Edgard Neuman

Just now, Strange said:

Because you haven't shown the math for it.

the math are the same of those of Feynman diagrams except instead of having a void empty, you have a void full and particle are not considered to "disappeare" and "appear" from nothing, but being real and still there.

3 minutes ago, Strange said:

You have that the wrong way round. The wave function came first and then the interpretation of it as the probability of finding the particle in a location. But the wave function does a lot more than that.

And, again, the diagrams were invented to make it easier to visualise the underlying math.

the underlying math is explained in the book.... the sumation of different possible histories.. thanks I know that. So why can't you understand that my model is equivalent ? Do you understand what the diagrams are ? 

and if the probability of presence of a photon is in "1/(x²...t²)" how the hell is this a wave ?

Read Feynman book called "the light".. in this book, he explain how the diagram work.. he doesn't talk about wave, but a complex number 
If I remember correctly, the probability, for a photon, for instance, is given by the formula 1/(x²..... -t²)...
(it's the square of the modulus of the complex number)..
I can't make this up, it's in the book.

What you call wave is a construct on the fact that for a given particle, of course the probability across all space integrate to 1 (I call it wavy behavior to speak about this abstract construct you use for a "real particle"...) . but you know the real description of things is the Feynman diagrams, right ? 

4 hours ago, Strange said:

You "think" there are no differences? How about proving the two model are mathematically equivalent?

As real particles and virtual particles are very different things I find this argument extremely unconvincing with either mathematics or evidence to back it up.

 

So, to make this testable, you need to quantify this "void density" and specify exactly what the "complexity a qubit can carry" is.

Unless these are quantitative predictions, they are not testable.

Of course, doing that would require a model. Which you don't have so ...

I don't know what you are trying to prove here, on a forum about science. You want a cookie for your years of study ? You want a nobel prize ? I don't pretend to know what I don't know, I'm just here to propose a different interpretation of things (yes without mathematics). You eather contradict it, or leave me alone.

You can throw complex mathematics at me, if they don't contradict my theories (and they don't, until now I've answered your objections, but you don't seem to care), I've no reason to trust you. 

On 8/26/2019 at 1:50 PM, MigL said:

Again...

"Wouldn't these properties of the 'void' be measurable, and provide some observational evidence ?"

and again, what is the EM fields  when you assume particle are not quantum wave ? (magnetic force and electric force is observational evidence, how do they exist) I know how it works in the quantum theories : virtual particles carry forces (virtual photons carry forces.. "actual" photons are quanta of the fields)
When you assume the quantum world isn't based on wavy object, then what would be those field ?
What about particles of the void , that are not virtual anymore ? 
If the void is full with some anisotropic field of real photons, their are quantities coming with that : their average direction (the electric field !) and its variations (the magnetic field...).. if a charge is constantly hit by real photon, variation in the field would manifest as forces.

I think their are no mathematical difference between a quantum model where you assume the result is the sum of hypothetical particles that you use in thought implitly in probability equations, and a model where you assume they are actual particles.. when you read how Feynman diagram works,  you see what is summed.. it's just a matter of understanding that those particle could be real as long as the particle filled void could be perfectly balanced so it has no effect. 

But if you really want a difference between my model and the actual model, there is one, that will come soon enough : 
In my model, the void has a real density, and this density while it's not directly mesureable, define the ability of space to carry parallel Feynman paths.. so when the quantum computers will work, we may encounter the limit of complexity a qubit can carry. If quantum computers are limited in the parallel state each part of the circuit can carry, it would be a sign this quantity is limited and it will be mesureable.

7 hours ago, Mordred said:

The only difference between antiparticle and particles is that they are opposite in charge.

I found this explanation (in some wikipedia discussion, so it need to be checked)

"The electron field really describes four different particles: the left-handed electron, the right-handed electron, the left-handed positron and the right-handed positron. Parity-reversal exchanges left-handed particles with right-handed ones. Charge-conjugation exchanges electrons with positrons. Due to CPT symmetry, if you apply parity-reversal, charge-conjugation and then flip the arrow of time, all reactions look exactly the same. This implies that left-handed electrons are "like" right-handed positrons. But neither is the parity-flip of the other. -- Xerxes 18:44, 10 June 2006"

but I also read that a free particle interacting with the void and propagating is a mix of both parity (through self interacting ?)

Quote

If you had equal quantities of each they would indeed annihilate one another.

