That’s the point.
And in terms of the mathematical realm and the physical realm:
That one statement tells your whole story.
“Thou shalt not question!”
We are definitely agreed regarding the point on which we disagree.
You said something about the continuum being made up not of points, but of little segments. Can you tell me more about that? I’d like to understand your model of the reals.
The real numbers are merely designated location points along a line segment. There can be an absolute infinity of them on even the smallest of segments because they occupy no space at all. And that is where Zeno comes in.
If you keep assigning location points by a pattern that forbids locating up to the end of the segment (such as 1/2 then 1/2 of the remainder then 1/2 then 1/2… or 90% then 90% then 90%…), you will never be able to get to the end because you have forbidden yourself from getting there by the chosen pattern (which is why 0.999… can never equal 1.0). But you will still be able to keep adding location points eternally because you can never run out of numbers.
The physical issue is that the universe is not made of location points, but of the space between them (in this case line segments). There is no zero length line segment yet they can be infinitely short.
We are in absolute agreement on that. A real number is the location of a point on a line.
What is the definition of absolute infinity? As you know in standard math there is an uncountable infinity of points between any two points on the real number line; and specifically, a cardinal number called 2^AlephNull. But there are infinities bigger than that, and bigger than that. I’d like to understand what your absolute infinity is.
I can see that this picture is very much like the hyperreal numbers. In fact if we were in the real numbers, I would say that your sequence must converge by the completeness of the reals … the property that there are “no holes” in the real number line.
However the hyperreals are not complete … by pretty much the same process as you describe. At the “end” of your sequence, there’s a hole, not a hyperreal.
Am I understanding your point of view?
The idea that the real numbers are always in a state of “becoming,” and that new points are brought into creation by the conscious choice of a subjective intelligence, is part of (my very humble understanding of) intuitionist mathematics. Is this part of your idea? That the real numbers are continually being created by choices?
This of course is our point of disagreement. I don’t think math necessarily has anything to say about the true nature of the real world; you do.
I don’t wish to argue this point at the moment. But I’m struck by your
certainty that you, personally, are possessed of a truth that still eludes modern physicists. The question of whether the world is continuous or discrete is not yet answered, or whether it’s made up of points or segments or something else. Some people (me for one) don’t think ultimate reality can be known. What makes you certain that only you know the truth?
In order to understand the thread and what is meant in the opening post of this thread it is favourable to not only consider mathematical aspects but also and especially, namely in relation to mathematics, physical aspects (often called “reality”).
I use the term “absolute infinity” to refer to the absolute highest possible order of infinity (even though there is no upper limit).
I don’t see any direct association to the hyperreals. And even though the real number line has no “holes”, it also has no zero lengths. 1 - 0.999… cannot equal zero.
A hole?? And what end? There is no end … ? 
I suspect not.
Of course numbers are created by conscious choice. They are not physical entities. But that is not implying any kind of magic. One merely assigns as many numbers as he wishes, endless if need be. There is nothing magical or supernatural about it.
Math would be pretty pointless if it had nothing to say about the real world. Math isn’t a fantasy free to just go off in any direction. Quantum physicists love to do that, apparently merely to bemuse the populous. Quantum physics is a fantasy (although not quantum mechanics).
First, I don’t consider modern physicists to be the brightest people on the planet (I have known and listen to too many of them and even Feynman was complaining of them). And truth is merely a matter of ontology. A good ontology is very mostly about clean logic. Empiricism is for the express purpose of verifying hopefully logic based hypotheses. One cannot empirically discover truth. That is nonsense. One must already have a hypothesis in mind to merely be capable of observing anything at all. Your neural network (“wetware” and senses) provides for that before you are born. Empirical data merely lets one know when his hypothesis or suspicion is possibly true or definitely false. And truth to be gained from empirical observation must be created and assembled by the observer or theorist.
As far as why I think that I know what I do concerning the universe, my Rational Metaphysics and Affectance Ontology explain my confidence. RM requires that literally everything involved in the construction of an ontology must be absolute logic (void of alternatives). And ever ontology begins with definitions of the concepts involved.
RM:AO begins by defining existence as “that which has affect”. That is the foundation of Affectance Ontology. If something is proposed to have no affect, it is declared (by rational choice) to be non-existent within the ontology. Why waste mind time on things proposed to have absolutely no affect on anything?
