, the reals are indecomposable (cannot be written as a union of two disjoint inhabited sets), in others they are not locally compact, etc.
So what are we to think of all these possibilities? One option is to take an absolute position and declare one possibility to be the correct one. This is the current situation, where the official position is that Zermelo-Fraenkel set theory with choice (ZFC) is the foundation of mathematics. I call this a dogma because I have seen many times the opposition to the idea that ZFC may be replaced by something else. The training of (almost) all mahtematicians is thoroughly within the context of ZFC. So much so that it becomes very hard for them to understand the other possiblities. It takes about two years of training for a graduate student to be able to start thinking outside of ZFC (I speak from personal experience as an advisor to such a PhD student). This training does not build upon existing knowledge, but is about de-programming and un-training the intuitions that the student has gained. His knowledge precludes him from obtaining new knowledge. This deserves to be called a dogma.
I do not deny the importance and usefulness of the unifying power of ZFC and the foundations of mathematics, as developed in the first half of the 20th century. However, one should always be aware of one's limitations and commitments. When we accept ZFC, we also accept certain ideas about real numbers. These ideas may not correspond exactly to someone's pre-mathematical intuition of what the real number are about, so he is entitled to re-examine the situation. Another valid reason for re-examining the existing monopoly of ZFC is to look for setups that better fit the indented applications. As computers have become very important, it makes sense to look for foundations that account for computability better and at a more fundamental level than in ZFC.
I know very well what the axiomatic method is, what a definition is, etc. When Georege Green states that I am throwing everything away, to me that just shows he does not understand at all what I am saying.
In any case, I encourage you to be a bit more polite and to use FEWER ALL CAPS.
Sincerely,
Andrej Bauer
I would just like make a couple of clarifications.
The real numbers of course have a well known and fixed definition, such sa the one given by Virgil above. It is technically slightly better to reformulate the usual axioms in a form which is classically equivalent, but also works intuitionistically. This gives more generality at no expense to classical mathematics. So my first point is that of course I am not changing definitions.
But even if I did, there would be nothing wrong with that. Often we start with a "pre-mathematical" understadning of some phenomenon and only later arrive at what counts as a mathematical definition. In fact, Dedekind and Cauchy did exactly this when they treated real numbers: they made imprecise ideas precise. However, there are many ways to make an imprecise idea mathematically precise. For instance, Dedekind and Cauchy found two ways of treating the real numbers. Those turn out to be equivalent, but that is hindsight. When you were taught mathematics you were given definitions as cast in stone, unchanging. But that is far from truth in research mathematics, where old definitions are revisited, criticized and improved upon.
But anyhow, this is not the case here, we can all agree on one precise definition of real numbers. The essential components are: arithmetic, order, and completeness. Completeness is tricikiest, both because there are several ways to express it and because it requires more complex logical language, for instance second-order (or higher order) logic, or some form of set theory.
A definition of reals, however, does not fix all of its properties. This is so because the reals do not exist in a vacuum, but within a larger context of a mathematical universe, be it a topos, a model of ZFC, or some form of type theory. For instance, within the context of set theory, it is possible to construct models of ZFC such that every subset of the reals is measurable.
If we widen the context and allow also non-classical mathematical universes, for instance toposes, then we see an even wider spectrum of possibilities. One and the same definition of reals gives us in different toposes objects of reals which all share the same basic properties (arithmetic, order, completeness), but differ in other aspects. In some toposes, the reals are indecomposable (cannot be written as a union of two disjoint inhabited sets), in others they are not locally compact, etc.
So what are we to think of all these possibilities? One option is to take an absolute position and declare one possibility to be the correct one. This is the current situation, where the official position is that Zermelo-Fraenkel set theory with choice (ZFC) is the foundation of mathematics. I call this a dogma because I have seen many times the opposition to the idea that ZFC may be replaced by something else. The training of (almost) all mahtematicians is thoroughly within the context of ZFC. So much so that it becomes very hard for them to understand the other possiblities. It takes about two years of training for a graduate student to be able to start thinking outside of ZFC (I speak from personal experience as an advisor to such a PhD student). This training does not build upon existing knowledge, but is about de-programming and un-training the intuitions that the student has gained. His knowledge precludes him from obtaining new knowledge. This deserves to be called a dogma.
I do not deny the importance and usefulness of the unifying power of ZFC and the foundations of mathematics, as developed in the first half of the 20th century. However, one should always be aware of one's limitations and commitments. When we accept ZFC, we also accept certain ideas about real numbers. These ideas may not correspond exactly to someone's pre-mathematical intuition of what the real number are about, so he is entitled to re-examine the situation. Another valid reason for re-examining the existing monopoly of ZFC is to look for setups that better fit the indented applications. As computers have become very important, it makes sense to look for foundations that account for computability better and at a more fundamental level than in ZFC.
I know very well what the axiomatic method is, what a definition is, etc. When Georege Green states that I am throwing everything away, to me that just shows he does not understand at all what I am saying.
In any case, I encourage you to be a bit more polite and to use FEWER ALL CAPS.
Sincerely,
Andrej Bauer
Albrecht
Dec 1, 2013, 7:11:25 AM12/1/13
to
On Sunday, December 1, 2013 10:52:53 AM UTC+1, andrej...@andrej.com wrote:
>[...]
You claim the ontological equivalence of several definitions of the reals. I think by myself that it isn't so easy. A strange thing of math is the applicability of math to reality. And I think, it is a desirable aspect of math. Not only for engineering and things like that. The application to reality is also the only 100% sure proof of correctness of math itself.
Nobody knows if math stays applicable if it would be further developed only within ZFC. Even if we accept the reals as a powerset of an infinite set yet, any powerset of a powerset of an infinite set is beyond any imagination and use. Is there any proof in ZFC which is not provable otherwise and which is useful in any way? I can't see one.
Regards
Albrecht
>[...]
>
> A definition of reals, however, does not fix all of its properties. This is so because the reals do not exist in a vacuum, but within a larger context of a mathematical universe, be it a topos, a model of ZFC, or some form of type theory. For instance, within the context of set theory, it is possible to construct models of ZFC such that every subset of the reals is measurable.
>
>[...]
> A definition of reals, however, does not fix all of its properties. This is so because the reals do not exist in a vacuum, but within a larger context of a mathematical universe, be it a topos, a model of ZFC, or some form of type theory. For instance, within the context of set theory, it is possible to construct models of ZFC such that every subset of the reals is measurable.
>
You claim the ontological equivalence of several definitions of the reals. I think by myself that it isn't so easy. A strange thing of math is the applicability of math to reality. And I think, it is a desirable aspect of math. Not only for engineering and things like that. The application to reality is also the only 100% sure proof of correctness of math itself.
Nobody knows if math stays applicable if it would be further developed only within ZFC. Even if we accept the reals as a powerset of an infinite set yet, any powerset of a powerset of an infinite set is beyond any imagination and use. Is there any proof in ZFC which is not provable otherwise and which is useful in any way? I can't see one.
Regards
Albrecht
George Greene
Dec 1, 2013, 7:46:56 AM12/1/13
to
On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
> I do not appreciate Gorge Green's the ad hominem attacks on me.
A strange complaint from one who accused the profession at large of dogma.
> I do not appreciate Gorge Green's the ad hominem attacks on me.
Don't dish it out if..
> I would just like make a couple of clarifications.
the clarifications. Nobody wants to give insult accidentally.
If the next presentation of the thesis clarifies these points to begin
with, the political battle may be won sooner. Not that anybody who
is already at IAS can be "losing".
> The real numbers of course have a well known and fixed definition,
> better to reformulate the usual axioms in a form which is classically
> equivalent, but also works intuitionistically.
Link to such a "better" formulation??
> equivalent, but also works intuitionistically.
> This gives more generality at no expense to classical mathematics.
> So my first point is that of course I am not changing definitions.
"of course", if, as you said, the following 8 questions are not
already locked into single answers, then OF COURSE definitions
would have to be changing. The traditional expectation would
be for the definition to answer the questions.
1> The reals as a set are uncountable and
> in bijection with the powerset of natural numbers.
2>The reals as an algebraic structure form a linearly ordered field.
3>The reals as a space are locally compact, Hausdorff, and connected.
4>The reals are a measurable space on which measure theory rests.
5>The reals of non-standard analysis contain infinitesimals.
6>The reals as understood by Leibniz contain nilpotent infinitesimals.
7>The reals as Brouwerian continuum cannot be
> decomposed into two disjoint inhabited subsets.
8> The reals are overt.
> We can have some of these properties but not all at once.
> History has chosen for us a combination that is taught today as a dogma.
Are you informing us that these don't follow FROM THE definition?
That THE known fixed definition doesn't in fact FORCE 1,2,3, and 4 to
be true? In the case of 5 and 6, you CONCEDE the question in advance!
You concede that infinitesimals are "non-standard", i.e, that the "real"
(standard) reals do not have them!
> However, there are many ways to make an imprecise
> idea mathematically precise. For instance, Dedekind
> and Cauchy found two ways of treating the real numbers.
> Those turn out to be equivalent, but that is hindsight.
> When you were taught mathematics you were given definitions
> as cast in stone, unchanging. But that is far from truth
> in research mathematics, where old definitions are revisited,
> criticized and improved upon.
If you conceded that you were revisiting, criticizing, and
improving upon THE DEFINITION OF THE REALS, then you would
concede too much. Isn't it *important* here for you to continue
to insist that you are taking ONE standard/classical/fixed "definition"
and examining how it plays out in alternative "universes"?
> But anyhow, this is not the case here,
> we can all agree on one precise definition of real numbers.
WithIN "standard, classical, 1st-order logic", it either decides or
doesn't decide EACH of those 8 questions. That is (for better or
for worse) NOT the end of the matter BECAUSE whatever-axiom-is-used-
to-ensure-"completeness" will HAVE to be SECOND-order.
> The essential components are: arithmetic, order, and completeness.
> Completeness is tricikiest,
> both because there are several ways to express it
> and because it requires more complex logical language,
> for instance second-order (or higher order) logic,
> or some form of set theory.
is going to get self-contradictory in the following way:
you complain that ZFC itself restricts us to affirming ONE
of the 2^8 subsets of these 8 questions, yet you then point
out that different models of ZFC answer them differently!
If the problem is intolerance of mathematical diversity then
ZFC is not what is causing it -- the fact that first-order
ZFC is a lame approximation of higher-order things is (as you
yourself are pointing out) INTRODUCING mathematical diversity.
> A definition of reals, however, does not fix all of its properties.
Surely the reals are not the only thing about which this is true.