 Why do you think I mentioned baryogenesis in my first reply ? There must be some assymetry between the two in order to get a higher percentage of positive matter. Charge is symmetric under rotation translation so charge itself cannot account for it.

Baryogenesis happened just after the bigbang, not now. You don't see matter pop out of the vaccum from nowhere. My theory doesn't require symmetry break.. If you add baryogenesis into this, the void would then all turn into matter... there wouldn't be apparent conservation of energy and charges.
You can't break symmetry here or it wouldn't work at all. 

In my idea, the photon that results the annihilation are just dense enough to cause creation reaction.. and the void is in a state of equilibrium between creation and annihilation. (the void is stable as we experience it, not changing)

I know what is the problem : it's hard to imagine a isotropic field of high energy photon , that is so isotropic that it can't be detected.  (unless we call it... the EM field)

In fact the only difference between my theory and the ordinary MQ, is that in my model : all virtual temporary particle you use in Feynman diagram are as real as the real ones we study and are just there in the void all cancelling out themselves globally (a lot of interaction without long term effect), but creating the "noise" and the randomness (and carrying the quantum information). 

31 minutes ago, MigL said:

This 'distinction', where you see particles with classical trajectories not really moving, but their properties translating due to the interactions of a multitude of intermediate particles already present in the 'void', would require an extremely large particle density, and an equally large void energy.
Wouldn't these properties of the 'void' be measurable, and provide some observational evidence ?

If the particles of matter and antimatter are really exact opposites, every interaction is perfectly symmetrical and the photons (and all the rest) are in perfect anisotropic distribution, why would they ? The law of large numbers would insure that in the long runs every interactions cancels out. There would only be a noise (some sort of brownian movement) at very small scale. And the important thing is that there is no "virtual" particle and "real particle". Imagine the electron emitted in that space : it would instantaneously hit a positron and become 2 photons. So now where is the positive charge ? The interaction did also remove a positron, so the void that was neutral now have a electron more than a positron. The electron is still there, it's now another one that was already here. while the momentum is carried now by the photons. The photons (that are asymmetrical) hit an other photon, and create a pair with a momentum.. those 2 particle then interact etc.. the momentum is shared between all particle.. Now you start to see how the whole thing behave like a wave, while no wave are used. 
The effect, would probably only be seen when accelerating (the unhru effect). And my next idea is that "this void", I mean the potentially varying density of particle IS actually the gravitational field. (It could also explain the expansion : this void would indeed spread like a gas)

28 minutes ago, Mordred said:

 You can get away with Unruh effect in classical terms but decoherence invariably does involve probability.

no, I don't think so. You use probability to describe what happen, but it's not a necessary property of the system.
(I'm sorry, i have no other way to explain my idea than with a bad metaphor. )
Imagine a town full of people. You throw somebody in it. At some point, because there's too many people, the village will decide to throw somebody out. Who will it be ? We can't know. But we know it's going to be a full somebody, and he is going to pop out from somewhere well defined.
Now the screen in the double slit can perfectly behave this way. It's a threshold effect.

What is tricky is the interference.. but If you assume the void is full of two type of particle that cancel each others, it maybe normal that when something is propagating into it, it alternates between two opposite extreme of the perturbation.. I don't know more details

31 minutes ago, Mordred said:

Sorry but that last doesn't make much sense.

A field is an abstract device describing a set of values/mathematical objects that have a coordinate basis. So a void with spatial components would have a set of values at each coordinate. Even if the value for say as an example the energy density is zero it is still treatable as a field as we can assign coordinates to any void.

I agree with that (the description of a what is a field), but what I'm saying in my model, is that - hypothetically - we could make a true "void" empty of energy (neither matter nor antimatter). In that space, a particle would stop to behave "quantumly" and behave like a classical particle (with a defined trajectory).. In that space, the particle wouldn't interact with anything being there, the momentum wouldn't be spreaded and it would behave exactly like a classical point.
So in my model "the void" is what explains it all. The problem with using fields (although I understand it's the only way of studying a nearly infinitely big set of objects of course) is that it's already a statistical version of the reality you describe. I know you work with fields, so you work with "probabilities" of particle being there or there.. but I try to find an other way to explain reality without starting to believe the field is actually the reality.(I don't know if I'm clear).. I'm on the side of those who think quantum theory is just a statistical tool, and that the strangeness can be explained by classical means.

