The Mathematical Substructure Underlying Quantum Field Theory, and Why I Like Anthony Zee's Book so Much
Jay R. Yablon
I have often made reference to the textbook by Anthony Zee, Quantum
Field Theory in a Nutshell. One of the things I find very attractive
about this book is the way in which he deals very simply with the
mathematical substructure which underlies quantum field theory. I
specifically have in mind Appendix 1 in Chapter 1.2 and his overall
Appendix A. At
http://jayryablon.files.wordpress.com/2009/11/zee-qft-substructure.pdf,
I have posted these two excerpts from Zee's book, as a backdrop for some
further rooting through the mathematical substructure of quantum field
theory, which I have also posted at:
In particular, I have offered some thoughts about resolving the "measure
problem" that shows up in efforts to quantize gravitation, and about how
using the Dirac delta as the D-->0 limit of a Gaussian integral provides
a possible connection between the mathematical substructures of gauge
theory and gravitational theory.
That is what I so much like about Zee's book: he does an excellent job
in laying out the mathematical substructure of QFT, rather than try to
calculate out everything under the sun, and provides an extremely
penetrating way to look at how everything ultimately hangs together.
I look forward to discussing all of this.
Thanks,
Jay
____________________________
Jay R. Yablon
Email: jya...@nycap.rr.com
co-moderator: sci.physics.foundations
Weblog: http://jayryablon.wordpress.com/
Web Site: http://home.roadrunner.com/~jry/FermionMass.htm
Koobee Wublee
> I have often made reference to the textbook by Anthony Zee, Quantum
> Field Theory in a Nutshell. One of the things I find very attractive
> about this book is the way in which he deals very simply with the
> mathematical substructure which underlies quantum field theory.
What you have quoted does not connect what you are writing. <shrug>
> I
> specifically have in mind Appendix 1 in Chapter 1.2 and his overall
> Appendix A. At
> http://jayryablon.files.wordpress.com/2009/11/zee-qft-substructure.pdf,
> I have posted these two excerpts from Zee's book, as a backdrop for some
> further rooting through the mathematical substructure of quantum field
> theory, which I have also posted at:
It remains your very wishful thinking. <shrug>
> http://jayryablon.files.wordpress.com/2009/11/zee-and-the-mathematica...
It looks like you are still having a hard on on carrying out the
integration of the Einstein-Hilbert action. <shrug>
> In particular, I have offered some thoughts about resolving the "measure
> problem"
There is no measurement problem but understanding problem. <shrug>
> that shows up in efforts to quantize gravitation,
There is no such thing as quantize gravitation. <shrug>
> and about how
> using the Dirac delta as the D-->0 limit of a Gaussian integral provides
> a possible connection between the mathematical substructures of gauge
> theory and gravitational theory.
The gauge thing was a brainchild of Rosen who was so chickshit about
publishing that nonsense under his sole name but had to drag Einstein
the nitwit, the plagiarist, and the liar into co-authorship to gain
acceptance. <shrug>
> That is what I so much like about Zee's book: he does an excellent job
> in laying out the mathematical substructure of QFT, rather than try to
> calculate out everything under the sun, and provides an extremely
> penetrating way to look at how everything ultimately hangs together.
What you have quoted are merely the mechanical solving of certain
integrals. They don't offer any physical insights. Tell me if you
are not intoxicated when writing this post.
> I look forward to discussing all of this.
I doubt your sincerity. You mean you are looking for emotional
support from certain Einstein Dingleberries. <shrug>
Are you still "buddy" "buddy" with that senile salesman who thinks he
knows everything after staring at a supposedly pertinent book for over
40 years and still is clueless as to what a Lagrangian is?
Does Orwellian come in mind?
** FAITH IS THEORY
** LYING IS TEACHING
** NITWIT IS GENIUS
** OCCULT IS SCIENCE
** PARADOX IS KOSHER
** BULLSHIT IS TRUTH
** BELIEVING IS LEARNING
** IGNORANCE IS KNOWLEDGE
** MYSTICISM IS WISDOM
** CONJECTURE IS REALITY
** PLAGIARISM IS CREATIVITY
** MATHEMAGICS IS MATHEMATICS
<shrug>
Ken S. Tucker
Must you always be so emotional, it's a delicate subject,
can you please please control your hormones, use pills.
