Entropy can be reset to initial or previous state

[obsrvr524]

The equations don't allow for a value of infinity is what I was saying. I didn't mean to start an argument over it.

Sure, no equations allow for a value of infinity by definition. They can only allow for tending towards infinity. It's not anything to start an argument over in the first place, it's just a fact. So no need to worry about that. And it's not merely an interpretation of the maths that I'm explaining.

x = 1/0
x = infinity

X isn't going toward infinity. If the denominator is 0 (as in zero K temperature), and the numerator is greater than zero (as positive heat flow), then x is infinite.

If you have learned some other maths, we are just at an impasse.

As for the rest of your argumentation, you don't seem to be able to separate the total of the universe from far more restricted and local events. I don't think that relativity has anything to do with the universe as a whole. Perhaps some people just can't comprehend infinity and the necessary logical consequences.

[Silhouette]

x = 1/0
x = infinity

X isn't going toward infinity. If the denominator is 0 (as in zero K temperature), and the numerator is greater than zero (as positive heat flow), then x is infinite.

If you have learned some other maths, we are just at an impasse.

Unfortunately it would appear that you are unaware that 1/0 is undefined.

Please do look things up more than you are before answering. Last time you did, you made it sound like I was causing you an inconvenience... - this is the opposite to what ought to be the case.

Infinity isn't a quantity, it can't "equal" something. If you studied maths at a higher level you would know that you only formulate tendencies to infinity, and infinities or division by zero is a serious problem if it's ever discovered in your formulations. The same goes for mathematical equations in physics, which you would know if you were familiar enough with either subject to be so certain about the topics we're discussing.

As for the rest of your argumentation, you don't seem to be able to separate the total of the universe from far more restricted and local events. I don't think that relativity has anything to do with the universe as a whole. Perhaps some people just can't comprehend infinity and the necessary logical consequences.

If you think you can comprehend infinity, you are either giving bounds to infinity or implying your comprehension is infinite. So are you contradicting yourself or professing godlike capabilities?
It's by definition only possible to comprehend the tendency towards infinity, which is why maths deals with tendencies instead of infinities. This shouldn't be hard to comprehend.

If you don't think relativity has anything to do with the universe as a whole then either you don't understand the extent to which it models the universe as a whole and holds up to experimentation, or you have some experimental evidence that the entire scientific community is not yet aware of, and/or some imminent thesis with extensive mathematical support to be the next revolutionary to Einstein with regards to his theories. So are you lacking knowledge or are you one of the greatest geniuses that the world has recently seen? I won't rule out the latter without you presenting anything to support this yet, but probabalistically and based on what I've seen so far I strongly suspect that I'm dealing with the former.

And yes, I can separate the total of the universe from local events. I was referring to both, so it's strange how you don't think I know the difference.

[obsrvr524]

x = 1/0
x = infinity

X isn't going toward infinity. If the denominator is 0 (as in zero K temperature), and the numerator is greater than zero (as positive heat flow), then x is infinite.

If you have learned some other maths, we are just at an impasse.

Unfortunately it would appear that you are unaware that 1/0 is undefined.

Please do look things up more than you are before answering. Last time you did, you made it sound like I was causing you an inconvenience... - this is the opposite to what ought to be the case.

I will grant you that they call it "undefined", but I could still argue that it has meaning anyway.

The greater issue is that you have merely made your case worse.

I was saying that the equation could not rationally produce an answer of "infinite". You have now confirmed that not only can it not produce a value of infinite, but the best it can do is to become completely undefined.

So your idea of the universe's entropy being infinite just got even more impossible. At zero K, entropy is undefined, void of meaning. And at anything above zero K, entropy is less than infinite.

If you think you can comprehend infinity, you are either giving bounds to infinity or implying your comprehension is infinite. So are you contradicting yourself or professing godlike capabilities?
It's by definition only possible to comprehend the tendency towards infinity, which is why maths deals with tendencies instead of infinities. This shouldn't be hard to comprehend.

People who say things like that are saying that only God can understand things that they don't.

[Silhouette]

I was saying that the equation could not rationally produce an answer of "infinite". You have now confirmed that not only can it not produce a value of infinite, but the best it can do is to become completely undefined.

It would not be able to define conditions that can't exist, sure. For all the conditions that can exist, we're ok.
Why can such conditions that make the equation produce undefined results not exist? Read on.