That one act alone, of declaring what existence is in a meaningful way, you will probably find to be unique. And you are certainly invited to find out if that is true. I am unaware of anyone throughout history who has defined existence in such a way or actually in any meaningful way.
From that beginning, the attributes of affect are not discovered but proclaimed such as to leave no room for question. From such fundamental proclamations, necessary logic consequences are apparent. And as it turns out, and as was the hope, the final consequence of the logic is that everything that science has observed to be true is fully explained from the very most fundamental thought (that existence of made of affect(ance)). All of the “fields” in physics, all of the particles in physics, all of the “forces” in physics (gravitation, electromagnetic, static potentials, inertia, momentum…) are the necessary logical consequence of affects upon affects (and void of any granular notion of reality). Modern physicists cannot tell you why opposite charges attract. But I can explain why in great detail, as well as why any of it exists in the first place.
The confidence came first through the logical purity of the assembly, but really became solid when my computer emulation of the fundamentals (“Jack”) inherently formed particles out of random chaos in a metaspace that behaved exactly as science now knows subatomic particle to behave. The probability of me constructing an ontology that is thoroughly solid and just happens to exactly fit all of the numerous requirements of subatomic physics and yet not be accurate is infinitesimal.
And even more importantly, every explanation throughout history has always been merely someone’s ontological construction. That is what Truth is. And there can be many ontologies that all fully explain the same reality. Thus mine is merely one of many possible ontologies, some of which could be more useful than mine. An ontology is merely a mental language. As long as the ontology is logically clean and reflects the behavior of physical reality, it is true (because that is what “true” means). The empirical evidence provided by the common scientific method then proves that the ontology is not false with extreme probability of being exactly true. Thus a true unified theory for physics was formed.
…and other short clips from the video.
Do you mean the infinite set with the largest number of members ? But if there is no upper limit is the term just meaningless ?
In the poll seven agree with the proof but ten do not so there is doubt about its validity
Think of it as the asymptote of infinities (much like 1 is the asymptote for 0.999…).
By popular acclaim I’m happy to leave it at that.
Well when one can’t win a debate even with the admin and two mods helping him out, it might be time to rethink one’s ontology.
Or one’s debate partner. Surely winning is more than refusing to admit when you’re wrong?
What’s to admit, when one or the other point of view is embedded in the paradox? The paradoxical paradox, becomes a paradox, and so on, where the paradox is simply an inconsistent use of languages.
(Logical, mathematical, @semantic.
The debate seems to have been between the following positions:
For 1 = 0.999…
1• Ordained mathematicians have the authority to define them equal regardless of argument.
2• The sum of the series is defined as the “limit” of the series and the limit is 1.
3• Zeno’s paradox requires that it be accepted as true, else distance could never be achieved.
4• 1/3 = 0.333…, therefore 1/1 = 0.999…
For 1 ≠ 0.999…
1• “0.999…” is defined as an infinite series requiring an ever-present remainder between it an 1.
2• “1 - 0.999…” requires an ever present infinitesimal, as displayed by the infinite series summation.
3• 1 is a natural number whereas 0.999… is merely an indication of inability to reach a number.
4• 1 is a bound decimal whereas 0.999… is an unbound decimal. The same number cannot be both.
I provided refutation for each of the former, “for =”, arguments. The only refutation of the latter, “for ≠”, arguments was “that isn’t what they told me”.
Refutations against 1 = 0.999…
1• Even ordained mathematicians are restricted to logic (real mathematics isn’t a religion)
2• The “limit” is defined as the boundary of a series, not the sum of a series.
3• Distance is formed of distances not numbers. There can be an infinity of numbers along any distance.
4• “0.333…” also is an indication that the value of 1/3 cannot ever be established in decimal.
Refutation against 1 ≠ 0.999…
1• Ain’t so
2• Ain’t so
3• Ain’t so
4• Ain’t so
James, I’d direct you to Daniel Dennett’s summary of ‘Rapoport’s rules’ for civil discussion:
It seems like you’re not living up to the first rule. Defeating a caricature is no accomplishment.
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Well those “rules” were not stated, but if those are what you would like to go by, it seems that you are not meeting them any better than I.
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I can’t recall anyone on this entire site ever saying “Thanks, I wish I’d thought of putting it that way.” And I am certain there is a reason for that. What was your reason for never getting that response?