Ken
Jay R. Yablon
of the last week can be put into a one-equation, one page synopsis,
linked below:
http://jayryablon.files.wordpress.com/2009/11/one-equation-synopsis.pdf
I would like to get agreement that this one equation, which I would
maintain is a *mathematical identity*, is correct as a mathematical
matter.
Then, we can talk about the possible physics applications.
For those of you who have followed me "in labor" this past week, that
may not have been a pretty thing. Here, I think / hope, is the "baby."
master1729
>
> ** FAITH IS THEORY
> ** LYING IS TEACHING
> ** NITWIT IS GENIUS
> ** OCCULT IS SCIENCE
> ** PARADOX IS KOSHER
> ** BULLSHIT IS TRUTH
> ** BELIEVING IS LEARNING
> ** IGNORANCE IS KNOWLEDGE
> ** MYSTICISM IS WISDOM
> ** CONJECTURE IS REALITY
> ** PLAGIARISM IS CREATIVITY
> ** MATHEMAGICS IS MATHEMATICS
>
i love that :)
reminds me of ZFC and the alike
faith (in axioms) = theory
lying (axioms are true and the best) = teaching
ZFC teacher = genius
occult beliefs = science
ZFC paradoxes ( many examples ) = kosher
bullshit = truth
believing ( axioms ) = learning
etc
regards
tommy1729
A
> Kobee Wublee wrote :
>
>
>
> > ** FAITH IS THEORY
> > ** LYING IS TEACHING
> > ** NITWIT IS GENIUS
> > ** OCCULT IS SCIENCE
> > ** PARADOX IS KOSHER
> > ** BULLSHIT IS TRUTH
> > ** BELIEVING IS LEARNING
> > ** IGNORANCE IS KNOWLEDGE
> > ** MYSTICISM IS WISDOM
> > ** CONJECTURE IS REALITY
> > ** PLAGIARISM IS CREATIVITY
> > ** MATHEMAGICS IS MATHEMATICS
>
> i love that :)
>
> reminds me of ZFC and the alike
>
> faith (in axioms) = theory
> lying (axioms are true and the best) = teaching
> ZFC teacher = genius
> occult beliefs = science
> ZFC paradoxes ( many examples ) = kosher
What are some examples of paradoxes in ZFC?
glird
> For those who like conciseness, everything I come up with from my posts of the last week can be put into a one-equation, one page synopsis, linked below:
http://jayryablon.files.wordpress.com/2009/11/one-equation-synopsis.pdf
I would like to get agreement that this one equation, which I would
maintain is a *mathematical identity*, is correct as a mathematical
matter. >
I looked at the page and found in it this:
L_EH = R/2k + L_M.
If you want to be understood, Jay, please define what each symbol in
that equation represents.
To me, H denotes Plank's quantum of action, and EH may or may not
denote his quantum of energy. What does R, k, L and M denote?
glird
Ken S. Tucker
Not sure either, then R = 2k( L_EH - L_M).
Wherein the L are (sorta) defined as relative tensors.
Abbreviate to R = L_goth
So I suppose "R" is a relative tensor??
Don't know.
Ken
Jay R. Yablon
"glird" <gl...@aol.com> wrote in message
news:12c68a48-28b2-4d9d...@g1g2000vbr.googlegroups.com...
glird
No, Glird, this is the Einstein-Hilbert Lagrangian. See
http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action.
L is Lagrangian. Always is.
EH = Einstein-Hilbert. Thought that would be apparent.
R= Ricci Scalar.
k = the kappa from the Einstein equation.
L_M is the matter Lagrangian. Always is.
Maybe I tried to be too concise.
Jay.
Jay R. Yablon
"Ken S. Tucker" <dyna...@vianet.on.ca> wrote in message
news:0183fd93-1644-41d1...@z10g2000prh.googlegroups.com...
Ken see reply to Glird. See also
http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action. Jay
Ken S. Tucker
> "Ken S. Tucker" <dynam...@vianet.on.ca> wrote in messagenews:0183fd93-1644-41d1...@z10g2000prh.googlegroups.com...