I have an other example to illustrate the  idea that using abstract theory can be deceitful (ok it's not maths). you have for instance the fact that if you put 2 polarized filter opposed by 90° you cut all the light. But if you put a third one inclined by 45° between the 2, you get some light to pass.. everybody see this and think "it's the magic of quantum theory ! how can adding a filter increase the passing light ? It's impossible".. but it's in fact very easy to get the same result by a simple classical way. 
Let's represent the possible incoming light wave orientation as a 2d surface (all the possible vector of the wave on a plane perpendicular to propagation).. if the filter project the wave on a plane (parallel to the propagation vector), it would project the surface on a line. So for instance the first filter would project a circle into a horizontal line, and the 2 filter would project the horizontal line on the vertical axis, so a point at 0 (and so : no light pass). But if you put a 45° filter between the two, the first unity horizontal line would then be projected into the 45° line (a none zero length line). And then this, line, when projected into the vertical axis, would not be 0. There's no "quantum magic" here : it would work on classical waves as well.. (it would be funny to do it with sound waves)

2 minutes ago, Mordred said:

The term void isn't really applicable under either QFT or QM. For that matter not even under GR. Your far better off describing fields  than voids. Under field treatments the propogator action is spread out onto the field. The propogators propogate  the operators and the operators operate upon the propogators.

If you think about that statement it's much the same as spacetime tells particles  how to move while particles tells spacetime  how to curve.

no but my idea is to suppose any part of empty space where we see, for instance "1 electron"..
is in fact full (for instance, I've no idea of the density) of 100000001 classical electrons and 100000000 classical positrons (and many many more high energy photons and a lot of other things... all of this would interact, but all the effect of all those interactions would cancel out, and would only be some noise at very little scale.
The term field describe a function space => value (or vector)..  Here the void is really the important thing that explain the behavior of what we call matter.
 

13 minutes ago, Mordred said:

 The problem is you cannot have decoherence on particles unless you are describing their wave functions. So you wouldn't have quantum Darwism.

No, because the void can carry the information that is temporally missing. If the void is full of particle, it can move globally and fill the gaps.. we wouldn't see the difference, because, all particle of the void are the same. It's all about what is stable configuration and what is not, in the long terme. Collectively, particle can behave like wave, and finally one situation that is the simplest stable situation emerge.. (in the young slit experiment, the thing that make decoherence is that the energy of the photon can be absorbed and reemitted only by one atom of the screen, probably because of some conservation principle that insure the photon that is seen has the same energy level as the photon emitted, so it can't be divided)..
 

the three results seems to fit with my idea.... you won't see it until you understand it.. My theories have to respect results of experiments, not your models. The unruh effect is something you can measure. The fact that a void reacts to a body accelerating in it, tend to prove my idea that the void is not void, and that what you call particle are indeed just charges shared between a whole lot of void particles..
Particle and antiparticle are opposed by the charge and parity,  to get a particle from an antiparticle, you invert "C" and "P".. It's the other solutions for a given energy when you invert the frame in special relativity. So yes, I think I understand. 

What I try to explain to you, is very simple.
In the first times of quantum theory, people couldn't figure out how a particle propagate as a wave. So they described the wave and came to quantum mechanics. In my idea, the particle for instance from a emitter never reach the screen : it's interact very closely with the void. For my model to work, the density of the void must be enormous. I suppose, (no I have no proof, sorry) that particle truly travel in average no longer than the scale of planck length. But because the void is full of as much particle of matter and antimatter, each interaction cancels out, and the thing that travel is a average surplus of positive kinetic energy (and eventually a surplus of charge). 

So If you can show me how that wouldn't explain the quantum behavior, I'll be happy to read it.

24 minutes ago, Mordred said:

To answer an earlier question  of a post you edited later on. A is the creation operator under QFT. A with the dagger is the annihilation operator.

The B is the operators for antiparticles.

K corresponds to the momentum operator hence it is a vector.

https://en.m.wikipedia.org/wiki/Creation_and_annihilation_operators

The wiki article covers mainly the QM treatment but position is downgraded to a propogator and the field is upgraded to operator under QFT. Which uses the Klein Gordon and Dirac  equations to better handle Lorentz invariance. The Schrodinger equation of QM isn't Lorentz invariant.