So your idea of the universe's entropy being infinite just got even more impossible. At zero K, entropy is undefined, void of meaning. And at anything above zero K, entropy is less than infinite.

Cooling something down to absolute zero requires the cooling agent to also be at absolute zero, or below which appears impossible, because the energy of whatever is doing the cooling is always transferring to what it's cooling. You can get very close...

Again, I'm afraid I must inconvenience you to read up on absolute zero.

People who say things like that are saying that only God can understand things that they don't.

Not sure I follow the logic. Also, I'm an atheist, which would make me seem even worse if I was saying nobody can understand things that I don't - which isn't what I mean to say I can assure you!

I'm using the adjective "godlike" hyperbolically here, because logically nobody can have infinite comprehension: "comprehension" is by definition putting mental bounds around something, which contradicts infinity. Being godlike is a contradiction in the same way, so really what I'm saying is that your implication of understanding infinity is necessarily a contradiction regardless of how much I know or don't know. I'm sure many people understand things that I don't, but I'm more sure that nobody comprehends infinity by definition - and the best that can be done is comprehending tending towards infinity.

Does that dispel your suspicions about my arrogance or lack thereof?

[Jakob]

As for the rest of your argumentation, you don't seem to be able to separate the total of the universe from far more restricted and local events. I don't think that relativity has anything to do with the universe as a whole. Perhaps some people just can't comprehend infinity and the necessary logical consequences.

Its not necessarily very easy.
But Relativity seems to me to be a natural consequence of an infinitely extended potential for power concentration curvature, simply because infinity doesn't allow for a centre.

[obsrvr524]

Does that dispel your suspicions about my arrogance or lack thereof?

Your statement implied your arrogance, not mine. I would say instead that you attempt to argue almost any irrelevant issue even to your own demise. You like to argue. I prefer people who like to find agreement.

[Silhouette]

Does that dispel your suspicions about my arrogance or lack thereof?

Your statement implied your arrogance, not mine. I would say instead that you attempt to argue almost any irrelevant issue even to your own demise. You like to argue. I prefer people who like to find agreement.

Again I'm not sure I follow your logic, and again arrogance is not intended - the above was an invitation for you to explain how you arrived at that conclusion.

The fact that you did not accept this invitation, nor elaborate on the subject matter, and that you express distaste toward discussion with me indicates that we are done. Thank you for the debate.

[bahman]

If matter has existed from infinite past, then entropy is such that it can be reset to a previous state.

If this was not true, the world would be approaching much closer to a fully entropic state than what we experience right now. Or else perhaps we'd be in a fully entropic state.

This subject is on the list of seemingly new ideas that Jame S Saint introduced to the world. The idea that the universe has always existed or has an infinite past isn't new, but James explained why it must be true and also why the universe can never repeat. He explains that space is larger than time.

As the mathematics turns out, even given an infinite eternity of time, the 3D universe could never, ever exactly duplicate itself. Every single instant of time, throughout eternal time, is and will always be unique. The universe has no opportunity to repeat or cycle.

I'm still struggling through the maths, but the logic seems easy enough. Check the logic:

If you have 100 envelops to address but only 50 addresses, you know that you will have to repeat an address.

If you have 100 envelops and more than 100 addresses, you might accidentally repeat an address, but might not.

If you have 100 envelops and an infinite list of addresses, you cannot accidentally repeat an address because the probability becomes zero.

If the universe has more possible states to exist in than the timeline has moments in which to exist, the universe might be able to accidentally repeat itself.


The infinite timeline has an infinity of points in time, but the universe has more than a simple infinity of states in which it can exist. That means that it doesn't have to repeat itself even given an infinite amount of time.


Space apparently has infinitely more states in which to exist than the infinite timeline. Therefore, without intentionally arranging to do so, the universe has zero probability of ever repeating any one state even given an infinite past.


He also went through the maths about why the universe has more than a simple infinity of possible states but if I followed it right, the above is the end result.

Additionally he also explained that the universe is not actually falling into heat death or maximum entropy.

What you did say is only valid if the universe is defined as all possible states. Only in this case, you would have 2^ininifty (number of possible states) > infinite (number of points on time). Universe is however not all possible states, it is one point in 3 (space dimension) * infinity (number of particles)* infinite (number of point in each dimension) = infinite (number of point on time).