Thanks Jay, I'm on learning mode where Relative tensors
are concerned.
Regards
Ken S. Tucker
Koobee Wublee
> "glird" <gl...@aol.com> wrote in message
> > I looked at the page and found in it this:
> > L_EH = R/2k + L_M.
>
> L is Lagrangian. Always is.
>
> EH = Einstein-Hilbert. Thought that would be apparent.
>
> R= Ricci Scalar.
>
> k = the kappa from the Einstein equation.
It is merely a constant. It can be derived through satisfying
boundary conditions. <shrug>
> L_M is the matter Lagrangian. Always is.
It should merely be
** rho c^2
Where
** rho = Mass density or measured in kg/m^3
Thus, it should be independent of any metric.
> Maybe I tried to be too concise.
It can be a sign of covering weaknesses such as that angry, bitter,
and senile salesman who after staring at a textbook and still cannot
understand the basic principles. <shrug>
That so called Lagrangian (L = (K R + rho c^2) sqrt(-det(g))) was sewn
together like Dr. Frankenstein did to that monster. It is now alive
and wreaking havoc. <shrug>
Koobee Wublee
> "Ken S. Tucker" wrote:
> > Not sure either, then R = 2k( L_EH - L_M).
>
> > Wherein the L are (sorta) defined as relative tensors.
>
> > Abbreviate to R = L_goth
>
> > So I suppose "R" is a relative tensor??
> > Don't know.
>
> http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action
Jay, you certainly have a lot of patience for someone who stared at a
GR textbook for over 40 years and still does not know what a Ricci
scalar is. This is no under-statement. Sorry to be so blunt, but it
is true.
Anyhow, notice the Wikipedia article is under dispute. That is
supposed to be the very foundation of GR and still under dispute.
Wow, I smell fraud.
Notice the Lagrangian of mass is just thrown in without any
justification. Notice the Ricci scalar is just thrown in without any
explanation. Notice your sqrt(-det(g)) is thrown in grossly
misinterpreting what invariance is all about. There is actually no
invariance in the claim you made. The units do not even match. How
can you continue on this nonsense without reconciling the most basic
mathematical axiom of unit analysis? In reality, one banana plus one
apple does not equate to two bananas or two apples. This very
elementary axiom is eluding the self-styled physicists, and you too?
You have done more than any other self-styled physicists by digging
into the basics of GR, and yet are you equally mystified in elementary-
school level logics as the self-styled physicists are?
Do you know how the Riemann curvature tensor is derived? Almost all
self-styled physicists don't. <shrug>
Oh, one more thing. How do you reconcile the Einstein-Hilbert action
being null with the Euclidean coordinate system under flat spacetime?
The action is already zero. How can you extremize it and find a path
of trajectory and justifying that it is indeed a straight line through
local spacetime? The mathematics with the always-null action always
indicates unambiguous path for any trajectory. This is totally beyond
any common sense and known observed physics laws. <shrug>
Ken S. Tucker
> On Nov 24, 3:21 pm, "Jay R. Yablon" wrote:
>
> > "Ken S. Tucker" wrote:
> > > Not sure either, then R = 2k( L_EH - L_M).
>
> > > Wherein the L are (sorta) defined as relative tensors.
>
> > > Abbreviate to R = L_goth
>
> > > So I suppose "R" is a relative tensor??
> > > Don't know.
>
> >http://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action
>
> Jay, you certainly have a lot of patience for someone who stared at a
> GR textbook for over 40 years and still does not know what a Ricci
> scalar is. This is no under-statement. Sorry to be so blunt, but it
> is true.
Miss Woobee, (and Jay and all)...
Jay is an encouraging friend to me and others, and
I'm certainly studential on many aspects of math &
physics.
I hit some blank paper with a pencil and found a
mathematical means to justify Jay's result, and
I find the physics to be ok too, and I might do
a post on that if my confidence improves.
Regards
Ken
...
debaser
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The true grounds of ecclesiasticall regiment : set forth in
a ...
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though ^egts^ in quantum Reges fervium I>ftf,as Saint.