Well, maybe not "surely" --
if you want a FINE example of "dogma",
then one would SURELY be:
" OF COURSE a [correct] definition OF ANY thing
fixes all the thing's properties -- that's by the
DEFINITION of 'definition'! "
If you present the definition and it doesn't decide some
of the properties then the definition IS INcomplete!
Natural-language definitions fail this test all the time
but math was SUPPOSED to be different.
> This is so because the reals do not exist in a vacuum,
> but within a larger context of a mathematical universe, be it a topos,
> a model of ZFC, or some form of type theory.
exist, as different approaches to logic in general. Talking about getting
"different reals" in "different models of ZFC" is, as I hinted above,
problematic for YOUR thesis if you are going to call ZFC itself part of the
"dogma". For starters, ZFC means FIRST-order ZFC. Even all the
way back down at something as simple as Peano Arithmetic for
the natural numbers, there were truly-important
*SECOND*-order considerations afoot. Yet nobody alleges, simply
because the first-order "definitions" admit non-standard naturals, that
any questions about naturals are not fixed by the definition. Rather,
people simply concede that first-order treatments are necessarily incomplete.
> For instance, within the context of set theory, it is possible
> to construct models of ZFC such that every subset of the reals
> is measurable.
is measurable. My point being that ZFC is *not*, in and of itself,
the thing dogmatically insisting on a single answer in a context when
multiple answers ought to be tolerated.
But if you were just dealing with an axiomatic definition of the reals,
you would not necessarily even be able TO SAY "every subset of the reals".
That is a question that JUST PLAIN *DOES*NOT*ARISE* if you are talking about
the reals AS OPPOSED to about set theory. That is a question that IS AN
ARTIFACT of the choice of a set theory as a framework. To anyone who says
that "set theory" and "reals theory" are irretrievably enmeshed
because whatever-axiom[s]-you-use-for-completeness must unavoidably talk
about sets of reals being separable or having least-upper-bounds or whatever,
ONE COULD retort that that question is INHERENTLY SECOND-ORDER and that
RATHER than introduce A FIRST-order set theory, which, purely in virtue of
its BEING first-order, will be A LAME approximation, one must simply LEAVE
the defintion in a 2nd-order framework, IF one aspires to correctness.
In the 2nd-order treatment, rather than every "subset" of the reals, one
would have every 1st-order-predicate-over the reals. It remains a thornier
question as to whether that would resolve the issue, and, if so, whether
the answer that that approach gives would or would not be "dogma" (more
like "canon", if the answer were ever discovered).
> If we widen the context and allow also non-classical
> mathematical universes, for instance toposes, then
> we see an even wider spectrum of possibilities.
> One and the same definition of reals gives us in
> different toposes objects of reals which all share
> the same basic properties (arithmetic, order, completeness),
> but differ in other aspects.
> In some toposes, the reals are indecomposable
> (cannot be written as a union of two disjoint inhabited sets),
> in others they are not locally compact, etc.
different models of first-order ZFC. In the ZFC case, it is not clear why
this is good. It is not clear whether first-order ZFC has vs. lacks "an
intended" model. If it has one then the results from the others are sub-
standard.
> So what are we to think of all these possibilities?
on what "we" think.
> One option is to take an absolute position and declare
> one possibility to be the correct one.
and Peano Arithmetic as an axiomatization of it.
It turns out that there are tons of models of first-order PA.
ONE of those (up to isomorphism) *is* "the correct" one. That is NOT dogma.
That is simply the situation we CARE MOST about (the situation
regarding the factually-actual natural numbers). In THAT context,
there ALWAYS WAS ONE thing we wanted to investigate and deal with (the natural
numbers). We were blessed to discover some new things (non-standard hyper-
finite numbers) along the quest, but we continued to care about "the true"
naturals. In the case of PA, going up to second-order resolves the issue.
Nobody mourns the loss of "other mathematical universes" that occurs
when it turns out that the 2nd-order Peano axioms get you down to ONE
model up-to-isomorphism. Everybody who wants to investigate the other
objects remains perfectly free to do so.
> This is the current situation, where the official position
> is that Zermelo-Fraenkel set theory with choice (ZFC) is
> the foundation of mathematics.
your own argument to point out that *within* ZFC, within different models
of ZFC, you can *already* produce diversity of answers to your 8 questions.
> I call this a dogma because I have seen many times
> the opposition to the idea that ZFC may be replaced by something else.
was wrongly excluding the other 255 alternatives, but as you
yourself have been pointing out, it doesn't decide them all.
George Greene
Dec 1, 2013, 7:48:43 AM12/1/13
to
On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
> I was alerted to this discussion,
WHO SNITCHED?!?
Seriously, I had for a long time thought (especially after Chris Menzel
and the other professors left) that I would be safely under the radar here.
That the most important visitor we have recently had here would arrive
focused on what an asshole *I* am has GOT to be the SWEETEST victory the
cranks WILL EVER have.
George Greene
Dec 1, 2013, 8:14:02 AM12/1/13
to
From some wikipedia page (authors are multiple; I couldn't attribute this clearly)
"There is a theory of the real numbers in each topos,
and so no one master intuitionist theory."
There was also the implication that different models of ZFC could
yield different answers to some set-theoretical questions about "the"
reals, i.e., that you could get "different" reals "in" different models
of ZFC.
"There is a theory of the real numbers in each topos,
and so no one master intuitionist theory."
There was also the implication that different models of ZFC could
yield different answers to some set-theoretical questions about "the"
reals, i.e., that you could get "different" reals "in" different models
of ZFC.
George Greene
Dec 1, 2013, 9:17:17 AM12/1/13
to
On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
> But anyhow, this is not the case here,
> we can all agree on one precise definition of real numbers.
I don't buy this for a second.
> we can all agree on one precise definition of real numbers.
There is something beyond weird about claiming that one and "the same"
definition produces "different results" in different topoi or different
models of ZFC. If the thing defined really is 1 thing then it has
*1*CORRECT* constellation of properties. If different topoi or different
models of ZFC cause some of these properties to come out different then
I wouldn't be so harsh as to say that some of them are "right" and others
are "wrong", but I would also insist that saying that some of them are
"the properties we were TRYING to define [the reals] TO HAVE" and others
are SOMETHING ELSE. There is a sense in which all mathematical objects
are created equal but they are not all equally entitled to call themselves
"the reals".
> The essential components are: arithmetic, order, and completeness.
> A definition of reals, however, does not fix all of its properties.
It fixes at least the above 3 "properties". If "completeness" implies
the Archimedean property and the absence of infinitesimals then the things
WITH infinitesimals ARE NOT "the reals".
> This is so because the reals do not exist in a vacuum,
> but within a larger context of a mathematical universe,
> be it a topos, a model of ZFC, or some form of type theory.
In the traditional axiomatic paradigm, if you are trying to use some
axioms to define a kind or sort of thing (like numbers), then ALL
models of the axioms really are (or ought to be) created equal.
The x's are ANY things that satisfy the axioms. There are a great
many different "places they could be". There are a great many different
versions of them. But they are all supposed to agree on the properties
that can be proven from the axioms. If the axioms are incomplete and
are consistent with divergent interpretations (if the axioms don't
decide some questions) then either the axioms need extending or the
things aren't just 1 kind of thing after all.
> For instance, within the context of set theory,
> it is possible to construct models of ZFC such
> that every subset of the reals is measurable.
is or isn't decided by the axioms defining the reals.
If it isn't decided by the axioms then you do NOT need to go to
ALL THE TROUBLE of constructing "a model of ZFC" -- you could CONTENT
yourself with (instead) constructing THE MUCH SIMPLER
edifice of a model of THE AXIOMS DEFINING THE REALS.
George Greene
Dec 1, 2013, 11:31:28 AM12/1/13
to
> On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
>
> > I was alerted to this discussion,
>
On Sunday, December 1, 2013 7:48:43 AM UTC-5, George Greene wrote:
>
> > I was alerted to this discussion,
>
>
> WHO SNITCHED?!?
I REPEAT,
*WHO* SNITCHED ?!?
If anybody had told me that anybody from IAS would ever show up HERE,
I would've laughed hard enough to burst a blood vessel. If I had known
that sending something to the likes of Andrej Bauer would get taken
seriously, I MIGHT HAVE TRIED THAT instead of this. I find it far
beyond intriguing that there is somebody still reading here who can
get the likes of Andrej Bauer to answer his emails. I mean, it IS NOT
LIKE AB could afford to "lose time" by actually discussing any of this
with the likes of ME.
But SOMEbody reading here DID get that level of respect from him!
WHO is THAT?!?
Ross A. Finlayson
Dec 1, 2013, 1:47:31 PM12/1/13
to
"A definition of reals, however, does not fix all of its properties. This is so
because the reals do not exist in a vacuum, but within a larger context of a
mathematical universe, be it a topos, a model of ZFC, or some form of type
theory. For instance, within the context of set theory, it is possible to
construct models of ZFC such that every subset of the reals is measurable. "
please, ignore incivility there's really no purchase to it. It's as simple to
write "standard" as pick a side here where Virgil ignores the field axioms
defined for the doubly-unit interval.
Dr. Bauer you'll find I quite agree with you, or at least, with this.
Finding the univalent sensible, then the notion of the null-valent here is then
where the origin of the sweep carries through: for example as that the reals
are as not a set in ZF or of that GCH is undecideable in ZFC, and that they
still have all the properties of the real numbers here as all their "classical"
properties, here those being "modern" (18th c.
Cauchy/Weierstrass/Dedekind/Eudoxus), the classical or here "standard". The
systems of higher order are constructive and constructible, here that they're
original and Skolem/Levy. The various branches of mathematics are filling
these as they are in the usually constructible.
Then: "... declare one possibility to be the correct one" ... "deserves to be
called a dogma.", that is dogma, to work to the exhaustion of declared
definitions. Cardinal arithmetic is a strong dogma from uncountability of the
reals, here then I generally refer to the sweep in principle as then the
functional component of a declarative description of a counterexample to the
standard that is still classical, then that it is interesting for its
consequent properties, in both the standard and classical, and the
extra-classical, as to: standard.
And that's dogma.
So, I'd be interested in this generally or the general development, basically
put in terms of "n/d".
Thanks, Dr. Bauer, we're interested in your opinion.
Regards, Ross Finlayson
~
WM
Dec 1, 2013, 2:32:14 PM12/1/13
to
Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> In article <66e51715-2199-475b...@googlegroups.com>,
>
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
> >
>
> >
>
> > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
>
> > > infininty (OR FINITUDE) AT ALL,
>
> >
>
> > That means the proof is cheating since ZF needs completed it everywhere.