[obsrvr524]

What you did say is only valid if the universe is defined as all possible states. Only in this case, you would have 2^ininifty (number of possible states) > infinite (number of points on time). Universe is however not all possible states, it is one point in 3 (space dimension) * infinity (number of particles)* infinite (number of point in each dimension) = infinite (number of point on time).

I was working from James' posting of this:

Using a Cartesian system, there are 3/4 * Pi * infinity^6 points in the entire universe.

Space has infA^6 first order points (or more precisely: 4/3 Pi*((infA^2)/2)^3) whereas
time only has infA^2 first order points.

That is an extremely significant difference, especially concerning why the universe has substance, "exists".

He proposed that a single line (1 dimension) has infA^2 segments (using the real number set),
a single plane (2 dimensions) has infA^2 * infA^2 = infA^4, and
a single volume (3 dimensions) has infA^2 * infA^4 = infA^6 segments

whereas a timelime (not using his 3 dimensional time) has only infA^2 segments.

thus "space is bigger than time"

Or when he used merely the whole number system:

Okay, now given that you have 10 cups with the random possibility of each cup having as many as 10 coins in it, what is the possibility that you have the same number of coins in all 10 cups?

Mathematically that would be (1/10)^10 or 0.0000000001.

The state of nothingness and the state of absolute homogeneity are actually the same thing. If there is no distinction in affect at all in every point in space, there is no universe. Thus for a universe to exist, there must be distinction or variation in affect between the points in space. What is the possibility that every point in space is of the exact same value of PtA (potential-to-affect)?

Well, just a single infinite line would give us infA points on that line. And each point on the line, assuming nothing is forcing any particular PtA value, might have a value anywhere from infinitesimal to infinite, a range of infA PtA.

So the possibility for every point on the line to have the same PtA value (given steps of 1 infinitesimal) would be;
Possibility of homogeneous line = (1/infA)^(infA).

That is 1 infinitesimal reduce by itself an infinite number of times. And right there is the issue. Also in 3D space, you actually have;
Possibility of homogeneous space = (1/infA)^(infA^3)

Normally in mathematics if your number has reached 1 infinitesimal, it is accepted as zero and is certainly close enough to zero for all practical purposes but we are literally infinitely less than infinity less than 1 infinitesimal. For 3D space, we are looking at 1 infinitesimal times itself an infinite number of times, times an infinite number more times, times an infinite number more times.

Given an infinite amount of time and with or without causality, the possibility of running across homogeneity of space is;
Possibility of homogeneity through all time = infA * (1/infA)^(infA^3)
Possibility of homogeneity through all time = (1/infA)^(infA^2)

With a possibility being that degree of infinitely small, not only can it never randomly end up homogeneous even through an infinite number of trials (an infinite time line, never getting up to even 1 infinitesimal possibility), but it can't even be forced to be homogeneous. A force is an affect. If all affects are identical, the total affect is zero. What would be left in existence to force all points to be infinitely identical?


But if that isn't good enough for you, realize that those calculations are based on stepped values of merely 1 infinitesimal using a standard of infA. In reality, each step would be as close to absolute zero as possible without actually being absolute zero using a standard of as close to absolute infinity as possible, "AbsInf".

Possibility of homogeneity through all time = (1/AbsInf)^(Absinf^2)

Now we get truly absolute zero possibility. If we are already as close to absolute zero as possible with "1/AbsInf", as soon as we multiply that by any fraction, we have breached absolute zero. And we have breached absolute zero by a factor of AbsInf^2, well, well beyond absolute zero possibility of homogeneity.

Thus Absolute Homogeneity, "Nothingness", is absolutely impossible.

Note that at each location, there is a measure of affect, called PtA, that can range from 0 to infinite. For existence to replicate, that measure of PtA must be identical from the first time moment to the replicated time moment in every of the infA^6 locations in space.

And then the possibility of space being at any specific chosen state is absolute zero. That is to say that the actual state of the universe at any given time cannot replicate or even be known under any circumstances.

[Silhouette]

You can tell the amateurish level of mathematics that we're dealing with by its presentation.

He's brazenly multiplying infinities all over the place. What do you get when you multiply, or perform any arithmetical operation on something that's undefined? Something that's undefined. Even adding 1 to quantity that has no bound still has no bound - his is the realm where you can make mathematical nonsense such as 0=1. E.g. "Infinity + 1 = Infinity, subtract infinity from both sites of the equation and bam" - that's what you get when you treat infinity like a finite quantity.