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(hail not now conioync,and UDitc them again out ofa ...... THenext
Argument therefore of truer force is this : ...
https://qspace.library.queensu.ca/.../1974/.../truegroundsofecc00parkuoft.pdf
6.
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In no one of the sciences is thb truer than in medicine. ......
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7.
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File Format: PDF/Adobe Acrobat
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Results 1 - 10 for zub truer quantum theory. (0.11 seconds)
1.
» Best Of 2009 1/2: Albums Of The Year So Far :: The Rawking ...
... may be worn out with Animal Collective by now, but on MPP,
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Zub" · Tour Dates: The Parson Red Heads ...
www.rawkblog.net/2009/07/best-of-2009-12-albums-of-the-year-so-far/
2.
Full text of "The true grounds of ecclesiasticall regiment :
set ...
zUb (hail not now conioync,and UDitc them again out of a fond
pragmacicall ..... herein the (7^) the nobler office '.quantum qui
ftAvem temper 4t, anteit-. ... THe next Argument therefore of truer
force is this : There can be no office ...
www.archive.org/stream/.../truegroundsofecc00parkuoft_djvu.txt
3.
Jennings-Rosicrucians Rites and Mysteries
... even of his little quantum of truth, being perverted. ......
and the whole theory requires a key of explanation to render it
intelligible ; which ...... seeming a further and truer, because a
more solid and a more sensible, glory. ...... zub The “Prophets,” or
The “Gospel” (Woman). W. Because it came through —. ...
www.scribd.com/doc/16677110/JenningsRosicrucians-Rites-and-Mysteries
4.
Full text of "A BOOK OF QUOTATIONS PROVERBS AND HOUSEHOLD WORDS"
There is no truer truth obtainable By man, than comes of
music. ..... Ib. A. thing may look specious in theory, and yet be
ruinous in practice; a thing may look evil in ..... 1 waive the
quantum o the sin, The hazard of concealing ; But, och ! it hardens
a ...... I. ZUb. Elephants endorsed with towers. I. 329. ...
www.archive.org/stream/.../bookofquotations000382mbp_djvu.txt
5.
[PDF]
The true grounds of ecclesiasticall regiment : set forth in
a ...
File Format: PDF/Adobe Acrobat
though ^egts^ in quantum Reges fervium I>ftf,as Saint.
Auguflirje fayes , yet in quantum pti Reges^zhcy fcrve ...... zUb
(hail not now conioync,and UDitc them again out ofa ...... THenext
Argument therefore of truer force is this : ...
https://qspace.library.queensu.ca/.../1974/.../truegroundsofecc00parkuoft.pdf
6.
Full text of "Makers of modern medicine"
In no one of the sciences is thb truer than in medicine. ......
cum Baglivio dioturum esse spero: O quantum difficile est diagnoeoere
moiboe ...... His cell theory has now become the 256 MAKERS OF MODERN
MEDICINE cell ...... »CMi:§^ soBtL ^ 'tt^ ansszan. fr is^ • IE X.
Snaenmir <^ iiesiaf jb msor IT ZUB^ M^jwyyt ...
www.archive.org/stream/.../makersmodernmed01walsgoog_djvu.txt
7.
Jennings - The Rosicrucians (4th edition)
... even of his little quantum of truth, being perverted. ......
To recur for a moment to the theory of human sacrifices which once
largely prevailed. ...... in seeming a further and truer, because a
more solid and a more sensible, glory. ...... }yk!y / 'The Law' (Man)
Boaz (Left Pillar) zub The 'Prophets or The ...
www.scribd.com/doc/3013648/Jennings-The-Rosicrucians-4th-edition
8.
[PDF]
THE ROSICRUCIANS
File Format: PDF/Adobe Acrobat
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1 2 Next
glird
> "glird" wrote
> On Nov 24, 11:31 am "Jay R. Yablon" wrote:
><<< For those who like conciseness, everything I come up with from my posts of the last week can be put into a one-equation, one page synopsis, linked below:
http://jayryablon.files.wordpress.com/2009/11/one-equation-
synopsis.pdf
I would like to get agreement that this one equation, which I would
maintain is a *mathematical identity*, is correct as a mathematical
matter. >
>
><< I looked at the page and found in it this:
L_EH = R/2k + L_M.