>
>
>
> What is "completed it" supposed to mean?
>
Let S_n be the set {2, 4, 6, ..., 2n}.
> In article <66e51715-2199-475b...@googlegroups.com>,
>
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
> >
>
> >
>
> > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
>
> > > infininty (OR FINITUDE) AT ALL,
>
> >
>
> > That means the proof is cheating since ZF needs completed it everywhere.
>
>
>
> What is "completed it" supposed to mean?
>
Let M_n be the set of humers in S_n that are larger than Card(S_n).
The sequence Card(M_n) has limit oo.
It is calculated by the usual means from the potentially infinite sequence (i.e., for every n there exists n+1.}
Only the notion of completed infinity supplies a cardinal number aleph_0 of the limit set of all even numbers that is larger than every finite cardinal number such that the above limit is 0.
Regards, WM
Ben Bacarisse
Dec 1, 2013, 3:01:15 PM12/1/13
to
WM <wolfgang.m...@hs-augsburg.de> writes:
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
<snip>
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
>> What is "completed it" supposed to mean?
>
> Let S_n be the set {2, 4, 6, ..., 2n}.
> Let M_n be the set of humers in S_n that are larger than Card(S_n).
>
> The sequence Card(M_n) has limit oo.
To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
>
> Let S_n be the set {2, 4, 6, ..., 2n}.
> Let M_n be the set of humers in S_n that are larger than Card(S_n).
>
> The sequence Card(M_n) has limit oo.
floor function. The sequence [(n+1) / 2] has limit oo (or is unbounded
-- take you pick) in mathematics as well as in WMathematics.
> It is calculated by the usual means from the potentially infinite
> sequence (i.e., for every n there exists n+1.}
>
> Only the notion of completed infinity supplies a cardinal number
> aleph_0 of the limit set of all even numbers that is larger than every
> finite cardinal number such that the above limit is 0.
seriously flawed. Don't palm it off on anyone else.
--
Ben.
WM
Dec 1, 2013, 4:15:06 PM12/1/13
to
Am Sonntag, 1. Dezember 2013 21:01:15 UTC+1 schrieb Ben Bacarisse:
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
>
>
> > Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
>
> <snip>
>
> >> What is "completed it" supposed to mean?
>
> >
>
> > Let S_n be the set {2, 4, 6, ..., 2n}.
>
> > Let M_n be the set of numbers in S_n that are larger than Card(S_n).
> WM <wolfgang.m...@hs-augsburg.de> writes:
>
>
>
> > Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
>
> <snip>
>
> >> What is "completed it" supposed to mean?
>
> >
>
> > Let S_n be the set {2, 4, 6, ..., 2n}.
>
>
> >
>
> > The sequence Card(M_n) has limit oo.
>
>
>
> To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
>
> floor function. The sequence [(n+1) / 2] has limit oo (or is unbounded
>
> -- take you pick) in mathematics as well as in WMathematics.
>
>
>
> > It is calculated by the usual means from the potentially infinite
>
> > sequence (i.e., for every n there exists n+1.}
>
> >
>
> > Only the notion of completed infinity supplies a cardinal number
>
> > aleph_0 of the limit set of all even numbers that is larger than every
>
> > finite cardinal number such that the above limit is 0.
>
>
>
> If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
>
> seriously flawed.
It is not my notion. Set theory says that the cardinality of the set of all even numbers, that are surpassing the cardinality of the set of all even numbers, is zero.
> >
>
> > The sequence Card(M_n) has limit oo.
>
>
>
> To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
>
> floor function. The sequence [(n+1) / 2] has limit oo (or is unbounded
>
> -- take you pick) in mathematics as well as in WMathematics.
>
>
>
> > It is calculated by the usual means from the potentially infinite
>
> > sequence (i.e., for every n there exists n+1.}
>
> >
>
> > Only the notion of completed infinity supplies a cardinal number
>
> > aleph_0 of the limit set of all even numbers that is larger than every
>
> > finite cardinal number such that the above limit is 0.
>
>
>
> If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
>
> seriously flawed.
In mathematics we have no completed infinity. There we can only calculate limits using the finite terms of the sequence. The (improper) limit in this case is oo.
By the way, this used to be the same in set theory: Of course Cantor concluded from the fact that the diagonal number differs from the first n entries of the list that it differs from all entries of the list. Magic transformations and baffling tricks like above have only been introduced after it was clear that otherwise finished infinity is self-contradictory.
Regards, WM
Ross A. Finlayson
Dec 1, 2013, 4:20:57 PM12/1/13
to
be as of cardinal arithmetic, where here that there is completeness of
combination in ordinal arithmetic. An infinite ordinal for an unbounded
variable, that is linearly related to another unbounded variable, sees ordinal
arithmetic with transfinite ordinals as simple representing the order in that.
Then, constructively, there are objects as of cardinals in their arithmetic,
those are then of cardinal forms or a model of cardinals, as to applying
cardinal arithmetic quite high in the structure instead of as of the (direct)
foundation. Properties of cardinal arithmetic are simply less than of ordinal
arithmetic while each has an initial ordinal.
Then, the idea here that there are no functions between the naturals and reals,
then that for all the properties it would fulfill there is exactly one of them,
is to build a constructive framework in ordinal arithmetic that works out to
cardinal arithmetic.
Then just as vectors and tensors are as of their form, then cardinal arithmetic
and cardinals, _of their form_, would have for their measure as sets of reals
topologically, then systems as to conditions that what maintain cardinal
arithmetic, would work out to have some application, thus constructive form.
This is where the construction in the countable ordinals is usually paramount
in the definitive of the constructible.
Regards, Ross Finlayson
WM
Dec 1, 2013, 4:25:27 PM12/1/13
to
Am Sonntag, 1. Dezember 2013 13:46:56 UTC+1 schrieb George Greene:
>
> Well, sure, but the reals are old, settled, stuff.
Yes. But "the reals that cannot be defined in English language" are clearly new. - And only those are in question here.
>
> Well, sure, but the reals are old, settled, stuff.
Regards, WM
andrej...@andrej.com
Dec 1, 2013, 4:26:40 PM12/1/13
to
I've forgotten what an amazing place usenet newgroups are.
A couple of corrections:
1. I am not at IAS, I was there last year.
2. I should have said "there is a model of ZF in which every subset of the reals is measurable" (i.e., no choice, with choice we can prove existence of non-measurable sets).
I think I know what bothers George Greene (by the way, congratulations on being such a masterful troll). The giveaway was when he spoke about how there are the *true* natural numbers. That indicates that he holds a Platonistic view of mathematics, i.e., he believes there is an objective universe of mathematics, the one we call "true". Under such conviction the alternative universes get a secondary status. And so, one would be upset by suggestions that certain properties of reals may not be decided, as obviously they must be in the objective universe.
Ah, before I forget. George Green asked for a reference. An intuitionistically better definition of real numbers can be found in one of the following places:
1. The HoTT book, chapter 11, http://homotopytypetheory.org/book/
2. Troelstra & van Dalen's "Constructivism in mathematics, vol. 1"
These use a rich logic, for instance the HoTT book uses dependent type theory. It is possible to define the real numbers in a very weak system, using only equational reasoning and lambda calculus. This is technically more advanced, but it shows that you do not actually need second-order logic or set theory to get hold of the reals:
4. A. Bauer, P. Taylor: "The Dedekind Reals in Abstract Stone Duality", http://www.paultaylor.eu/ASD/dedras/
The point about 4 is that there is no logic to speak of. Just lambda calculus extended with a couple of features.
In all cases 1-4 above one can show that the object of reals is unique up to isomorphism. So this means that the definition pins down the reals as well as we can hope. But, contrary to George Green's expectations, this is *not* sufficient to answer all questions about the reals. Perhaps a more familiar example will help explain this.
Consider the following question about the reals: "Is every infinite subset of the reals either countable or in bijection with the reals?" This of course is just famous Cantor's hypothesis, and it is known to be undecidable in set theory. It does not help to pass to second-order logic, or higher-order logic, or intuitionisitic logic, or whatever. Cantor's hypothesis remains undecidable.
So, even though the real numbers are defined very precisely, and unique up to isomorphism, that does not answer all questions about them. For a Platonist such as George Green seems to be, that ought to present a problem.
And by the way, Gödel's incompleteness theorems show that the same happens with the natural numbers, even if you use second-order logic or whatnot. There will always be undecided statements about natural numbers.
To me that says something about believing in a One True World of Mathematics, namely that such a belief is a form of mysticism. As all good forms of mysticism, this one too is self-consistent and soothing to the soul. That makes it hard to let go off it.
You people have fun SHOUTING at each other (you really should stop that George, it is annoying as hell), I've got real work to do. But this was fun.
With kind regards,
Andrej
A couple of corrections:
1. I am not at IAS, I was there last year.
2. I should have said "there is a model of ZF in which every subset of the reals is measurable" (i.e., no choice, with choice we can prove existence of non-measurable sets).
I think I know what bothers George Greene (by the way, congratulations on being such a masterful troll). The giveaway was when he spoke about how there are the *true* natural numbers. That indicates that he holds a Platonistic view of mathematics, i.e., he believes there is an objective universe of mathematics, the one we call "true". Under such conviction the alternative universes get a secondary status. And so, one would be upset by suggestions that certain properties of reals may not be decided, as obviously they must be in the objective universe.
Ah, before I forget. George Green asked for a reference. An intuitionistically better definition of real numbers can be found in one of the following places:
1. The HoTT book, chapter 11, http://homotopytypetheory.org/book/
2. Troelstra & van Dalen's "Constructivism in mathematics, vol. 1"
These use a rich logic, for instance the HoTT book uses dependent type theory. It is possible to define the real numbers in a very weak system, using only equational reasoning and lambda calculus. This is technically more advanced, but it shows that you do not actually need second-order logic or set theory to get hold of the reals:
4. A. Bauer, P. Taylor: "The Dedekind Reals in Abstract Stone Duality", http://www.paultaylor.eu/ASD/dedras/
The point about 4 is that there is no logic to speak of. Just lambda calculus extended with a couple of features.
In all cases 1-4 above one can show that the object of reals is unique up to isomorphism. So this means that the definition pins down the reals as well as we can hope. But, contrary to George Green's expectations, this is *not* sufficient to answer all questions about the reals. Perhaps a more familiar example will help explain this.