You can sum up what you've quoted of him in a couple of lines:
1) Conventionally we use 3 dimensions to measure space and 1 dimension to measure time, therefore space is bigger than time.
2) There's more ways to be heterogenous than homogenous, therefore heterogeneity is more likely i.e. everything being the same has lower entropy than everything not being the same.
The first line says nothing, and the second line just says "entropy".

So I take it back, you can sum up all that nonsense in one word: "entropy". There's no need to make a fool out of yourself just to explain what one word already explains.

[promethean75]

Yeah and tell James to stop brazenly multiplying infinities all over the place, too! Fucking affectance ontologists. They're almost as bad as value ontologists.

[obsrvr524]

What do you get when you multiply, or perform any arithmetical operation on something that's undefined? Something that's undefined. Even adding 1 to quantity that has no bound still has no bound - his is the realm where you can make mathematical nonsense such as 0=1. E.g. "Infinity + 1 = Infinity, subtract infinity from both sites of the equation and bam" - that's what you get when you treat infinity like a finite quantity.

Speaking of one who should look things up before posting.

James addressed that issue nicely by first acknowledging what happens when you try to use "infinity" in maths. He explains it doesn't work. He explains that "infinity" is insufficiently defined for mathematical use. Then he gives infA precise definition.

RM:Affectance Ontology: InfA
In order to deal with relative values of infinity, Rational Metaphysics:Affectance Ontology defines a particular infinite value;

infA ≡ (1+1+1+...+1)
ismlA ≡ 1 / infA

And thus;
infA * ismlA = (1+1+1+...+1) / (1+1+1+...+1) = 1
2 * infA = (2+2+2+...+2)

And;
infA / (2*infA) = (1+1+1+...+1) / (2+2+2+...+2) = 1/2


==================================================================

Rationale

(Identity A) / (Identity A) = 1

(Identity A) / 1 = (Identity A)

(Identity A) = (Identity A)

A = A



A common confusion came in the form of;
infinity / infinity = indeterminate,

which is what you were taught in elementary education. But "infinity" isn't an identity. It is merely one property of a potential identity revealing too little information.

It is like saying;
length / length = indeterminate.


Until you specify a particular length, you haven't enough information and thus, "indeterminate".

But if someone specifies a particular length;
(1 meter length) / (1 meter length) = 1, you have a different syllogism.

And when someone specifies a particular infinity ("infA");
(1+1+1+ ... +1) / (1+1+1+ ... +1) = 1, as;
infA / infA = 1,
to deny it is denying that A = A

Actually, what you are suggesting is almost exactly what I did back in the 80's to resolve the operations for infinities. I just annotated such things (math is just annotation) in such a way so as to maintain the rationale. This forum doesn't permit subscripting, so this might look a little confusing, but;

inf[a] = the base definition of the infinite in use.
inf[a+1] = inf[a] + 1
inf[a^2] = inf[a]^2

All of that might seen tautological, but it leads to usefulness;

inf[a] / inf[a] = 1 (as opposed to "indeterminate").
inf[a] / inf[b] = a/b wherein b has some known relation to a such as the number of points in two lines wherein one line is twice as long as another, perhaps b = 2π/a for circle issues.

The relational math is maintained within the cardinality designator [ ].

Calculus works in that same manner but merely deals with relations between infinity and infinitesimals with known relationships. My method expands calculus such as to allow for all of the hyper-reals and many otherwise "undefined or indeterminate" equations.

(I finally got that link thing working )


In short, it appears that James knew what he was talking about.

[Silhouette]

James addressed that issue nicely by first acknowledging what happens when you try to use "infinity" in maths. He explains it doesn't work. He explains that "infinity" is insufficiently defined for mathematical use.

And yet:

Using a Cartesian system, there are 3/4 * Pi * infinity^6 points in the entire universe.

3/4 * Pi * infinity^6 is just as infinite as merely saying "infinity". And as I covered, multiplying however many indetermined quantities gives you an indetermined quantity.

Then he gives infA precise definition.

He's defined infA the same as you would define infinity, just in an amateur format.

Admittedly writing in unicode for posts here is limited as far as I can work it, but something similar to the following (where i=1 should be below the sigma, and ∞ above) is how you represent how he presented his infA - in terms of an infinite sum:
i=1 ∞∑ 1ᵢ
Yet this is the same as representing infinity. Swapping the term out for "infA" does nothing.