Thanks a lot for this answer.
And do you confirm that an "operator" is a function of the state of the field to an other state of the field (the field of particles for instance). I understand how  creation and annihilation are operator (the effect of the "interactions" on the values of the field).. 
OK but the problem is, in my model, particle are not wave.. (so when you start to speak about frequencies, it's already not in my model)  they are classical particles with classical trajectories.. you may think "but how could it fit the Young slit experiment for instance.. ??" because in my model, what you call particle (the wavy thing) is in fact the surplus of momentum and charges that are statistically emerging from local perturbation of the otherwise neutral void.. 

If you can't understand my idea, why do you answer ? I put this in "speculation" for a good reason. I'm not here to receive a lesson on the classical accepted theory. I'm here to propose an alternative. In my theory, particle are not wave. They are particle. The wave behavior emerge from the statistical properties of a void "full" of particles.  
As always, you people are not able to understand me.. and decide you don't even want to try. How should I react ? I'm just sorry.
 

39 minutes ago, Mordred said:

That article certainly hit a huge media blitz but I wouldn't take it too far until you can review the pdf with the mathematics. 

it might as well be written in chinese.. somebody should explain with simple words.

For instance, you write : 
"Now accepting that baryogenesis leaves a slight higher density of positive frequency parts (cause unknown)"..
That's not coming from your equations (they describe the creation anhiliation events as an operator of the fields https://en.wikipedia.org/wiki/Creation_and_annihilation_operators)..
You are referring to an hypothesis for the existence of matter over antimatter, supposedly because of CPT symmetry violation (in neutrinos).. you can't use that here (it's not part of my model, and it's not a even proven fact)
In my model, the number of particle and antiparticle doesn't come from differences in the operators, but in the fact that the "positive" and "negative" values are somewhat constants.. In the beginning you have m > a  (m particles of matter > a particle of antimatter).. because operator leaves the relation unchanged, we still have matter m+1> a+1 or m-1 > a-1.. (my model doesn't speak about the origin of matter)

I'll explain further : let's have a cube meter of space (we can make it topologically closed by identifying the opposite faces).. inside let's put a ton of matter and a ton of antimatter.. the classical theory says that they would annihilate into high energy radiation. But now let's imagine the radiation is so high that the probability of pairs of matter and antimatter coming from photons interacting photons become high enough. Let's suppose that everything is real (no "virtual"  things).. now you put some more matter. The matter would interact very often with high energy photon, some of it would interact with locally created antimatter to get into high energy, but overall, there would still be more matter than antimatter.. but because of the constant interactions and the high density of particles in the void, it would constantly pop in and out. More : the charges of the particle would dissolve into the void, and possibly be constantly shared with the void.. for instant the kinetic energy would be spread between different particle of the void, and other charges too. So here you have : the quantum waves. The particles don't have to be "wavy" at the start. The waves are waves of real particle in the void. 

I think this should interest you : (and it's well explained)

https://www.newscientist.com/article/mg24332440-600-quantum-weirdness-isnt-real-weve-just-got-space-and-time-all-wrong/?utm_medium=SOC&utm_source=Facebook#Echobox=1566571542

On 8/10/2019 at 7:34 AM, Mordred said:

 

a^†(k⃗ )a^(k⃗ ) the former is creation the latter annihilation operators for particles 

 

ok so for me (and people like me) to begin to understand, here, what is the meaning of the letter "a" and what is the meaning of the letter "k" ? (you can just provide a wikipedia link)

On 8/10/2019 at 7:34 AM, Mordred said:

OK so you want particles and and antiparticles popping in and put. 

So let's use the field treatment of QFT. Now the positive frequency modes form the annihilation operators for particles with the negative frequency modes for the creation operators of anti particles.

a^†(k⃗ )a^(k⃗ ) the former is creation the latter annihilation operators for particles 

For antiparticles 

b^†(k⃗ )b^(k⃗ )

Hence a^†(k⃗ ) creates a particle of energy ℏω and momentum ℏk same applies to the antiparticles

Now with the above you sum up the positive frequency parts with the negative frequency parts. 

ψ^(x)=∫d3k(2π)322ωk√ a^(k⃗ )e−i(ωkx0−k⃗ ⋅x⃗ ) +b⃗ †(k⃗ )e−i(ωkx0−k⃗ ⋅x⃗ )

How's that for a start to your model you now have the a scalar spacetime complex field of particles and antiparticles.

Now accepting that baryogenesis leaves a slight higher density of positive frequency parts (cause unknown) I will let you think about this in terms of the energy density values for different observers for the Unruh effect in your link above. Naturally the mass density valued will vary accordingly to field potential which will affect the path integrals described by the Feynman lines you referred to.