Consider the following question about the reals: "Is every infinite subset of the reals either countable or in bijection with the reals?" This of course is just famous Cantor's hypothesis, and it is known to be undecidable in set theory. It does not help to pass to second-order logic, or higher-order logic, or intuitionisitic logic, or whatever. Cantor's hypothesis remains undecidable.
So, even though the real numbers are defined very precisely, and unique up to isomorphism, that does not answer all questions about them. For a Platonist such as George Green seems to be, that ought to present a problem.
And by the way, Gödel's incompleteness theorems show that the same happens with the natural numbers, even if you use second-order logic or whatnot. There will always be undecided statements about natural numbers.
To me that says something about believing in a One True World of Mathematics, namely that such a belief is a form of mysticism. As all good forms of mysticism, this one too is self-consistent and soothing to the soul. That makes it hard to let go off it.
You people have fun SHOUTING at each other (you really should stop that George, it is annoying as hell), I've got real work to do. But this was fun.
With kind regards,
Andrej
andrej...@andrej.com
Dec 1, 2013, 4:29:54 PM12/1/13
to
On Sunday, December 1, 2013 10:26:40 PM UTC+1, andrej...@andrej.com wrote:
>
> 1. The HoTT book, chapter 11, http://homotopytypetheory.org/book/
> 2. Troelstra & van Dalen's "Constructivism in mathematics, vol. 1"
>
> 1. The HoTT book, chapter 11, http://homotopytypetheory.org/book/
> 2. Troelstra & van Dalen's "Constructivism in mathematics, vol. 1"
> 4. A. Bauer, P. Taylor: "The Dedekind Reals in Abstract Stone Duality", http://www.paultaylor.eu/ASD/dedras/
Oops, that should have been 1, 2, 3.
P.S. I don't understand half a sentence of what Ross Finlayson "the snitch" says.
Ben Bacarisse
Dec 1, 2013, 4:48:18 PM12/1/13
to
WM <wolfgang.m...@hs-augsburg.de> writes:
> Am Sonntag, 1. Dezember 2013 21:01:15 UTC+1 schrieb Ben Bacarisse:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
>> <snip>
>> >> What is "completed it" supposed to mean?
>> >
>> > Let S_n be the set {2, 4, 6, ..., 2n}.
>> > Let M_n be the set of numbers in S_n that are larger than Card(S_n).
>> >
>> > The sequence Card(M_n) has limit oo.
>>
>> To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
>> floor function. The sequence [(n+1) / 2] has limit oo (or is unbounded
>> -- take you pick) in mathematics as well as in WMathematics.
>>
>> > It is calculated by the usual means from the potentially infinite
>> > sequence (i.e., for every n there exists n+1.}
>> >
>> > Only the notion of completed infinity supplies a cardinal number
>> > aleph_0 of the limit set of all even numbers that is larger than every
>> > finite cardinal number such that the above limit is 0.
>>
>> If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
>> seriously flawed.
>
> It is not my notion. Set theory says that the cardinality of the set
> of all even numbers, that are surpassing the cardinality of the set of
> all even numbers, is zero.
Bait and switch. This is not the assertion which I was refuting.
> Am Sonntag, 1. Dezember 2013 21:01:15 UTC+1 schrieb Ben Bacarisse:
>> WM <wolfgang.m...@hs-augsburg.de> writes:
>>
>> > Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
>> <snip>
>> >> What is "completed it" supposed to mean?
>> >
>> > Let S_n be the set {2, 4, 6, ..., 2n}.
>> > Let M_n be the set of numbers in S_n that are larger than Card(S_n).
>> >
>> > The sequence Card(M_n) has limit oo.
>>
>> To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
>> floor function. The sequence [(n+1) / 2] has limit oo (or is unbounded
>> -- take you pick) in mathematics as well as in WMathematics.
>>
>> > It is calculated by the usual means from the potentially infinite
>> > sequence (i.e., for every n there exists n+1.}
>> >
>> > Only the notion of completed infinity supplies a cardinal number
>> > aleph_0 of the limit set of all even numbers that is larger than every
>> > finite cardinal number such that the above limit is 0.
>>
>> If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
>> seriously flawed.
>
> It is not my notion. Set theory says that the cardinality of the set
> of all even numbers, that are surpassing the cardinality of the set of
> all even numbers, is zero.
<snip>
--
Ben.
fom
Dec 1, 2013, 6:27:56 PM12/1/13
to
On 12/1/2013 12:47 PM, Ross A. Finlayson wrote:
> On Sunday, December 1, 2013 8:31:28 AM UTC-8, George Greene wrote:
>>> On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
>>
>>>
>>
>>>> I was alerted to this discussion,
>>
>>>
>>
>> On Sunday, December 1, 2013 7:48:43 AM UTC-5, George Greene wrote:
>>
>>>
>>
>>> WHO SNITCHED?!?
>>
>>
Since I cannot read George's posts,
> On Sunday, December 1, 2013 8:31:28 AM UTC-8, George Greene wrote:
>>> On Sunday, December 1, 2013 4:52:53 AM UTC-5, andrej...@andrej.com wrote:
>>
>>>
>>
>>>> I was alerted to this discussion,
>>
>>>
>>
>> On Sunday, December 1, 2013 7:48:43 AM UTC-5, George Greene wrote:
>>
>>>
>>
>>> WHO SNITCHED?!?
>>
>>
it was not me.
<snip>
>> I find it far
>>
>> beyond intriguing that there is somebody still reading here who can
>>
>> get the likes of Andrej Bauer to answer his emails.
George underestimates the kindness and interest of
Professor Bauer. I wrote him once concerning a
topological representation of proof methods. He
answered promptly and considerately.
>
> Thanks, Dr. Bauer, we're interested in your opinion.
>
Ross A. Finlayson
Dec 1, 2013, 8:12:15 PM12/1/13
to
detrimental.
Now I can add that to Kolker's "No opinion, period."
"The central new idea in homotopy type theory is that types can be regarded as
spaces in homotopy theory, or higher-dimensional groupoids in category theory."
"By contrast, type theory is its own deductive system: it need not be
formulated inside any superstructure, such as first-order logic. Instead of the
two basic notions of set theory, sets and propositions, type theory has one
basic notion: types. Propositions (statements which we can prove, disprove,
assume, negate, and so on1) are identified with particular types, via the
correspondence shown in Table 1 on page 11. Thus, the mathematical activity of
proving a theorem is identified with a special case of the mathematical
activity of constructing an object—in this case, an inhabitant of a type that
represents a proposition."
Here for the constructivist in set theory that would not be so different.
Saying that inferential statements aren;t constructive would ignore that types
as for example of a table may be modeled as regular sets (as to finite and
constructible sets, or here as types).
Then the major statement as to establishing those ranges as space elements is
typically not different than that they are real numbers or points for types of
ranges over numbers, or as to spaces and subspaces. These are so far all
constructible as sets, as they are.
Then, basically as to establishing _deduction_ besides _induction_ as the or a
primary tool, simply promotes the usual in results in the contrapositive from
one to the other. This is explained in set theory as the transfer principle.
Provability in set theory and here as to types, Russell's types for example are
still as to the propositional besides the resultant, the point is to maintain
deductive results from the universal and inferential results from the
existential. So, changing types still sees the usual completeness results of
ordinary set theory apply.
"However, as in set theory", ....
"When we say that A is a type, we mean that it inhabits some universe Ui."
Then, I see useful developmental constructions in defining types of among
"universes" of a type, where the properties of the type as of a value simply
forms the same constructivist notions as of those properties in the products
that are instead with types unbuilt, they are still generally preserved notions
from where types are built from collections, or partitions: either. The
product forms then are as standard.
It seems a mathematics written for the expression of computer programs.
Generally in the world of "types" in computer science, modern type theory and
the pi-calculus for combinatorial completeness then again is maintaining
interpretability of regular/ordinary sets or types.
From "to specify a type":
"an optional uniqueness principle8, which expresses uniqueness of maps into or
out of that type"
"For some types, the uniqueness principle characterizes maps into the type, by
stating that every element of the type is uniquely determined by the results of
applying eliminators to it, and can be reconstructed from those results by
applying a constructor— thus expressing how constructors act on eliminators,
dually to the computation rule."
Then, this axiom 5 of the definition of a type from page 27 is each a
definition, that it's an axiom schema over types. In ZF they might get to
axiom schema then generally as over types where the ordering in higher-order
arithmetic is along resource lines, again typically, that NBG for schema not
classes is equi-interpretable with ZF. Here then it is of general exceptions
to the rule. This doesn't seem to conclude as a well-founded universe of
mutually consistent types as they are so defined to be.
Further on duality there is basically the concern that to each branch there is
its concern, from topology here with duality over topoi or geometry as to
Riemmanian or the infinitesimals as Leibniz or Newton, this is all the
classical that then results in area or product space are concretely maintained.
Then, the notion is to at least define the uniqueness principle as to, of
everything that it is not, it's not the exception but the correction.
It would be interesting to hear how you might explain what you meant, or
rather, how it could be, from the "Am I a constructive mathematician?", that:
"It may happen that the reals are in 1-1 correspondence with a subset of the
natural numbers, while at the same time they form an uncountable set."
Are the reals, the real numbers, having as a type, particularly of the defined
type as noted, this unique mapping?
This is where, I've written out a general notation as to why that is so.
Regards, Ross Finlayson
Virgil
Dec 1, 2013, 9:46:03 PM12/1/13
to
In article <0bbbbe5e-b9d0-4829...@googlegroups.com>,
limit of anything.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> > In article <66e51715-2199-475b...@googlegroups.com>,
> >
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> >
> >
> >
> > > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
> >
> > >
> >
> > >
> >
> > > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT
> > > > mention
> >
> > > > infininty (OR FINITUDE) AT ALL,
> >
> > >
> >
> > > That means the proof is cheating since ZF needs completed it everywhere.
> >
> >
> >
> > What is "completed it" supposed to mean?
> >
>
> Let S_n be the set {2, 4, 6, ..., 2n}.
> Let M_n be the set of humers in S_n that are larger than Card(S_n).
>
> The sequence Card(M_n) has limit oo.
But, accoding to WM, there cannot be any such number available to be a
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> > In article <66e51715-2199-475b...@googlegroups.com>,
> >
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> >
> >
> >
> > > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
> >
> > >
> >
> > >
> >
> > > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT
> > > > mention
> >
> > > > infininty (OR FINITUDE) AT ALL,
> >
> > >
> >
> > > That means the proof is cheating since ZF needs completed it everywhere.
> >
> >
> >
> > What is "completed it" supposed to mean?
> >
>
> Let S_n be the set {2, 4, 6, ..., 2n}.