He then goes on to treat infinity algebraically like a finite quantity, and then performs some more algebra to represent a logical tautology...

Next is an attempt to vitiate actual definitions that you use at all levels of education as merely elementary education, as his best attempt to legitimise what looks like a merely elementary education of his own - presumably to make it seem as though there's no point wasting your time with actually learning higher education, which would not incidentally reveal the nonsense behind his formulations, and potentially draw in amateur level mathematicians to his sophistry, who never wanted to get into higher education in the first place.

"infinity / infinity = indeterminate" is not like saying "length / length = indeterminate".

Length is a unit, and letting the numerator be x and the demoninator be y, you get x/y, which is determined in that form. Dividing one length by another just means the result has no units, not that it has "indetermined" units: it is determined as having no units.

Then he finishes off your first quote with a conclusion based on his invalid premises.

The second quote is more of the same infinity-algebra, pretending that things like an indeterminable quantity + 1 can be determined with precision, like you would do with determined quantities.

In short, it appears that James knew what he was talking about.

In short, your quotes show nothing of the sort.

[Jakob]

It is like saying;
length / length = indeterminate.


Until you specify a particular length, you haven't enough information and thus, "indeterminate".

I really miss this guy, crackling sharp.

Of course he's right, the number of real numbers is greater than the numbers of integers, even though of both there are infinitely many.

In the same way an unlimited three dimensional space is greater than an infinite line.

I like the argument that time is smaller than space, but I am not sure what is meant by the universe as consisting of all possible states. Except if all possible states means all necessary states. Possible states would comprise a meta universe that is never fully attained.

[Meno_]

Entropy can be reset to initial or previous state ?


Initial or original - appears here that this needs a real distinction. There may not be one, or even, it may not even necessitate a proof for one. That is the first introduced fallacy.

Second, states are a continuum. Prior states are infinitely divisible, and correspond to particular variables. Passing over this leads to a circularity with which to argue space/time is invalid. Invalidity and boundary problems are relativistic .

The reduction will appear in 3 forms , then, phenomenological, eidectic - ontological, and entropycal , - all part of the continuum modally. - changing timespace in terms of preception, understanding and representation.

At that point visualizing absolute space/time will not become feasable.with the problem approaching irresolutability.

That it is so embedded, is the problem.
Other way- absolute, and infinitely regressed fallibility. They may become obsolete by definition, including mathematical ones.

[Jakob]

Also entropy isn't technically a state because A=A doesn't apply to it.

Entropy means the absence of order, not a particular constellation of disorder, which would be a weird thing.

The laws of thermodynamics presuppose order, where heat as well as heath death are derivates of it.
Existence isn't actually of an expansive but rather a contracting nature.

[Jakob]

Other way- absolute, and infinitely regressed fallibility. They may become obsolete by definition, including mathematical ones.

The question James' work first of all evokes, is how there can be different qualities of infinitesimals?
Which actually addresses the deeper question of how one could compute one infinitesimal with another if they have no distinct features. So we do this by using different orders of infinity.

An infinitesimal of the rational order is smaller than one of the integer order. It is the noise within the noise.
I don't know if there is an infinite number of classes of infinity, but I don't think so. So thats a start.

[Jakob]

I prefer to give an infinitesimal a quality of affect - namely, valuing. Which draws it out of strict analytical infinitesimally - it just has no size or mass requirement to it, it is a minimal concept of a being rather than a concept of a minimal being.

For something to affect it must also be affected. Resistance.

In the end resistance, or friction, is the real cauldron of logical being.

(and traction is what's referred to as momentum)

"the moon grinding through space" - Capable

[obsrvr524]

He then goes on to treat infinity algebraically like a finite quantity, and then performs some more algebra to represent a logical tautology...

Next is an attempt to vitiate actual definitions that you use at all levels of education as merely elementary education, as his best attempt to legitimise what looks like a merely elementary education of his own - presumably to make it seem as though there's no point wasting your time with actually learning higher education, which would not incidentally reveal the nonsense behind his formulations, and potentially draw in amateur level mathematicians to his sophistry, who never wanted to get into higher education in the first place.

You haven't impressed me as someone who should be spouting accusations of amateurishness.

Apparently the hyperreal maths which address this exact issue were formalized professionally back in 1948. I image James had plenty of time to read about them and probably before you or I were born.