If you want to be understood, Jay, please define what each symbol in
that equation represents.
To me, H denotes Plank's quantum of action, and EH may or may not
denote his quantum of energy. What does R, k, L and M denote? >
>
L is Lagrangian. Always is.
EH = Einstein-Hilbert. Thought that would be apparent. >
I looked, and after awhile found this:
"Hamilton's principle states that the true evolution \mathbf{q}(t)
of a system described by N generalized coordinates \mathbf{q} = \left
( q_{1}, q_{2}, \ldots, q_{N} \right) between two specified states
\mathbf{q}_{1} \ \stackrel{\mathrm{def}}{=}\ \mathbf{q}(t_{1}) and
\mathbf{q}_{2} \ \stackrel{\mathrm{def}}{=}\ \mathbf{q}(t_{2}) at two
specified times t1 and t2 is an extremum (i.e., a stationary point, a
minimum, maximum or saddle point) of the action functional
\mathcal{S}[\mathbf{q}] \ \stackrel{\mathrm{def}}{=}\ \int_{t_{1}}^
{t_{2}} L(\mathbf{q}(t),\dot{\mathbf{q}}(t),t)\, dt
where L(\mathbf{q},\dot{\mathbf{q}},t) is the Lagrangian function for
the system. In other words, any first-order perturbation of the true
evolution results in (at most) second-order changes in \mathcal{S}. It
should be noted that the action \mathcal{S} is a functional, i.e.,
something that takes as its input a function and returns a single
number, a scalar.
"In terms of functional analysis, Hamilton's principle states that
the true evolution of a physical system is the solution of the
functional equation
\frac{\delta \mathcal{S}}{\delta \mathbf{q}(t)}=0
[edit] Euler-Lagrange equations for the action integral
Requiring that the true trajectory \mathbf{q}(t) be a stationary point
of the action functional \mathcal{S} is equivalent to a set of
differential equations for \mathbf{q}(t) (the Euler-Lagrange
equations), which may be derived as follows.
"Let \mathbf{q}(t) represent the true evolution of the system between
two specified states \mathbf{q}_{1} \ \stackrel{\mathrm{def}}{=}\
\mathbf{q}(t_{1}) and \mathbf{q}_{2} \ \stackrel{\mathrm{def}}{=}\
\mathbf{q}(t_{2}) at two specified times t1 and t2, and let \boldsymbol
\varepsilon(t) be a small perturbation that is zero at the endpoints
of the trajectory
\boldsymbol\varepsilon(t_{1}) = \boldsymbol\varepsilon(t_{2}) \
\stackrel{\mathrm{def}}{=}\ 0
To first order in the perturbation \boldsymbol\varepsilon(t), the
change in the action functional \delta\mathcal{S} would be
\delta \mathcal{S} = \int_{t_{1}}^{t_{2}}\; \left[ L(\mathbf{q}+
\boldsymbol\varepsilon,\dot\mathbf{q} +\dot\boldsymbol\varepsilon)- L
(\mathbf{q},\dot\mathbf{q}) \right]dt = \int_{t_{1}}^{t_{2}}\; \left
( \boldsymbol\varepsilon \cdot \frac{\partial L}{\partial \mathbf{q}}
+ \dot\boldsymbol\varepsilon \cdot \frac{\partial L}{\partial \dot
\mathbf{q}} \right)\,dt
where we have expanded the Lagrangian L to first order in the
perturbation \boldsymbol\varepsilon(t).