> Let M_n be the set of humers in S_n that are larger than Card(S_n).
>
> The sequence Card(M_n) has limit oo.
limit of anything.
--
Virgil
Dec 1, 2013, 9:56:10 PM12/1/13
to
In article <4d8403ba-18bf-4247...@googlegroups.com>,
but have not always been recognized as being there.
The distiction is important, at least outside of WM's wild weird world
of WMytheology.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Sonntag, 1. Dezember 2013 13:46:56 UTC+1 schrieb George Greene:
>
>
> >
> > Well, sure, but the reals are old, settled, stuff.
>
> Yes. But "the reals that cannot be defined in English language" are clearly
> new.
On the contrary, they have always been there,
> Am Sonntag, 1. Dezember 2013 13:46:56 UTC+1 schrieb George Greene:
>
>
> >
> > Well, sure, but the reals are old, settled, stuff.
>
> Yes. But "the reals that cannot be defined in English language" are clearly
> new.
but have not always been recognized as being there.
The distiction is important, at least outside of WM's wild weird world
of WMytheology.
--
Virgil
Dec 1, 2013, 10:04:59 PM12/1/13
to
In article <83c42af0-8af5-46d6...@googlegroups.com>,
> Set theory says that the cardinality of the set of all
> even numbers, that are surpassing the cardinality of the set of all even
> numbers, is zero.
Set theory outside of WM's wild weird world of WMytheology says that
that description is far to ambiguous to have any set theoretical meaning.
mathematicians, call mathematics, we do have a number of completed
infinities!
>
> By the way, this used to be the same in set theory
Never believe anything that WM says about set theory, or anything else.
for the matter, unless one has reliable evidence of its truth from a
source that WM cannot command.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
>
> > If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
> >
> > seriously flawed.
>
> It is not my notion.
It is certrainly not anyone else's.
>
> > If your notion of "completed" makes lim [(n+1)/2] = 0, then it is
> >
> > seriously flawed.
>
> It is not my notion.
> Set theory says that the cardinality of the set of all
> even numbers, that are surpassing the cardinality of the set of all even
> numbers, is zero.
that description is far to ambiguous to have any set theoretical meaning.
>
> In mathematics we have no completed infinity.
In what we, as part the vast majority of those who call themselves
> In mathematics we have no completed infinity.
mathematicians, call mathematics, we do have a number of completed
infinities!
>
> By the way, this used to be the same in set theory
for the matter, unless one has reliable evidence of its truth from a
source that WM cannot command.
--
George Greene
Dec 1, 2013, 10:20:41 PM12/1/13
to
On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
> Yes. But "the reals that cannot be defined in English language"
> are clearly new.
They are NOT clearly new.
> Yes. But "the reals that cannot be defined in English language"
> are clearly new.
It has always been obvious that there have to be undefinable reals.
Sentences of the English language may be infinite in number if there
is no upper limit to their length, but every last one of them IS FINITE
and there are therefore at most countably infinitely many of them.
Therefore there are at most countably infinitely many definitions.
Therefore THE VAST majority of reals are not definable in English.
It was EVER thus. This is HARDLY new, not since 1888 anyway.
George Greene
Dec 1, 2013, 10:22:15 PM12/1/13
to
On Sunday, December 1, 2013 6:27:56 PM UTC-5, fom wrote:
> George underestimates the kindness and interest of
> Professor Bauer. I wrote him once concerning a
> topological representation of proof methods. He
> answered promptly and considerately.
The only thing that will prove that *I* have underestimated this
> George underestimates the kindness and interest of
> Professor Bauer. I wrote him once concerning a
> topological representation of proof methods. He
> answered promptly and considerately.
is if *I* get a reply. Just because he was kind or understanding
TO YOU does not necessarily warrant a universal generalization.
George Greene
Dec 1, 2013, 10:59:51 PM12/1/13
to
On Sunday, December 1, 2013 4:26:40 PM UTC-5, andrej...@andrej.com wrote:
> I think I know what bothers George Greene
I thought you weren't going to engage!
> I think I know what bothers George Greene
I will have to forgo my usual style here since it is more than a little
shocking to me (as a failed PhD student in Comp.Sci.) to be being talked
to by the likes of anyone at IAS.
> The giveaway was when he spoke about how there are
> the *true* natural numbers.
what I say. I have to make concessions to what other people will understand.
My personal opinion is just that, personal. Sharing it accurately is not
necessarily even important.
> That indicates that he holds a Platonistic view of mathematics,
> i.e., he believes there is an objective universe of mathematics,
> the one we call "true".
> Under such conviction the alternative universes get a secondary status.
Yes, among the community you were calling dogmatic, that happens,
but if Google went back far enough, you could see me arguing against
the late Torkel Franzen (and Aatu Koskensilta and any number of other
people) that IF we are going to do a (standard/classical)FIRST-order(logic)
treatment of the natural numbers, the so-called "non-standard" models of the
first-order Peano Axioms should NOT be deprecated as "sub"-standard (which
they in practice often are -- as a primary example I used to object to my opponents' insistence on locutions like "proves a false theorem").
THEY had no qualms about calling an undecidable univ.gen. "true"[simpliciter] even if, by virtue of being undecidable, it WAS ALSO FALSE in SOME models of first-order PA! They wanted to call it "true" because it was true of all the natural numbers. I was adamant that they needed to say "true in the standard
model" when that was what they meant. They were equally adamant that the
community's common parlance simply left that understood and that the only
reason I needed it spelled out every time was that I was new! That was NOT
my only reason!
> And so, one would be upset by suggestions that
> certain properties of reals may not be decided,
> as obviously they must be in the objective universe.
WILL NOT define/decide all the properties. This problem ALREADY ARISES
AT THE LOWER LEVEL than the reals -- this problem ALREADY ARISES FOR
Peano Arithmetic and the naturals!
> It is possible to define the real numbers in a very weak system,
> using only equational reasoning and lambda calculus.
> This is technically more advanced,
> but it shows that you do not actually
> need second-order logic or set theory to get hold of the reals
2nd-order logic is ONE thing. Set theory is ANOTHER DIFFERENT thing.
In particular, the usual set theory is FIRST-order ZFC and PURELY because
it is FIRST-order, IT MUST OBVIOUSLY FAIL to do a decent job. That is
one way of phrasing Godel's thm.
> The point about 4 is that there is no logic to speak of.
for having DIFFICULTY with an approach that brags about having
NO logic to speak of!
> Consider the following question about the reals:
> "Is every infinite subset of the reals either countable
> or in bijection with the reals?"
> This of course is just famous Cantor's hypothesis,
> and it is known to be undecidable in set theory.
before, I really thought there was 1 intended model, then the question
would necessarily be decided in THAT model.
> It does not help to pass to second-order logic,
there REALLY IS "one intended model" in the 2nd-order-logic case,
namely the one with "full" powersets. This fundamentally is NOT MERELY
a question about "the reals". It's a question about the nature of "powerset"
in general. It's a question about just how rich the powerset of an infinite
set can "truly" get. The 2nd-order consequence relation in general is
not amenable to syntactic proof-as-we-know-it but that does not mean that
we won't eventually deal with enough fragments of it to get some traction.
I *guess* that the pessimistic result will be that as long as the piece-of-
2nd-order-logic-that-we-have-some-proof-theory for remains recursive, it will
always be too weak to decide the question, and once it's more complicated than
that, it will not be finitarily specifiable any more and so will not be of
practical use. But there is still an answer. I am not willing to read
Godel's 1st Incompleteness thm as applying to 2nd-order anything.
> or higher-order logic, or intuitionistic logic, or whatever.
Well I am obviously too dumb to even envision a proof (what logic
would it be proved in??) that higher-order logic can't decide
something.
> And by the way, Gödel's incompleteness theorems show that
> the same happens with the natural numbers, even if you use
> second-order logic or whatnot.
Godel's 1st incompleteness theorem involves Godel numbers for formulas.
If you are using FIRST-order PA then the induction axiom is a schema
with instances that are formulas, so those instances will have Godel
numbers. At 2nd-order you don't even have a recursive proof-system TO
BEGIN with. If you call yourself doing 2nd-order logic because you took
some recursive collection of of comprehension axioms (a la Henkin) then
they will NOT exhaust the 2nd-order consequence relation (they will not
satisfy Godel's COMPleteness theorem) so the whole thing truly becomes
apples vs. oranges.
> I There will always be undecided statements about natural numbers.
From a recursive axiomatization and finitary methods for generating theorems
from them, of course. That doesn't mean that anything that happens to be
undecided currently by current methods won't become decidable later by deeper
better methods. There will always be undecidable questions, but it is not
clear that any particular example (even yours of Cantor's Hypothesis) will
always remain one of them.
> To me that says something about believing in a
> One True World of Mathematics, namely that such a
> belief is a form of mysticism.
I repeat, around HERE, *I* have a history of insisting that
even the standard model of first-order PA did NOT deserve to be
as-privileged-as-it-is above the other models of PA.
The place where I *would* get convicted of thinking there was
1 preferred model would be in 2nd-order logic, would be regarding
the notion of a quantifier over "all possible" collections (of
a denumerably infinite set) as coherent, as quantifying over a
unique specific intended larger collection.
> With kind regards,
The judgment of the quality of the regards is
than just arguably in the eye of the regardee.
But, seriously, thanks for any&everything.
This is certainly more than I had any right to expect!
WM
Dec 2, 2013, 2:51:16 AM12/2/13
to
Am Sonntag, 1. Dezember 2013 22:48:18 UTC+1 schrieb Ben Bacarisse:
>
> Bait and switch. This is not the assertion which I was refuting.
Why do you never say what you want to say?
>
> Bait and switch. This is not the assertion which I was refuting.
Regards, WM
WM
Dec 2, 2013, 2:55:20 AM12/2/13
to
Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
> On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
>
> > Yes. But "the reals that cannot be defined in English language"
>
> > are clearly new.
>
>
>
> They are NOT clearly new.
>
> It has always been obvious that there have to be undefinable reals.
Mathematics in English-speaking countries is done in English. Undefinable reals in English (or any other language spoken by men) do not belong to mathematics.
> On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
>
> > Yes. But "the reals that cannot be defined in English language"
>
> > are clearly new.
>
>
>
> They are NOT clearly new.
>
> It has always been obvious that there have to be undefinable reals.
Regards, WM
Virgil
Dec 2, 2013, 3:42:05 AM12/2/13
to
In article <bf894613-7815-4f4e...@googlegroups.com>,
> Undefinable
> reals in English (or any other language spoken by men) do not belong to
> mathematics.