Hyperreal number
"*R" and "R*" redirect here. For other uses, see R* (disambiguation).
Infinitesimals (ε) and infinites (ω) on the hyperreal number line (1/ε = ω/1)

The system of hyperreal numbers is a way of treating infinite and infinitesimal quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers greater than anything of the form

1 + 1 + ⋯ + 1 {\displaystyle 1+1+\cdots +1} {\displaystyle 1+1+\cdots +1} (for any finite number of terms).

Such numbers are infinite, and their reciprocals are infinitesimals. The term "hyper-real" was introduced by Edwin Hewitt in 1948

Well, you are arguing with the entire world of mathematics (not merely science). Hyper-real mathematics was logically proven by Edwin Hewitt back 70 years ago. But regardless of that, I have my own proof that it all makes perfect logical sense as long as you understand the concepts properly (which you do not seem to).

I prefer to give an infinitesimal a quality of affect - namely, valuing. Which draws it out of strict analytical infinitesimally - it just has no size or mass requirement to it, it is a minimal concept of a being rather than a concept of a minimal being.

I might look into that "value ontology" someday but one ontology at a time (I'm still not even comfortable with that word). And at my current pace this could take years just to catch up with James' posts. I barely have any more time to read them than the pace at which he apparently was writing them. And then there is all of the parsing, categorizing and so on. My parsing program didn't like the downloaded file from this server. I have a friend who might get around to fixing something for me (someday).

[obsrvr524]

The question James' work first of all evokes, is how there can be different qualities of infinitesimals?
Which actually addresses the deeper question of how one could compute one infinitesimal with another if they have no distinct features. So we do this by using different orders of infinity.

An infinitesimal of the rational order is smaller than one of the integer order. It is the noise within the noise.
I don't know if there is an infinite number of classes of infinity, but I don't think so. So thats a start.

I think the issue was merely tying down exactly what was meant by a chosen infinity, such as the whole number set. And then the math is merely adding to or multiplying that set. The issue seem to be simply that if you have two identical whole number sets (perhaps in two different languages) then you have twice as many as merely one whole number set:
infA + infA = 2 * infA

I really don't see the complication with it.

With infinitesimals, it would be the same excepting using only the real numbers that are less than 1. He proposed that one infinetisimal = 1/infA. I started to go try to verify that with professional maths but immediately realized that he is the one declaring the definitions, so obviously he is right about them.

[Jakob]

The significance of the type of infinity is in the density of coordinates.
This density is lesser in the integer set than it is in the real numbers set.
It doesn't matter if you multiply the set of infinite integers, the density of coordinates doesn't increase. Whereas if you take an infinity of one of the more branched sets, you get a higher density and a deeper infinity.

Not sure if this is how James thought.

[Silhouette]

Apparently the hyperreal maths which address this exact issue were formalized professionally back in 1948. I image James had plenty of time to read about them and probably before you or I were born.

Oh yes, the hyperreals are a legitimate number set, but that doesn't mean you can just do whatever you want with them and expect it to be valid.

There's plenty of introductory material on the subject floating around on the internet, but actual examples of their proper use seems to be sparse.

Most of the actual usage of hyperreals in arithmetic seems to be done through sets, for example here where every term in the set is operated on sequentially with the corresponding term in the second set. So it would be legitimate to represent a specific divergent series within a set, and operate mathematically upon that. Dealing with specific infinite series (tending to infinity) is done all the time and is fine.
Other usage seems to be in conjunction with sets of numbers like the real numbers, in order to get a real number result. Concepts like 'dx' are used all the time in calculus, which is fine because they represent a tendency towards infinites.

James may have been perfectly aquainted with their usage at the time he wrote what you quoted, but none of it shows. He said he had his "own proof" so apparently intentionally deviated from the legitimate pathway, and what he came up with is extremely dubious.

You say you don't see the complication with infA + infA = 2 * infA because you're using your intuitions about finite quantities. Infinities easily result in nonsense and have to be dealt with very carefully.

I'm hearing things like "densities" in posts on this thread, but it doesn't matter how relatively closer numbers are when they're added to infinity, for example, the sum is still divergent and not-finite i.e. can't be defined. It goes on forever so never reaches a point at which it can be compared with another way of going on forever, such that one can be called "bigger" than another. Size presupposes finitude: something "infinitely big" has no bounds to compare to anything else - it can't even be ascertained to be a specific thing. The only thing it can be ascertained to be is a set of smaller things - maybe even a specific infinite series, which like I said is fine. Care is needed.