Applying integration by parts to the last term results in
\delta \mathcal{S} = \left[ \boldsymbol\varepsilon \cdot \frac
{\partial L}{\partial \dot\mathbf{q}}\right]_{t_{1}}^{t_{2}} + \int_{t_
{1}}^{t_{2}}\; \left( \boldsymbol\varepsilon \cdot \frac{\partial L}
{\partial \mathbf{q}} - \boldsymbol\varepsilon \cdot \frac{d}{dt} \frac
{\partial L}{\partial \dot\mathbf{q}} \right)\,dt
The boundary conditions \boldsymbol\varepsilon(t_{1}) = \boldsymbol
\varepsilon(t_{2}) \ \stackrel{\mathrm{def}}{=}\ 0 causes the first
term to vanish \delta \mathcal{S} = \int_{t_{1}}^{t_{2}}\; \boldsymbol
\varepsilon \cdot \left(\frac{\partial L}{\partial \mathbf{q}} - \frac
{d}{dt} \frac{\partial L}{\partial \dot\mathbf{q}} \right)\,dt
"Hamilton's principle requires that this first-order change is zero
for all possible perturbations \boldsymbol\varepsilon(t), i.e., the
true path is a stationary point of the action functional \mathcal{S}
(either a minimum, maximum or saddle point). This requirement can be
satisfied if and only if [snip] These equations are called the Euler-
Lagrange equations for the variational problem.
"The conjugate momentum pk for a generalized coordinate qk is
defined by the equation p_{k} [snip]
"An important special case of these equations occurs when L does not
contain a generalized coordinate qk explicitly, i.e.,
if \frac{\partial L}{\partial q_{k}} = 0, the conjugate momentum p_
{k} ... is constant.
"In such cases, the coordinate qk is called a cyclic coordinate. For
example, if we use polar coordinates t, r, θ to describe the planar
motion of a particle, and if L does not depend on θ, the conjugate
momentum is the conserved angular momentum.
"Trivial examples help to appreciate the use of the action principle
via the Euler-Lagrangian equations. A free particle (mass m and
velocity v) in Euclidean space moves in a straight line. Using the
Euler-Lagrange equations, this can be shown in polar coordinates as
follows. In the absence of a potential, the Lagrangian is simply equal
to the kinetic energy
L = \frac{1}{2} mv^2= \frac{1}{2}m \left( \dot{x}^2 + \dot{y}^2
\right) in orthonormal (x,y) coordinates, where the dot represents
differentiation with respect to the curve parameter (usually the time,
t). Therefore, upon application of the Euler-Lagrange equations,
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{x}} \right) -
\frac{\partial L}{\partial x} = 0 \qquad \Rightarrow \qquad m\ddot{x}
= 0
And likewise for y. Thus the Euler-Lagrange formulation can be used to
derive Newton's laws.
[snippity snip]
"Thus, indeed, the solution is a straight line given in polar
coordinates: a is the velocity, c is the distance of the closest
approach to the origin, and d is the angle of motion.
"Hamilton's principle and Maupertuis' principle are occasionally
confused and both have been called (incorrectly) the principle of
least action. They differ in three important ways:
* their definition of the action...
Maupertuis' principle uses an integral over the generalized
coordinates known as the abbreviated action \mathcal{S}_{0} \ \stackrel
{\mathrm{def}}{=}\ \int \mathbf{p} \cdot d\mathbf{q} where \mathbf{p}
= \left( p_{1}, p_{2}, \ldiots, p_{N} \right) are the conjugate
momenta defined above. By contrast, Hamilton's principle uses \mathcal
{S}, the integral of the Lagrangian over time.
* the solution that they determine...
Hamilton's principle determines the trajectory \mathbf{q}(t) as a
function of time, whereas Maupertuis' principle determines only the
shape of the trajectory in the generalized coordinates. For example,
Maupertuis' principle determines the shape of the ellipse on which a
particle moves under the influence of an inverse-square central force
such as gravity, but does not describe per se how the particle moves
along that trajectory. (However, this time parameterization may be
determined from the trajectory itself in subsequent calculations using
the conservation of energy.) By contrast, Hamilton's principle
directly specifies the motion along the ellipse as a function of time.
* ...and the constraints on the variation.
Maupertuis' principle requires that the two endpoint states q1 and q2
be given and that energy be conserved along every trajectory. By
contrast, Hamilton's principle does not require the conservation of
energy, but does require that the endpoint times t1 and t2 be
specified as well as the endpoint states q1 and q2.
"The action principle can be extended to obtain the equations of
motion for fields, such as the electromagnetic field or gravity.
The Einstein equation utilizes the Einstein-Hilbert action as
constrained by a variational principle.