Individually, they are inacessible, but collectively they must exist,
unles one wants to revise the real number system as it is usuallly
defined.
So far, WM has not have any definition of a real number system, much
less one that works.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
> > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
> >
> > > Yes. But "the reals that cannot be defined in English language"
> >
> > > are clearly new.
> >
> >
> >
> > They are NOT clearly new.
> >
> > It has always been obvious that there have to be undefinable reals.
>
> Mathematics in English-speaking countries is done in English.
Usually!
> Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
> > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
> >
> > > Yes. But "the reals that cannot be defined in English language"
> >
> > > are clearly new.
> >
> >
> >
> > They are NOT clearly new.
> >
> > It has always been obvious that there have to be undefinable reals.
>
> Mathematics in English-speaking countries is done in English.
> Undefinable
> reals in English (or any other language spoken by men) do not belong to
> mathematics.
unles one wants to revise the real number system as it is usuallly
defined.
So far, WM has not have any definition of a real number system, much
less one that works.
--
Virgil
Dec 2, 2013, 3:43:53 AM12/2/13
to
In article <76643d8f-8e43-4bef...@googlegroups.com>,
He quite frequently says what HE wants to say, but does not often say
what WM wants him to say, for which more power to him.
--
what WM wants him to say, for which more power to him.
--
Ben Bacarisse
Dec 2, 2013, 7:15:51 AM12/2/13
to
was zero and said it was not. Your reply had nothing to do with your
previous incorrect assertion so I pointed that out.
There's no point in my repeating myself -- if you don't want to reply to
the points someone makes that's your business.
--
Ben.
Ben Bacarisse
Dec 2, 2013, 7:24:04 AM12/2/13
to
Virgil <vir...@ligriv.com> writes:
<snip>
usual way, though the definition is, as might be expected in an
introductory text, rather cursory. The only reasonable conclusion ithat
it is the consequences of that definition that he is not happy with.
The book defines injections, surjections and bijections in the normal
way but it does not, as far as I can see (my German is poor), define the
power set of a set, and it avoids the question of bijections between N
and R.
--
Ben.
<snip>
> So far, WM has not have any definition of a real number system, much
> less one that works.
I'd say he has. He's written a book that defines and uses R in the
> less one that works.
usual way, though the definition is, as might be expected in an
introductory text, rather cursory. The only reasonable conclusion ithat
it is the consequences of that definition that he is not happy with.
The book defines injections, surjections and bijections in the normal
way but it does not, as far as I can see (my German is poor), define the
power set of a set, and it avoids the question of bijections between N
and R.
--
Ben.
WM
Dec 2, 2013, 7:44:37 AM12/2/13
to
Am Montag, 2. Dezember 2013 09:42:05 UTC+1 schrieb Virgil:
> In article <bf894613-7815-4f4e...@googlegroups.com>,
>
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
>
> > > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
>
> > >
>
> > > > Yes. But "the reals that cannot be defined in English language"
>
> > >
>
> > > > are clearly new.
>
> > >
>
> > >
>
> > >
>
> > > They are NOT clearly new.
>
> > >
>
> > > It has always been obvious that there have to be undefinable reals.
>
> >
>
> > Mathematics in English-speaking countries is done in English.
>
> Usually!
>
>
>
> > Undefinable
>
> > reals in English (or any other language spoken by men) do not belong to
>
> > mathematics.
>
>
>
> Individually, they are inacessible, but collectively they must exist,
>
This "must" is erroneously concluded from the idea that Cantor's arguments force the existence of undefineable numbers. But they don't. Every application of mathematics, including Cantor's argument, on defineable numbers produce defineable numbers. Application of the diagonal argument on undefineable numbers must necessarily fail.
> In article <bf894613-7815-4f4e...@googlegroups.com>,
>
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
>
> > > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
>
> > >
>
> > > > Yes. But "the reals that cannot be defined in English language"
>
> > >
>
> > > > are clearly new.
>
> > >
>
> > >
>
> > >
>
> > > They are NOT clearly new.
>
> > >
>
> > > It has always been obvious that there have to be undefinable reals.
>
> >
>
> > Mathematics in English-speaking countries is done in English.
>
> Usually!
>
>
>
> > Undefinable
>
> > reals in English (or any other language spoken by men) do not belong to
>
> > mathematics.
>
>
>
> Individually, they are inacessible, but collectively they must exist,
>
>
> So far, WM has not have any definition of a real number system,
Another error. Read my book
> So far, WM has not have any definition of a real number system,
http://www.degruyter.com/search?t1=MA&type_0=textbooks&pageSize=10&sort=datedescending&searchTitles=true&authorCount=5&q1=M%C3%BCckenheim&t2=
Regards, WM
WM
Dec 2, 2013, 7:53:43 AM12/2/13
to
Am Montag, 2. Dezember 2013 13:15:51 UTC+1 schrieb Ben Bacarisse:
> Your reply had nothing to do with your
>
> previous incorrect assertion so I pointed that out.
Your pointing out has nothing to do with my posting. But may be you have not understood it correctly: I said that the limit of cardinalities in mathematics is oo. The limit of the same cardinalities in set theory is 0.
> Your reply had nothing to do with your
>
> previous incorrect assertion so I pointed that out.
You can read an elaborated essay on the same topic on p. 242 of our research report:
http://www.hs-augsburg.de/medium/download/oeffentlichkeitsarbeit/publikationen/forschungsbericht_2012.pdf
Regards, WM
WM
Dec 2, 2013, 7:59:09 AM12/2/13
to
Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> In article <66e51715-2199-475b...@googlegroups.com>,
>
> > segments of the even natural numbers
>
> >
>
> > |{2}| = 1
>
> > |{2, 4}| = 2
>
> > |{2, 4, 6}| = 3
>
> > ...
>
> >
>
> > are never larger than all elements of the initial segments.
>
>
>
> Being larger than all elements of the initial segments means being
>
> actually infinite
I was just explaining this to GG. Thanks for support.
Regards, WM
> In article <66e51715-2199-475b...@googlegroups.com>,
>
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
> >
>
> >
>
> > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
>
> > > infininty (OR FINITUDE) AT ALL,
>
> >
>
> > That means the proof is cheating since ZF needs completed it everywhere.
>
>
>
> What is "completed it" supposed to mean?
>
> >
>
> > Without completed infinity we can prove that the cardinalities of the initial
> WM <wolfgang.m...@hs-augsburg.de> wrote:
>
>
>
> > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
>
> >
>
> >
>
> > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT mention
>
> > > infininty (OR FINITUDE) AT ALL,
>
> >
>
> > That means the proof is cheating since ZF needs completed it everywhere.
>
>
>
> What is "completed it" supposed to mean?
>
> >
>
>
> > segments of the even natural numbers
>
> >
>
> > |{2}| = 1
>
> > |{2, 4}| = 2
>
> > |{2, 4, 6}| = 3
>
> > ...
>
> >
>
> > are never larger than all elements of the initial segments.
>
>
>
> Being larger than all elements of the initial segments means being
>
> actually infinite
I was just explaining this to GG. Thanks for support.
Regards, WM
FredJeffries
Dec 2, 2013, 11:54:26 AM12/2/13
to
On Monday, December 2, 2013 4:44:37 AM UTC-8, WM wrote:
> Read my book
>
> http://www.degruyter.com/search?t1=MA&type_0=textbooks&pageSize=10&sort=datedescending&searchTitles=true&authorCount=5&q1=M%C3%BCckenheim&t2=
I have asked this before but never received a reply: Just how many schmucks are there who have shelled out 248 euros (or 350 US dollars) for the electronic version of your "bestseller"?
> Read my book
>
> http://www.degruyter.com/search?t1=MA&type_0=textbooks&pageSize=10&sort=datedescending&searchTitles=true&authorCount=5&q1=M%C3%BCckenheim&t2=
http://www.degruyter.com/view/product/216861
Ben Bacarisse
Dec 2, 2013, 3:59:19 PM12/2/13
to
WM <wolfgang.m...@hs-augsburg.de> writes:
> Am Montag, 2. Dezember 2013 13:15:51 UTC+1 schrieb Ben Bacarisse:
>> Your reply had nothing to do with your
>> previous incorrect assertion so I pointed that out.
>
> Your pointing out has nothing to do with my posting. But may be you
> have not understood it correctly: I said that the limit of
> cardinalities in mathematics is oo. The limit of the same
> cardinalities in set theory is 0.
No it isn't.
> Am Montag, 2. Dezember 2013 13:15:51 UTC+1 schrieb Ben Bacarisse:
>> Your reply had nothing to do with your
>> previous incorrect assertion so I pointed that out.
>
> Your pointing out has nothing to do with my posting. But may be you
> have not understood it correctly: I said that the limit of
> cardinalities in mathematics is oo. The limit of the same
> cardinalities in set theory is 0.
You:
|| Let S_n be the set {2, 4, 6, ..., 2n}.
|| Let M_n be the set of humers in S_n that are larger than Card(S_n).
||
|| The sequence Card(M_n) has limit oo.
|| The sequence Card(M_n) has limit oo.
|| Only the notion of completed infinity supplies a cardinal number
|| aleph_0 of the limit set of all even numbers that is larger than every
|| finite cardinal number such that the above limit is 0
me:
|| aleph_0 of the limit set of all even numbers that is larger than every
|| finite cardinal number such that the above limit is 0
| To be concrete about it, Card(M_n) is [(n+1) / 2] using [] to mean the
| floor function. The sequence [(n+1) / 2] has limit oo (or is
| unbounded -- take you pick) in mathematics as well as in WMathematics.
The limit of the sequence of cardinalities in question, Card(M_n), is
| floor function. The sequence [(n+1) / 2] has limit oo (or is
| unbounded -- take you pick) in mathematics as well as in WMathematics.
not zero in anything but your peculiar misconception of what set theory
says. If you want to argue with someone who says that "the above limit
is 0", you have to find one first. Currently you are arguing with
yourself.
--
Ben.
Virgil
Dec 2, 2013, 4:40:55 PM12/2/13
to
In article <f0a7ae3a-a7a3-412a...@googlegroups.com>,
Note that a non-empty subset of any well-ordered set that is not
actually infinite must have both first and a last member.
So in WM's wild weird world of WMytheology every non-empty ordered set
must have a specific and unchangeable last member.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> > In article <66e51715-2199-475b...@googlegroups.com>,
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> > > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
> > > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT
> > > > mention
> > > > infininty (OR FINITUDE) AT ALL
,
> > > That means the proof is cheating since ZF needs completed it everywhere.
> > What is "completed it" supposed to mean?
