Even you’re link says they’re rounding the numbers to prove the theorem. Come on James, you can do better than that. Well… of course T/F/ - N/A, I was giving you the paradox grid.
Rounding what numbers? The hyperreals?
Do I have to give you a lecture on hyperreal numbers too? 
Wikipedia:
“This put to rest the fear that any proof involving infinitesimals might be unsound, provided that they were manipulated according to the logical rules which Robinson delineated.”
“for an infinitesimal , where st(·) denotes the standard part function, which “rounds off” each finite hyperreal to the nearest real. Similarly, the integral is defined as the standard part of a suitable infinite sum.”
It’s apparent that you didn’t understand what that meant. I used the standard called “infA” to signify what all other powers of infinity are going to mean. And one infinitesimal, raised to no power, is merely 1/infA. His standard was called “st(.)” which refers to what one infinitesimal (not raised to any power) is to mean.
Regardless of the notation standard, any single first order infinitesimal (being not raised to any power) ignores all of the higher powers, thus “rounds off” any of those left over if you are using them. It is no different than rounding off anything below 0.00001, except infinitely smaller.
In my explanation, I first went through a derivation using merely Georg Cantor’s infinity-squared real numbers, from one first order infinitesimal up to infinity (“infA^2”). But that leaves room for speculation below one infinitesimal, so I continued. The second portion explains the proof in terms of the absolute highest and lowest possible numbers so as to prove that there can be no homogeneous state because it requires numbers less than absolute lowest possible, aka “impossible”. That is how you get to “absolutely no possibility”, not merely an infinitesimal possibility.
You forget the biggest problem with infinitesimals James, and it’s a mathematical fringe, VERY fringe, like 1 dude (there’s this one professor that believes in infinitesimals), is that there are an infinite number of zeroes before the 1… the 1 never gets expressed.
You also don’t understand the concept of dimensional flooding, which I invented, and tried to teach you, when dealing with infinities.
As for the hyper-reals, infinity doesn’t express itself as a totality… I tried to explain this to you with a tree analogy. There is no infinity tree that contains that set of all trees in it… though there are an infinite number of trees that are bigger than another tree (and smaller). I understand this is you’re proof… that infinity has no biggest or largest but exists. You cannot prove infinity necessarily exists using your mathematical techniques. You have to use more basic logical argumentation to do this.
We know there is basically a grid…
we have
true
false
N/A
true/true
true/false
false/true
and
false/false
It’s true that it’s true. = true
It’s true that it’s false. = false
It’s false that it’s true. = false
It’s false that it’s false. = true / paradox / false
A perfect example is “This statement is false”
The only truth value for the statement is it’s existential value, which isn’t explicit in the statement… We know the statement exists, therefor we know the statement is true on one level as existing, and false on another level, as claiming it doesn’t exist. So we can divide the paradox into different claims and resolve the paradox… it’s true in this sense, and false in this other sense.
What people generally do, is they say, if the statement is true, then it must be false, and if it’s false it must be true. The reason false and false have this effect is because “false” is OTHER THAN… so what happens with false being false is that it’s itself and OTHER THAN itself… this is why false/false on the grid creates paradoxes. The word false itself is a paradox in it’s own context… it’s OTHER THAN, but it’s OWN IDENTITY at the same time… the IDENTITY of OTHER THAN / a paradox…
False can’t exist as a word, and yet there it is for all of us to see…
Now we generally use context… other than this (instead of other than), and false doesn’t run into the paradox, but when it feeds back on itself or doesn’t have this context, it falls into the paradox.
But all you have to do to get out of the paradox is just look at it as a contradiction
It is what it is and isn’t what it is… it’s true/false ---- true that it’s false and false that it’s true, which both yield false. which only leaves you with the true/true grid, which leaves you with existence exists… it’s true that existence is existence, and all other possible solutions are false.
None of that is the slightest bit relevant and merely tells me that you have a hole typical in you understanding of logic. There is a saying concerning that sort of thing.
An “ass” is a stubborn, non-thinking creature. The ground is one’s under-standing. And you can’t tell the difference between your ass and a hole in the ground.
No I don’t. You keep reminding me. The biggest problem with infinity an infinitesimals is that stubborn, simple minded people can’t handle the concepts.
Surely even you can handle simple arithmetic (I dubiously hope).
If you count all of the whole number lengths along an infinite line, you would count an infinity of them, right? And if you count all of the real numbers below one (the decimals), you would also count an infinity of those. So if you then add your two counts together, what do you get? And before you answer, try to realize that the very meaning and definition of “add” is to put MORE into something than before, assuming that you aren’t merely adding nothing, zero.