(I am going to up vote you +1 for coming up with a viable speculation model though we can improve your descriptive as we go )

Now continuing from above a complex field has an adjoint. 

ψ^†(x)=∫d3k(2π)322ωk√ a^(k⃗ )e−i(ωkx0−k⃗ ⋅x⃗ ) +b⃗ †(k⃗ )e−i(ωkx0−k⃗ ⋅x⃗ )

Now assuming you want bosons for an uncharged field. We can incorporate the Pauli exclusion symmetry to the Bose Einstein statistics 

So first we have normalize the vacuum to unity. In Dirac notation

〈0|0〉=1 . The ket |〉 is the initial state the bra 〈| is the final state.

So we need to compute the normalization  to an arbitrary state.

|k⃗ 〉 for that we need the inner product

〈k⃗ |k⃗ ´〉

So a^†(k⃗ )|0〉=|k⃗ 〉

with adjoint  〈0|=〈k⃗ |a^(k⃗ )

Without going through all the steps

〈k⃗ |k⃗ ´〉=δ(k⃗ −k⃗ ´)

Now each k state represents the momentum of a single particle.

If they are bosons  then the following relation holds 

|k1→,k2→〉=|k2→,k1→〉  

Now you have a complex spin zero boson field in momentum space.

Next lesson we apply thus to the four momentum of GR.

As we're dealing with a complex scalar boson field of particles/antiparticles  let's start with the Minkowskii metric

ds2=−c2dt2+dx2+dy2+dz2=ημνdxμdxν

Apply coordinate notation (x0,x1,x2,x3x4)=(ct,x,y,z)=xμ Where the indice range is 0 to 3.

Four momentum is given here

https://en.m.wikipedia.org/wiki/Four-momentum

In the format I have provided using a particle number operator I will just latex the positive frequency modes as the negative frequency modes simply switch a to b.

N^=a^†(k⃗ )a^(k⃗ )

The Hamilton is

H^=∫d3kωk[N^(k⃗ )+12] includes the harmonic oscillator.

The field momentum

P^=∫kk⃗ [N^(k⃗ )+12]

Now there is an interesting consequence of this when you compute the energy of the field...it is related to the energy of the harmonic oscillator.

Can you guess what it is ? I will show the answer tomorrow.

PS to OP I hope your descriptive above needs considerable work however you have several details with  accuracy that although poorly described are applicable. So I hope you don't mind if I run the modelling gauntlet with the basis of particle/antiparticle creation and annihilation and it's effects. I will get to your network analogy later on as I can employ that analogy

 

 

 

To be honest : I have no idea what you are talking about. That seems interesting, but I'm into explaining reality we are living in. So can you explain each part of you post ?
When I write with my simple english, my simple mind, I have no idea if I'm right. So how can you be sure you are write with such complicated ideas ?

Hi,

I read this article, and I have a theory   
https://www.zmescience.com/science/physics/physicists-are-a-step-closer-to-a-theory-of-quantum-gravity/

Let me explain : in model, 

- quantum theory use "virtual particles". In my model : particle are not virtual, but what we call "void" is filled with two kinds of real particles : matter and antimatter (or maybe some other kind of symmetry).. think of it like electric charges in a metal.. you have  two opposite charges but they globally cancel out each other. If the void is indeed filled with particles of matter and antimatter interacting, (but ! stable of course, because the high energy of the photons would constantly turns back into particles, for it to work, it has to come to some equilibrium state between annihilation and creation of particle that are not photons).. 
In that case : the void is full of particle that actually carries the information of all quantum scenaris, but in a limited quantity. This quantity is the "density of the void".. Because there would be a equilibrium of positive and negative particles, we can imagine that the void can contains more of less of both.
- that would also imply something else : the quantity of quantum parallel "versions" of systems would be finite : the void can't carry has much information as we want. That would perfectly fit with an other theory called Quantum Darwinism. Think about the Feynman diagrams. In this model, instead of exploding into infinite trees of possibilities, it would fit into a finite quantity of reality per units of void (because the void carries the information).  In that case, it would look like a limited width genealogy tree : some branch would constantly die out, will only one remain from the distant past. Decoherence would be explain like this : the one particle we see in the Young double slit experiment (for instance) would be like the common ancestor of all realities that remains (the other possibilities having died out, from the moment the wave hit the screen to it became macroscopic for us to see). The key idea is this : the quantity of parallel stories in each part of space is limited (because it's indeed carried by real particle in what we call the void).