> > > Without completed infinity we can prove that the cardinalities of the
> > > initial
> > > segments of the even natural numbers
> > > |{2}| = 1
> > > |{2, 4}| = 2
> > > |{2, 4, 6}| = 3
> > > ...
> > > are never larger than all elements of the initial segments.
> > Being larger than all elements of the initial segments means being
> > actually infinite
> I was just explaining this to GG. Thanks for support.
But only WM has any problem with sets being actually infinite.
> Am Freitag, 29. November 2013 23:15:33 UTC+1 schrieb Virgil:
> > In article <66e51715-2199-475b...@googlegroups.com>,
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> > > Am Freitag, 29. November 2013 02:14:37 UTC+1 schrieb George Greene:
> > > > The proof of Cantor's Theorem in ZF is VERY SHORT and it does NOT
> > > > mention
> > > > infininty (OR FINITUDE) AT ALL
,
> > > That means the proof is cheating since ZF needs completed it everywhere.
> > What is "completed it" supposed to mean?
> > > Without completed infinity we can prove that the cardinalities of the
> > > initial
> > > segments of the even natural numbers
> > > |{2}| = 1
> > > |{2, 4}| = 2
> > > |{2, 4, 6}| = 3
> > > ...
> > > are never larger than all elements of the initial segments.
> > Being larger than all elements of the initial segments means being
> > actually infinite
> I was just explaining this to GG. Thanks for support.
Note that a non-empty subset of any well-ordered set that is not
actually infinite must have both first and a last member.
So in WM's wild weird world of WMytheology every non-empty ordered set
must have a specific and unchangeable last member.
--
Virgil
Dec 2, 2013, 4:49:18 PM12/2/13
to
In article <3ce044cd-c8ca-4b60...@googlegroups.com>,
It is the definition of the standard field of real numbers which forces
the existence of more numbers than can be individually defined.
But they don't. Every
> application of mathematics, including Cantor's argument, on defineable
> numbers produce defineable numbers.
Cantor's argument shows that no counting/listing of real numbers can be
complete, but WM claims that one can have a complete litsing/counting of
individual number definitions, so it is WM who is causing his own
problems.
> Application of the diagonal argument on
> undefineable numbers must necessarily fail.
Even the supposedly defineable ones are too many to count.
>
> > So far, WM has not have any definition of a real number system,
>
> Another error. Read my book
I wouldn't touch that book with a ten foot slav.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Montag, 2. Dezember 2013 09:42:05 UTC+1 schrieb Virgil:
> > In article <bf894613-7815-4f4e...@googlegroups.com>,
> >
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> >
> >
> >
> > > Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
> >
> > > > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
> >
> > > >
> >
> > > > > Yes. But "the reals that cannot be defined in English language"
> >
> > > >
> >
> > > > > are clearly new.
> >
> > > >
> >
> > > >
> >
> > > >
> >
> > > > They are NOT clearly new.
> >
> > > >
> >
> > > > It has always been obvious that there have to be undefinable reals.
> >
> > >
> >
> > > Mathematics in English-speaking countries is done in English.
> >
> > Usually!
> >
> >
> >
> > > Undefinable
> >
> > > reals in English (or any other language spoken by men) do not belong to
> >
> > > mathematics.
> >
> >
> >
> > Individually, they are inacessible, but collectively they must exist,
> >
>
> This "must" is erroneously concluded from the idea that Cantor's arguments
> force the existence of undefineable numbers.
Wrong!
> Am Montag, 2. Dezember 2013 09:42:05 UTC+1 schrieb Virgil:
> > In article <bf894613-7815-4f4e...@googlegroups.com>,
> >
> > WM <wolfgang.m...@hs-augsburg.de> wrote:
> >
> >
> >
> > > Am Montag, 2. Dezember 2013 04:20:41 UTC+1 schrieb George Greene:
> >
> > > > On Sunday, December 1, 2013 4:25:27 PM UTC-5, WM wrote:
> >
> > > >
> >
> > > > > Yes. But "the reals that cannot be defined in English language"
> >
> > > >
> >
> > > > > are clearly new.
> >
> > > >
> >
> > > >
> >
> > > >
> >
> > > > They are NOT clearly new.
> >
> > > >
> >
> > > > It has always been obvious that there have to be undefinable reals.
> >
> > >
> >
> > > Mathematics in English-speaking countries is done in English.
> >
> > Usually!
> >
> >
> >
> > > Undefinable
> >
> > > reals in English (or any other language spoken by men) do not belong to
> >
> > > mathematics.
> >
> >
> >
> > Individually, they are inacessible, but collectively they must exist,
> >
>
> This "must" is erroneously concluded from the idea that Cantor's arguments
> force the existence of undefineable numbers.
It is the definition of the standard field of real numbers which forces
the existence of more numbers than can be individually defined.
But they don't. Every
> application of mathematics, including Cantor's argument, on defineable
> numbers produce defineable numbers.
complete, but WM claims that one can have a complete litsing/counting of
individual number definitions, so it is WM who is causing his own
problems.
> Application of the diagonal argument on
> undefineable numbers must necessarily fail.
>
> > So far, WM has not have any definition of a real number system,
>
> Another error. Read my book
--
Virgil
Dec 2, 2013, 4:52:28 PM12/2/13
to
Virgil
Dec 2, 2013, 4:56:31 PM12/2/13
to
In article <055c3b13-8deb-4c1b...@googlegroups.com>,
Ben's pointing out.
> But may be you have not
> understood it correctly: I said that the limit of cardinalities in
> mathematics is oo. The limit of the same cardinalities in set theory is 0.
So that in WM's wild weird world of WMytheology 0 = oo.
Elsewhere a sequence of cardinalities will have at most one limit.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:
> Am Montag, 2. Dezember 2013 13:15:51 UTC+1 schrieb Ben Bacarisse:
> > Your reply had nothing to do with your
> >
> > previous incorrect assertion so I pointed that out.
>
> Your pointing out has nothing to do with my posting.
Wm s usual has things backwards, WM's posting has little to do with
> Am Montag, 2. Dezember 2013 13:15:51 UTC+1 schrieb Ben Bacarisse:
> > Your reply had nothing to do with your
> >
> > previous incorrect assertion so I pointed that out.
>
> Your pointing out has nothing to do with my posting.
Ben's pointing out.
> But may be you have not
> understood it correctly: I said that the limit of cardinalities in
> mathematics is oo. The limit of the same cardinalities in set theory is 0.
Elsewhere a sequence of cardinalities will have at most one limit.
--
Virgil
Dec 2, 2013, 8:17:09 PM12/2/13
to
WM wrote
there must for each n in |N a term in the seqeunce in which the
cardinaity is less than n.
So for which terms in the sequnce is the cardinality of that term even
finiye, much less less than , say, a googleplex?
--
> > Your pointing out has nothing to do with my posting. But may be you
> > have not understood it correctly: I said that the limit of
> > cardinalities in mathematics is oo. The limit of the same
> > cardinalities in set theory is 0.
In order for the limit of such a sequence of cardiinaities to be zero
> > have not understood it correctly: I said that the limit of
> > cardinalities in mathematics is oo. The limit of the same
> > cardinalities in set theory is 0.
there must for each n in |N a term in the seqeunce in which the
cardinaity is less than n.
So for which terms in the sequnce is the cardinality of that term even
finiye, much less less than , say, a googleplex?
--
Ross A. Finlayson
Dec 2, 2013, 8:59:23 PM12/2/13
to
WM
Dec 3, 2013, 1:52:14 AM12/3/13
to
It is not. In mathematics we can calculate the limit. The number of even numbers that are larger than the number of numbers of the set is infinite. That proves that the assertion of set theory differs from mathematics. Simple as that.
Regards, WM
Virgil
Dec 3, 2013, 3:21:02 AM12/3/13
to
In article <0afca7ea-48dd-4f34...@googlegroups.com>,
WM <wolfgang.m...@hs-augsburg.de> wrote:> Am Dienstag, 3.
"google" equal to 10^100.
And an even larger number, 10^google, called a googleplex.
> It is not. In mathematics we can calculate the limit.
Wm cannot do anything in mathematics until he exits the anti-mthematical
world of WM's wild weird world of WMytheology.
> The number of even
> numbers that are larger than the number of numbers of the set is infinite.
> That proves that the assertion of set theory differs from mathematics.
Since set theory is a part of the world of mathematics everywhere
outside of WM's wild weird world of WMytheology, and a great deal of
mathematics can be done within set theory, WM's claim is just more
evidence of how totally WM has exiled himself from actuall mathematics.
--
WM <wolfgang.m...@hs-augsburg.de> wrote:> Am Dienstag, 3.
Dezember 2013 02:17:09 UTC+1 schrieb Virgil:
> > WM wrote
> > > > Your pointing out has nothing to do with my posting. But may be you
> > > > have not understood it correctly: I said that the limit of
> > > > cardinalities in mathematics is oo. The limit of the same
> > > > cardinalities in set theory is 0.
> > In order for the limit of such a sequence of cardinaities in set theory to be zero
> > WM wrote
> > > > Your pointing out has nothing to do with my posting. But may be you
> > > > have not understood it correctly: I said that the limit of
> > > > cardinalities in mathematics is oo. The limit of the same
> > > > cardinalities in set theory is 0.
> > there must for each n in |N a term in the seqeunce in which the
> > cardinaity is less than n.
> > So for which terms in the sequnce is the cardinality of that term even
> > finite, much less less than , say, a googleplex?
> > cardinaity is less than n.
> > So for which terms in the sequnce is the cardinality of that term even
> Google has entered the business of big numbers too?
Before there was a business named google there was a number named
"google" equal to 10^100.
And an even larger number, 10^google, called a googleplex.
> It is not. In mathematics we can calculate the limit.
world of WM's wild weird world of WMytheology.
> The number of even
> numbers that are larger than the number of numbers of the set is infinite.
> That proves that the assertion of set theory differs from mathematics.
outside of WM's wild weird world of WMytheology, and a great deal of
mathematics can be done within set theory, WM's claim is just more
evidence of how totally WM has exiled himself from actuall mathematics.
--
Ben Bacarisse
Dec 3, 2013, 10:32:57 AM12/3/13
to
Virgil <vir...@ligriv.com> writes:
> In article <0afca7ea-48dd-4f34...@googlegroups.com>,
> WM <wolfgang.m...@hs-augsburg.de> wrote:> Am Dienstag, 3.