The first count, I call “infA”. And as it turns out, that second count would be equal to infA. So if you can add, you should end up with 2infA, or twice as infinite. All infinities are not equal. And if you count all “real numbers”, you get infAinfA or infA^2, much larger than infA called the “second order”.
Georg Cantor went to his grave trying to explain that to people who did nothing but demean him for being insane. They even posted how foolish he was on his tombstone. Yet today, you would have a hard time finding a mathematician that doesn’t agree with his conclusion. They just mostly know not to confess any association with him.
James… if you have an infinite number of zeroes before a 1, the 1 never gets expressed, if you have an infinite number of zeroes after a 1, the last zero never gets expressed.
That’s basic logic.
You’re trying to sound like some monster of a mind that expresses that one or zero, and I’m just the feeble minded one who can’t comprehend a bound infinity. INFINITY IS UNBOUND… if you put something UNBOUND before something that is bound… you NEVER get to the bound thing, EVER, NEVER, EVER.
You might as well call yourself God, James… and try to pull that trick on us too. How feeble minded I am… seriously James.
I ask you a question Ecmandu. Can you answer it or not?
Adding is a process of before and after… you CAN’T add them. I’m sorry, but that’s the way it ACTUALLY works. You’re the one who’s always going on about people and their proagandic bullshit… and you can’t even see the basic error of logic here. Infinity is a process not a thing!!! To add two infinities together, means that one has to precede the additive process… which means you never get to the second one. We can only add finites!!!
And no. Infinity is NOT a “process” (an error you keep making). Infinity is a quality. And it is a quality that can become greater than before, twice as great as it was.
Let me put it to you this way James…
0.3… + 0.6… = 0.9… right?
Wrong!!!
0.3… NEVER ENDS… you can’t even get to the plus sign to add it.
what you have is:
0.3…
and 0.6…
and 0.9…
but NO 0.3… + 0.6… = 0.9…
That’s nonsense. Use your logical brain. I know you’ve been brainwashed by mathematics to this regard, I’m trying to un-brainwash you… it’s psychosis to believe you can add infinities.
You are still thinking of infinity of a PROCESS such that it has a beginning and an end. It doesn’t “end” because it never began because it is NOT a process occurring through time. It is the quality of a situation.
And yes,
0.3… + 0.6… = 0.9…, because the entire infinite series is added simultaneously, because it is only two numbers, not an infinite series of numbers to be added sequentially.
You very seriously need to reconsider ever trying to “un-brainwash” me or anyone.
Let me go a little further to explain how arbitrary mathematics is James…
1/2 = .5 right?
NO !!! actually it still equals 1!!! The other half is STILL THERE!!!
But lets dig a bit deeper here… since you’re such the math expert and I’m the fool…
1/2 equal 1 right? NO!!!
actually it equals 3 or 4!!!
when you divide one thing 2 times you end up with either 3 or 4 objects.
It’s in determinant.
But that’s right, I’m a moron and you’re the omniscient math wizard.
Okay, that’s enough of that.
Infinity is an object? Honestly James. You think 0.3… is actually an object that exists as bound? Do you really think you’re saying 0.3… when you say 0.3… ?
I know when I say Hello… I’m saying hello…
But when I say 0.3… I know I’m not really saying an infinite string… I’m using 6 points to describe something that goes on forever… apparently that makes sense to your huge brain!!
You can’t comprehend the difference between an object and a quality of an object/situation?
You had expressed two objects (numbers), both of which had an infinite quality. When you add the objects, you double the quality = 2*infA.
Ok, let’s work with the word quality for a bit then… infinity has the QUALITY of being unbound… so how do you propose to add one to the other?
Try to say the QUALITY of 0.3… before you get to the plus sign.
Try as hard as you can James.
Yup, that’s right, even the great unbrainwashable James, has been brainwashed.
No.
Infinity IS the quality of being unbound.
The two objects were two numbers that cannot be expressed in decimal form without inferring an infinite series of decimals to represent EACH SINGLE NUMBER.
You are conflating the representation of an object with the object. The numbers were not at all infinite, merely the mono-decimal representation itself. The two numbers merely represented two unbound objects.
There is no “before”. There is no process other than conceiving of double the quality that you originally had.
For one… under your theory 1 followed by an infinite number of zeroes and 0 followed by an infinite number of ones equals 2
I understand that the referent is not the object… LOL you just called infinity an object!!! James, you really don’t know what you’re getting into here. Wake up man!!
You did exactly what you said you weren’t going to do. I know your whole philosophy of ethics revolves around infinitesimals as well… so this is basically your whole life here, but man, you don’t know as much about math as you think you do… you can parrot, but you can’t question with authority, you’re even afraid of it.