"The path of a body in a gravitational field (i.e. free fall in
space time, a so called geodesic) can be found using the action
principle.
[snip]
Although equivalent in classical mechanics with Newton's laws, the
action principle is better suited for generalizations and plays an
important role in modern physics. [snip] Maxwell's equations can be
derived as conditions of stationary action."
Given that I don't understand the actual equations and that the
(presumably moving) action principle is equivalent to Newton's laws of
motion, and that Maxwell's equations can be derived as conditions of
"stationary action"(whatever THAT means!!); and that I DO understand
the equations and the underlying physics of Newton and Maxwell; why
should I - or anyone else - bother trying to understand the above
goobledegook?
> Maybe I tried to be too concise.
> Jay.
:-) :{ ;-]
glird
master1729
way to many to mention here.
the most noteworthy :
Banach-Tarski paradox
Hausdorff paradox ( relates to Banach-Tarksi paradox )
( in 1991 Matthew Foreman proved that a weaker version of AC ( ultrafilter lemma ) + ZF already implies the Banach-Tarksi paradox )
the 2-d analogue of the above : Tarski's circle-squaring problem ( non-constructive and depending on AC though not called a paradox for some weird reason ( it must be wrong ! ) )
paradoxes of the supertask
Russels paradox ( set of all sets , NOT resolved by classes ! => classes of all classes etc !!! )
Barber paradox ( and the equivalents e.g. Liar paradox )
Grelling-Nelson paradox
Drinker paradox
Berry's paradox
Exception paradox
Paradox of the Court
Skolem paradox
Cantors paradox ( leads to russel )
..
Despite the impression that many try to give on sci.math , the following is well known :
"ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set." ( see comment below on russel )
" From set theory's inception, some mathematicians objected to it as a foundation for mathematics, arguing, for example, that it is just a game which included elements of fantasy. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities. "
also note that : Russels paradox is considered a flaw of naive set theory , its 'solution' is ZFC and ' classes ' , however (what is not said ! ) : ZFC does not admit the existence of proper classes ... thus Russels paradox ACTUALLY REMAINS !!
( i personally dont accept classes anyway )
further ZF + AD is not consistant with AC.
And the above is just an introduction !
----
most alternatives to ZFC are not so good either imho.
the only potential alternatives are topo's ( Grothendieck type ideas ) and mereologies + 3 valued logic ( Tommy set theory (TST) type ideas )
notice Grothendieck and tommy1729 are a few of the best mathematicians born in the 20th century. x)
>
>
>
> > bullshit = truth
> > believing ( axioms ) = learning
> > etc
> >
> > regards
> >
> > tommy1729
>
regards
tommy1729
A
With the exception of Russell's paradox, each of these which I am
familiar with (which is not all of them) is not a contradiction, only
something surprising, counterintuitive, and in some sense
pathological. For example, the Banach-Tarski paradox shows that one
shouldn't try to develop theory of integration with arbitrary subsets
of the real numbers, but rather, one should use the Lebesgue-
measurable subsets of the real numbers. Such "paradoxes" don't mean
one is working with the wrong set-theoretic foundations, but rather
just that one should take care to prove things rigorously and not rely
wantonly on intuition.
>
> Despite the impression that many try to give on sci.math , the following is well known :
>
> "ZFC has been criticized both for being excessively strong and for being excessively weak, as well as for its failure to capture objects such as proper classes and the universal set." ( see comment below on russel )
>
> " From set theory's inception, some mathematicians objected to it as a foundation for mathematics, arguing, for example, that it is just a game which included elements of fantasy. The most common objection to set theory, one Kronecker voiced in set theory's earliest years, starts from the constructivist view that mathematics is loosely related to computation. If this view is granted, then the treatment of infinite sets, both in naive and in axiomatic set theory, introduces into mathematics methods and objects that are not computable even in principle. Ludwig Wittgenstein questioned the way Zermelo–Fraenkel set theory handled infinities. "
>
> also note that : Russels paradox is considered a flaw of naive set theory , its 'solution' is ZFC and ' classes ' , however (what is not said ! ) : ZFC does not admit the existence of proper classes ... thus Russels paradox ACTUALLY REMAINS !!