> Dezember 2013 02:17:09 UTC+1 schrieb Virgil:
>> > WM wrote
>> > > > Your pointing out has nothing to do with my posting. But may be you
>> > > > have not understood it correctly: I said that the limit of
>> > > > cardinalities in mathematics is oo. The limit of the same
>> > > > cardinalities in set theory is 0.
>
>> > In order for the limit of such a sequence of cardinaities in set theory to be zero
>> > there must for each n in |N a term in the seqeunce in which the
>> > cardinaity is less than n.
>
>> > So for which terms in the sequnce is the cardinality of that term even
>> > finite, much less less than , say, a googleplex?
>> Google has entered the business of big numbers too?
>
> Before there was a business named google there was a number named
> "google" equal to 10^100.
> And an even larger number, 10^google, called a googleplex.
The number was called a googol. Google is a pun on that older name.
> In article <0afca7ea-48dd-4f34...@googlegroups.com>,
> WM <wolfgang.m...@hs-augsburg.de> wrote:> Am Dienstag, 3.
> Dezember 2013 02:17:09 UTC+1 schrieb Virgil:
>> > WM wrote
>> > > > Your pointing out has nothing to do with my posting. But may be you
>> > > > have not understood it correctly: I said that the limit of
>> > > > cardinalities in mathematics is oo. The limit of the same
>> > > > cardinalities in set theory is 0.
>
>> > In order for the limit of such a sequence of cardinaities in set theory to be zero
>> > there must for each n in |N a term in the seqeunce in which the
>> > cardinaity is less than n.
>
>> > So for which terms in the sequnce is the cardinality of that term even
>> > finite, much less less than , say, a googleplex?
>> Google has entered the business of big numbers too?
>
> Before there was a business named google there was a number named
> "google" equal to 10^100.
> And an even larger number, 10^google, called a googleplex.
The name of Google's headquarters -- the Googleplex -- is a similar pun.
Us computer folk sure know how to have a laugh.
<snip>
--
Ben.
WM
Dec 3, 2013, 11:22:26 AM12/3/13
to
Am Dienstag, 3. Dezember 2013 09:21:02 UTC+1 schrieb Virgil:
>
> > > So for which terms in the sequnce is the cardinality of that term even
>
> > > finite, much less less than , say, a googleplex?
>
> > Google has entered the business of big numbers too?
>
>
>
>
>
> Before there was a business named google there was a number named
>
> "google" equal to 10^100.
>
> And an even larger number, 10^google, called a googleplex.
>
>
http://en.wikipedia.org/wiki/Googol
>
> > > So for which terms in the sequnce is the cardinality of that term even
>
> > > finite, much less less than , say, a googleplex?
>
> > Google has entered the business of big numbers too?
>
>
>
>
>
> Before there was a business named google there was a number named
>
> "google" equal to 10^100.
>
> And an even larger number, 10^google, called a googleplex.
>
>
http://www.hs-augsburg.de/~mueckenh/GU/GU01c.PPT#300,19,Folie 19
>
>
> > It is not. In mathematics we can calculate the limit.
>
>
>
> > It is not. In mathematics we can calculate the limit.
>
>
> > The number of even
>
> > numbers that are larger than the number of numbers of the set is infinite.
>
> > That proves that the assertion of set theory differs from mathematics.
>
>
>
> Since set theory is a part of the world of mathematics everywhere
the everywhere people should explain this contradiction.
>
> > numbers that are larger than the number of numbers of the set is infinite.
>
> > That proves that the assertion of set theory differs from mathematics.
>
>
>
> Since set theory is a part of the world of mathematics everywhere
Regards, WM
Virgil
Dec 3, 2013, 6:33:01 PM12/3/13
to
In article
<0.5936d349f45299dee398.2013...@bsb.me.uk>,
My spelling error!
I suppose seeing "Google" so often drove the "googol" and "googolplex"
spellings out of my mind.
--
<0.5936d349f45299dee398.2013...@bsb.me.uk>,
I suppose seeing "Google" so often drove the "googol" and "googolplex"
spellings out of my mind.
--
WM
Dec 4, 2013, 1:44:20 AM12/4/13
to
Regards, WM
Virgil
Dec 4, 2013, 4:16:37 AM12/4/13
to
Ross A. Finlayson
Dec 16, 2013, 11:18:35 PM12/16/13
to
you, for your standard, it's that an infinite set doesn't.
Or rather, that's possibly concrete in Muckenheim's infinite, and compact as
infinite, infinite.
Then the Big Infinity is the universe and world and everything around us, the
little infinities are just usual scalar ordinal infinities that are all in our
heads. But, the continuum of reals would go to the Big Infinity, you see, as
the continuum, as to how it is. It doesn't, it's what it is. At the same time
it is as to why it's so for example, usual: the real numbers are a continuum
and all of continuum analysis uses that.
"Then just as vectors and tensors are as of their form, then cardinal arithmetic
and cardinals, _of their form_, would have for their measure as sets of reals
topologically, then systems as to conditions that what maintain cardinal
arithmetic, would work out to have some application, thus constructive form.
This is where the construction in the countable ordinals is usually paramount
in the definitive of the constructible. "
Regards, Ross Finlayson
Virgil
Dec 17, 2013, 12:06:35 AM12/17/13
to
In article <250f6065-e213-412d...@googlegroups.com>,
--
"Ross A. Finlayson" <ross.fi...@gmail.com> wrote:
> On Monday, December 2, 2013 1:40:55 PM UTC-8, Virgil wrote:
> >
> > Note that a non-empty subset of any well-ordered set that is not
> > actually infinite must have both first and a last member.
> > So in WM's wild weird world of WMytheology every non-empty ordered set
> > must have a specific and unchangeable last member.
> > --
> "Note that a non-empty subset of any well-ordered set that is not
> actually infinite must have both first and a last member.
> So in WM's wild weird world of WMytheology every non-empty ordered set
> must have a specific and unchangeable last member. "
Why duplicate what has been already posted?
> On Monday, December 2, 2013 1:40:55 PM UTC-8, Virgil wrote:
> >
> > Note that a non-empty subset of any well-ordered set that is not
> > actually infinite must have both first and a last member.
> > So in WM's wild weird world of WMytheology every non-empty ordered set
> > must have a specific and unchangeable last member.
> > --
> "Note that a non-empty subset of any well-ordered set that is not
> actually infinite must have both first and a last member.
> So in WM's wild weird world of WMytheology every non-empty ordered set
> must have a specific and unchangeable last member. "
--
Ross A. Finlayson
May 19, 2014, 1:14:46 AM5/19/14
to
On 11/27/2013 12:18 AM, Ross A. Finlayson wrote:
> On Tuesday, November 26, 2013 11:30:54 PM UTC-8, fom wrote:
>> On 11/27/2013 12:43 AM, Virgil wrote:
>>
>>> Note that for standard sets of reals as defined below,
>>
>>> ...
>>
>>
>> Elsewhere, FredJefferies provided this link:
>>
>>
>>
>> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
>>
>>
>>
>> It is a nice discussion and adds to what WM
>>
>> cannot prove.
>
> Heh,
>
> Heh: the Institute for Advanced Study.
>
> "It may happen that the reals contain nilpotent infinitesimals, which validate
> the 17th century calculations that physicists still use because, luckily, they
> did not subscribe entirely to the ϵδ-dogma of analysis."
>
> That may not be exactly so, where, physicists do subscribe to delta-epsilonics, and, they do subscribe to Leibniz' notation.
>
> https://groups.google.com/forum/#!msg/sci.math/4RBNLj-Q4Mo/hsgK3usvIAcJ
>
> On Tuesday, November 26, 2013 11:30:54 PM UTC-8, fom wrote:
>> On 11/27/2013 12:43 AM, Virgil wrote:
>>
>>> Note that for standard sets of reals as defined below,
>>
>>> ...
>>
>>
>> Elsewhere, FredJefferies provided this link:
>>
>>
>>
>> http://math.andrej.com/2012/10/03/am-i-a-constructive-mathematician/#more-1247
>>
>>
>>
>> It is a nice discussion and adds to what WM
>>
>> cannot prove.
>
> Heh,
>
> "It may happen that the reals are in 1-1 correspondence with a subset
> of the natural numbers, while at the same time they form an uncountable set."
> -Andrej Bauer, visiting the Institute for Advanced Study
> "It may happen that the reals are in 1-1 correspondence with a subset
> of the natural numbers, while at the same time they form an uncountable set."
>
> Heh: the Institute for Advanced Study.
>
> "It may happen that the reals contain nilpotent infinitesimals, which validate
> the 17th century calculations that physicists still use because, luckily, they
> did not subscribe entirely to the ϵδ-dogma of analysis."
>
> That may not be exactly so, where, physicists do subscribe to delta-epsilonics, and, they do subscribe to Leibniz' notation.
>
> https://groups.google.com/forum/#!msg/sci.math/4RBNLj-Q4Mo/hsgK3usvIAcJ
>
George Greene
May 22, 2014, 4:34:45 PM5/22/14
to
On Monday, May 19, 2014 1:14:46 AM UTC-4, Ross A. Finlayson wrote:
> > "It may happen that the reals are in 1-1 correspondence with a subset
>
> > of the natural numbers, while at the same time they form an uncountable set."
>
> > -Andrej Bauer, visiting the Institute for Advanced Study
No, that MAY NOT happen, and the thing that forbids it from happening is not,
> > "It may happen that the reals are in 1-1 correspondence with a subset
>
> > of the natural numbers, while at the same time they form an uncountable set."
>
> > -Andrej Bauer, visiting the Institute for Advanced Study
as Bauer wrongly implies, a "dogma" -- it is a DEFINITION. At best he is a little unclear on the difference between the two.
I have already engaged at length but he already said he wouldn't.
"Mathematical relativism" is not going to get him out of the pickle of "a 1-1 correspondence between a set and N" CONSTITUTING, BY DEFINITION, a counting of the set (and thereby precluding its being reasonably referred to as UNcountable).
Ross A. Finlayson
May 22, 2014, 7:25:21 PM5/22/14
to
numbers are, in a set theory,
these real numbers of our analysis
are with methods that treat
the real numbers as if they have
properties according to their
definition. According to "the"
definition(s) of the real numbers,
they have these properties.
Here, it's a comment, that Andrej
Bauer, a fellow at the Institute
for Advanced Study, i.e., affiliated
with Goedel, Einstein, and
modern dogma, notes this as so or
variously.
Then, though it's rough on him, and
as he doesn't much directly
defend the statement beyond not
retracting it, it only matters
for people who care about
approbation of authority, the
mathematics more directly of course
would be there irrelevant to the
institution.
At least, that's what I've written
here.
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