- the density of the void would so define "permittivity" of information by the space. Relativity insure that a system is by itself organised by information (the speed of light). The reason the speed of light is constant is because light is the lightest possible information or the quantum of causality.. Information defines "distance" and "time" in the systems.. not the other way around. So if a system is in a space with some a certain permittivity of information, and we change the permittivity, the only difference is the speed of time.. the story would be the same, so we can't directly measure this (as we are "inside" the story).

I have a nice metaphor for that : think about "matter" and "relativity" as some kind of self organizing software running on a network, the network being the particle of void in equilibrium. The void, would then by the physical level of the network. The structure that the software take can only be influenced by the topology of the network (the software is self organizing : creating relative distance and relative reference frames between its parts), but not its average speed. A difference in the density of the void (the global speed of the network) wouldn't change what the software do and how it behave (and itself would be able to measure anything).. byt the speed of the computation, the speed of the story (relative to an outside observer) 

- So now, we can suppose that the density of the void is the gravitational field : (the curvature)... the variations of it would manifest exactly as general relativity does : a variation of time. (the gravitational force would be the local effect of gradient of efficiency : particle probabilities (for instance) would be greater in the direction of the more density : so the matter running on it would be in a accelerated frame, and information would take more time to go from a relative "fast network" to a slow network outside (causing what we see as curvature)

ok i see the problem now : once you rearrange the new number, it could be already in the rows..
 

But I found a fix :

Instead of writing the number a and b in sums of 1, you write it in product of primes 
For instance 
275/12 

275 =2^0 * 3^0  * 5 ^0 * 7^0 * 11 ^ 1
12  =  2^2 * 3^1  * 5 ^0 * 7^0 * 11 ^ 0 

so you get in the row 

275/12  = >     (0;2)  (0;1)  (0;0)  (0;0)  (1;0) ... (0;0)

and then to get a new row, you just add 1 to the bigger number (up or down) in each column

now you are sure the new number wasn't in any row
that a least would prove that there is more rational numbers than prime numbers 

 

Hi,

Ok, I was looking for a way to apply the cantor argument to rational number (because I'm masochist) and I think I found one. I'm sure somebody will be happy to disprove my idea :

I write rational numbers in a special way. Let's say you have a/b
I decompose a and b into sums of 1s..

a= 1 + 1 + .... + 1+1+1 
b= 1+ 1+ .. +1+1..
first I simplify, then I use it to create a pseudo number like this : when there's a 1 in the first line, and 1 in the second : it's 3. When 0 and 0 => 0, when 1 and 0 => 2 and when 0 and 1 => 1
So for instance 

4 / 5 = > 33331000000....
12/6 = 2/1 => 3200000...
I then use it to fill a cantor-like array 
now I use the diagonal to create a new row 
If it's a 0, i put a 1 in the new row
If it's a 1, i put a 2 in the new row
If it's a 2, i put a 3 in the new row 
If it's a 3, i put a 0 in the new row
Now we are sure that are new number is either   +1 or   -1 or  - 1 -1 on a or b from each previous row.. So it's a new row. 
(ok there may be a special case when 1+1 / 1+1 = 1/1 but we can obviously exclude 1 from rows from the beginning)
Did I just prove that Card (ℚ)> Card (ℕ) ? 

Hi,

https://en.wikipedia.org/wiki/Logistic_map

here is my question : does the logistic sequence for some choosen irrational parameter reach every real number inside a real interval, or is it always just a subset ? 

(I hope i'm in the right section)

thanks ! 

well on that sad note, I will leave now.. I'll come back in six month with another amazing but uncommon idea to share, as usual 

42 minutes ago, Strange said:

Simpler but less useful.

So tell me :
if 
A= 1+2+3+4 + ...
B=  r^1+ r ^2 + r^ 3 + r^4 .... 

is A > B or A < B when r = 1.5 ?

and what is the difference equal to  ?