[obsrvr524]

From Dr Math, the article that you linked to:

"There are good reasons to believe that nonstandard analysis, in some version or another, will be the analysis of the future." - Kurt Gödel

Euler made significant advances in infinitesimal calculus in his book Introduction to the Analysis of the Infinite. In his work, Euler freely used the notion of infinitesimals and tacitly assumed that almost all of the usual laws of arithmetic carried over without modification, plus a few extra properties that somehow seemed natural to him (and were later formalized and proven correct in an accepted manner).

That author points out that people have had a lot of trouble understanding the issues of infinity to the point of banning them from professional maths only to have to put them back in later.

I'm sure that I'm not in a position to argue with Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others who James seemed to have agreed with. You seem to have trouble with it all just as the author of that article, Hermoso, admitted to having. I wouldn't be able to competently take up the argument between you and them. But I do have a couple of questions for you.

1) Do you have trouble with the idea that one infinite set can be known to be larger than another?

2) Do you find something specific, very specific, that you consider to be invalid reasoning or usage from James? Please exactly, precisely, quote an example of the error.

[Ecmandu]

James only believed in a six dimensional reality, I never saw him attempt a proof for this.

Personally, I think there is a 'divine' sequence that orders the reals in 1:1 correspondence.

An algorithm that is infinitely chaotic won't formulate an expansion, it'd be indecipherable... I could make a very long post to show this, that you can't prove higher orders of infinity. Maybe I'll have the energy later.

[Silhouette]

That author points out that people have had a lot of trouble understanding the issues of infinity

I'm sure that I'm not in a position to argue with Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others

You seem to have trouble with it all just as the author of that article, Hermoso, admitted to having.

The author of that article I linked, and both you and I aren't in a position to argue with Euler, Hewitt, Newton, Gödel, Robinson and probably hundreds of others too. How about James?

The difference between his usage and the last 70 years of the number set being accepted to the extent that it has been seems striking to me, as does the logic behind his usage. You, James and I may not be world famous mathematicians, but does that mean we are equally amateurish? It's not without problems for each of us to argue the relative legitimacy of our arguments, and regardless even of the full potential that we each achieved at any given point during our respective lives, the standard of what James wrote on his blog some years ago is of the sort I used to play around with when I was a child. Having achieved top marks throughout my mathematical education, and maintaining significant interest and exposure to much higher levels over the decades since, I know there are at least some objective measurements to justify at least a higher level of amateur ability within myself - and I have no interest in overstating this subjectively, but I do have interest in criticising content that appears to me to be exhibiting lower levels of amateur ability, yet is gaining traction and influence over others who may be susceptible to mathematical sophistry due to their own standard being insufficient to see past it. Hell, I might be wrong, but I know there's plenty of reason for me to not be. I will put what I have out there, and you can take it or leave it, though I recommend you take at the very least a healthy amount of skepticism with you that you don't yet appear to be exhibiting.

1) Do you have trouble with the idea that one infinite set can be known to be larger than another?

2) Do you find something specific, very specific, that you consider to be invalid reasoning or usage from James? Please exactly, precisely, quote an example of the error.

1) I have zero trouble with how people can think they understand one "infinite quantity" to be larger than another even though it's undefinable, using their intuitions about finite quantities. The Hyperreals meet the transfer principle with respect to the Reals, but that does not make them equivalent - especially in how to treat their results.
For example, I have zero trouble with representing two infinite sums added together as twice the initial sum, particularly if it is a convergent series. However, to gain meaning from doing the same to divergent series is not without problems that need to be approached with due respect and caution. You can "represent bigger infinities", giving a semblence of size comparison between two or more, but this still makes no real-world sense as they both diverge forever and therefore never get to the point where they can be compared. Any specificity in constructing and comparing infinites is in their means of construction, not in the end itself - which you can only physically get to, by definition, if it's a finite value. Do you have trouble accepting this logic?

2) I've been giving specifics all this time, in particular in this post, which starts off with the most glaring contradiction so far:

James addressed that issue nicely by first acknowledging what happens when you try to use "infinity" in maths. He explains it doesn't work. He explains that "infinity" is insufficiently defined for mathematical use.

Using a Cartesian system, there are 3/4 * Pi * infinity^6 points in the entire universe.

But I'm re-reading previous posts of mine as well and they too are specific, exact, precise and with quotes for reference - as you requested... Do you have trouble accepting their logic?