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We did (a bit to my surprise and only due to your cooperation) state a few things that we could agree upon (as I have always recommended and attempted = Resolution Debating). And that helped. But was insufficient when it came to not wanting to agree with a necessary conclusion that would completely undermine your argument (and thus resorting to distractions).
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I don’t think anyone ever said/says “Oh, I didn’t know that” after a debate begins, although at times they might infer it.
If you want to get serious about debating to resolution, I can provide even better rules of engagement. But as always, they depend upon the cooperation of the opponents and thus require a strict and impartial moderator experienced with strict logical discussion (since no one cooperates otherwise).
While I think this thread, like too many conversations where you and I butt heads, is a lost cause, I’d be interested to see other guidelines you know of intended to make discussions more productive. It would be worth keeping a list of such sets of guidelines, with an eye toward getting a critical mass of people committing to various strategies for improving dialogue.
And I agree that I frequently fall short of those ideals, including in this thread. But they are something to aspire to.
The problem with Resolution Debating is that the participants end up having to face the truth (not a popular theme during the new age). Truth and agreement are not in vogue during this era.
Basically the idea is that a topic is analyzed by both parties by addressing one assertion at a time. As soon as an assertion is not agreed upon and the issue isn’t merely poor wording, that assertion becomes a new topic to resolve. With many people participating, everyone finds out where within the Tree of Understanding they fit. A branching code can be used to designate the different branching topics, their relation, and upon what branch any one person currently sits.
Logic (informal) determines when the process may proceed to the next stage. There are two basic flow charts indicating the moderator’s task (put together years ago):
LA = Logic Moderator
In the long run, a mountain of understandings and open questions, along with who fits into which group, get generated. With the right participants, eventually there are no more questions left.
Is 0.999… a number or a pattern that can match numbers?
If it is a pattern then does it match only one number?
If it does not, then how can we say it equals to 1?
We might be able to say “it is possibly equal to 1”, and only under the condition that it matches 1, but not that it is strictly, unambiguously, equal to 1.
Does it match 1?
It matches numbers 0.9, 0.99, 0.999, etc but it never matches 1.
So it is neither strictly equal to 1 nor possibly equal to 1.
What can be said it is is approximately equal to 1.
The error can be said to be 0.111…
The other problem is that the pattern that is 0.999… is not clearly defined. Does it match 0, for example? One is intuitively inclined to say “no” but this isn’t very reliable.
We can define the pattern 0.999… in the following manner:
{0.9, 0.99, 0.999, …}
The list gives us the idea which numbers the pattern matches and which it does not. For example, defined in this way, we know the pattern does not match 0 and we also know it matches unlisted numbers such as 0.9999.
Patterns can match one or more numbers. There may be a set they can match or there may be no such set.
This pattern matches more than one number and it matches no set (the set that it matches is called “infinite set” because it does not exist.)
Pattern simply means “one of permitted values”.
It can be represented using sets.
An example:
sqrt(9) = {+3, -3}
2 + 2 + {0, 1} = {4, 5}
2 + 2 + {0, 1, 2, …} = {4, 5, 6, …}
So let’s apply this to one of the proofs from the Wiki. The proof goes something like this:
x = 0.999…
10x = 9.999…
10x = 9 + 0.999…
10x = 9 + x
9x = 9
x = 1
Let’s try and follow the steps.
x = {0.9, 0.99, 0.999, …}
10x = 10 * {0.9, 0.99, 0.999, …}
10x = {9, 9.9, 9.99, …}
On the other hand…
9.999… = {9.9, 9.99, 9.999, …}
10x, it appears, is not equal to 9.999…
9.999… does not match 9.
Whereas 10x matches it.
In reality:
10x = {9, 9.999…}
But if we say that 9.999… = {9, 9.9, 9.99, 9.999, …} this will no longer be a problem.
However, the problem will not go away, it will merely change its location.
Now, the problem is that 9 + x does not equal 9.999…
9 + {0.9, 0.99, 0.999, …} = {9.9, 9.99, 9.999, …}
It does not match 9 whereas 9.999… matches 9.
In reality:
{9, 9 + x} = 9.999…
If we redefine 0.999… to match 0 the problem will disappear from this place but it will move back to the first place.
In other words, no matter how we define the two patterns used in the proof (0.999… and 9.999…) the logic remains invalid.