A-B = (ω ² + ω) / 2   - (1 – 1.5^ω ) / (1 – 1.5)
= ω ²/2 + ω/ 2    + 2*(1-  1.5^ω)
= ω ²/2 + ω/ 2    + 2-  2* 1.5^ω

ok it's still complicated but it's much closer to be solved

I suppose that 1.5^ω > ω² 

Anyway If you really don't see how using a measurable value for infinite is usefull, i'm sorry I can't do more for you

And I really curious to see where your infinite has been usefull.. It's simply "not related" to integer numbers. It's just "bigger".. How is it usefull in anyway ? 
You have to use alternative like "limits" and you can even sum them or substract them. Yeah very usefull.
You can't even solve 

x + Card(N) = Card(N)

x is a number ? 

YEAH VERY USEFULL 

inf/inf = a number ! or maybe not.. WOW usefull ! 


No our intuition..

Cut a segment in 2 part . How many points does each part have ? is it more or less that the original segment ? When you cut a cake, how many particle of cake do you have ? Can you infinitely cut a cake and  feed the world ? 

9 minutes ago, Strange said:

That's what I said.

So you have defined a set of numbers where the usual rules of arithmetic don't apply. I'm not sure what the point is. As has been pointed out several times, nothing that you prove about these numbers has any relevance to the natural numbers used in the real world (ie outside this thread).

except everything become much simpler. and conform to our intuition about quantities.. I gave you several example of the usage
a measurable infinite (you want to say it's "finite" if you really have trouble understanding that), but ω is really used to simulate an infinite quantity that is still mesurable .. sorry for you on that..

I had no idea it would be so hard for people to use another axiomatic.. 

In fact I should have explain the philosophy of the idea before all.. I've done it on my blog.. the idea that Peano stop and some numbers are not allowed is very easy to understand for me, because I 'm among other things a software developper :
when you do math with 32bits integer you really know how math and arithmetics with limited range  works..  
 

19 minutes ago, Strange said:

I think you can achieve this, even with Edgar's finite set of integers, using modulo arithmetic. 

However, he doesn't seem to want to do that, just saying that c is undefined which doesn't really make sense to me.

hum no. a+b IS NOT in ω
and it is not  a+b modulo  ω that would be a totally different axiomatic

Just now, Strange said:

Maybe you should ask the mods to change the title to "Cardinality and Bijection of FINITE sets" so it describes what you are actually talking about.

OK i would agree with that, except it's FINITE with the boundary unkwown 

45 minutes ago, studiot said:

well no, (ω -1 ) is not defined.

 

you use Peano.
BUT at some time you have a number ω, it is not in ℕ_ω
the one before is ω-1. It is defined and in ℕ_ω 

Do you really have a problem with the concept of "unknown" value ? like n ..
is n-1 defined ? how do you write n-1 in Peano ? 

Some number are in ℕ_ω some are not by axiomatic
ℕ_ω is simply a set


[0;1;2;;3;4[
That is a set. Each element is constructible and "unique"

a+b>ω  is NOT in ℕ_ω, 

but is as unique as it is in Peano
 

1 hour ago, Edgard Neuman said:

ω is divisible by any number n in ℕ_ω  
only if n<sqrt(ω )

I think probably this would better be :

ω is divisible by any number n in ℕ_ω  
only if n!<ω 
i don't know

34 minutes ago, studiot said:

So the set of all sums (a+b)>ω is not finite.

(a+b)>ω is not finite
yes.. 

(ω -1 ) +1  is not in ℕ_ω

I consider that if ω  is the measurable "infinite", all that's below is finite, all that is above or equal  is infinite

34 minutes ago, studiot said:

Of what use is an arithmetic that omits such a large number of possible sums?

I found the usage of a measurable ω very useful.. it automatically include limits of sums as comparable numbers

The fact that ω is not in ℕ_ω  doesn't mean you can't use It.. 

ω  + ω  = ω *2

_{It's "infinite" numbers, but there, the sums works exactly like finite numbers>}
 

_{And for cardinals, it really preserve quantities.. Card(even numbers) =} ω /2  
so even with infinite sets,
Card(A union B) = Card(A)+Card(B) - Card(A inter B)

_Card(ℕ_{ω) = Card(evens) + Card(odds) =} ω

you can build a bijection between evens and odds (n->n+1)
but not between numbers and evens  (or odds)
In that case "bijection" really mean equality of cards and vice-versa

_{What use is a number that is equal to its double ??  (your Card(N))}

_{It's kind of symmetrical to 0.. it's the other end of N ... you have 0  1 2 .. ω-2 ω-1}

_{I tried to add an other axiom, but I'm really not very sure it works  :}

ω is divisible by any number n in ℕ_ω  
only if n<sqrt(ω )