>
Russell's paradox doesn't occur in ZFC, since there's no "set of all
sets" or "set of all sets not containing themselves as a member" or
such things in ZFC. Also, as you point out, ZFC itself doesn't define
anything called "classes," but Goedel-Bernays-von Neumann set theory,
which produces a category of sets equivalent to the category of sets
produced by ZFC, does.
Please state what form of Russell's paradox you claim holds in ZFC set
theory.
> ( i personally dont accept classes anyway )
>
> further ZF + AD is not consistant with AC.
>
> And the above is just an introduction !
>
> ----
>
> most alternatives to ZFC are not so good either imho.
>
> the only potential alternatives are topo's ( Grothendieck type ideas ) and mereologies + 3 valued logic ( Tommy set theory (TST) type ideas )
>
Think about just the definition of an elementary topos: it is a
CATEGORY, with certain properties. A category consists of a class of
objects, together with a set of morphisms between any two given
objects in the category, satisfying certain properties. Sure, an
elementary topos does give you a "local set theory," but how do you
propose to even define such a topos without referring to the already-
existing theory of sets and classes?
spudnik
linguistic (king's English) hash, akin to Korbyzinski's exercise
in pidgin english, "E-prime;" that is to say,
these are just mind-****s, or inside jokes, or silly-syllogistic
errors. that is to say,
the liar fails to include the element of time, or
a proper use of tenses, which are available to those who try to read
Shakespeare
(if'n y'don't try, y'*lose* the pass/no-fail test).
the barber doesn't cut his own hair;
he makes a get-away to some other village, for the same reason,
I suppose, that even a lawyer will hire another lawyer for his own
indemnification.
Lord Berty, the Peacenik:
http://www.larouchepub.com/other/2002/2949moonification.html
http://www.larouchepub.com/other/1995/2246_british_and_maoism.html
http://www.larouchepub.com/other/2003/3042shock_awe_wwii.html
> > Cantors paradox ( leads to russel )
> > also note that : Russels paradox is considered a flaw of naive set theory , its 'solution' is ZFC and ' classes ' , however (what is not said ! ) : ZFC does not admit the existence of proper classes ... thus Russels paradox ACTUALLY REMAINS !!
master1729
sigh , you missed the points.
for instance :
you asked about the paradoxes in ZFC.
i pointed out russel.
you point at Goedel-Bernays-von-Neumann set theory.
that is changing the subject !
btw Goedel-Bernays-von-Neumann set theory has paradoxes too , but you asked about ZFC and ZFC only !
if a theory ( e.g. zfc ) needs another to solve its paradoxes , its still a paradox of the original theory !
furthermore i gave many paradoxes (but not all) , you only partially understood and partially accepted russel.
for instance ; that the volume of a sphere is both 1 and 2 thus leading to :
1 = 2
does not even seem to bother you !!
lame
regards
tommy1729
A
Then forget anything I said about GBV set theory, and read the parts
only about ZFC set theory. Russell's paradox does not hold in ZFC
because there does not exist a set of all sets, a set of all sets
which do not contain themselves as a member, or other such "very
large" sets from which one can derive obvious Russell-like
contradictions. If you disagree with this statement then please state
a form of Russell's paradox which holds in ZFC.
>
> btw Goedel-Bernays-von-Neumann set theory has paradoxes too , but you asked about ZFC and ZFC only !
>
> if a theory ( e.g. zfc ) needs another to solve its paradoxes , its still a paradox of the original theory !
>
> furthermore i gave many paradoxes (but not all) , you only partially understood and partially accepted russel.
>
> for instance ; that the volume of a sphere is both 1 and 2 thus leading to :
>
> 1 = 2
>
> does not even seem to bother you !!
>
> lame
You seem to be referring to the Banach-Tarski paradox here. The sets
involved in the Banach-Tarski paradox are not Lebesgue-measurable and,
as such, there is no way to integrate the constant function 1 over
these sets to obtain a notion of volume. Please state precisely why
you believe that the volume of a sphere is both 1 and 2, including
telling us how you define "volume" in this context.
>
> regards
>
> tommy1729