"0.999... === 1" is at the very least internally consistent

[Magnus Anderson]

Nonsense - there were things that were 3 times as long as other things.

"3 times as long as other things" is signficantly broader than "3 meters". That can also be "3 centimeters".

The "meter" didn't exist until we decided for it to exist - before that - nothing was "1 meter long"

It is the word "meter" that didn't exist. The length denoted by it very likely did. It wasn't the length itself thas was invented by humans but the word representing it.

(and in the extreme precision - still nothing is exactly 1 meter long - despite our presumptuous declaration).

The word "meter" was defined as the length of some particular object or distance. As such, by its very definition, there at least were things that were exactly, 100% precisely, 1 meter long.

We aren't the ancient Greeks trying to discover the atom. We are modern day rationalists trying to understand the make of physical reality - and it isn't what you think (yet).

You are not trying to understand "the make of physical reality" if you say that the number of divisions is "an arbitrary thing" and if you're talking about "us choosing to divide it [a second] infinitely into imagined infinitesimals". That has nothing to do with reality and everything to do with our models of reality (which can contain entirely arbitrary and imaginary entities.)

And we aren't trying to discover the atom here. We're talking about what follows if we accept a number of premises (one of them being that time is infinitely divisible.) The point that I'm trying to convey is that we're not talking about how many times time can be divided in our heads but how many times it can be divided in reality. That's what the term "infinite divisibility" refers to. It has nothing to do with human imagination.

That is the issue I was trying to get to with the questions that I have been asking (and not getting a response for).

So what's your response?

[Magnus Anderson]

Are you familiar with the terms "bending space" and "gravity well" as they are associated with relativity and motion?

I am not familiar with special relativity.

Can we all agree that there can be no absolute highest cardinality or degree of infinite?

What's the relevance of this question?

[obsrvr524]

Are you familiar with the terms "bending space" and "gravity well" as they are associated with relativity and motion?

I am not familiar with special relativity.

That's going to make it hard - if not impossible - to clearly explain the difference between linear maths and physical motion.

Can we all agree that there can be no absolute highest cardinality or degree of infinite?

What's the relevance of this question?

Does your answer change based on the relevance?

The issue is that because there can be no upper limit to numbers or infinities - there is no absolute infinite - there also can not be a lower limit --

If there is no "absinf" limit - there can be no "1/absinf" limit.

[Magnus Anderson]

Does your answer change based on the relevance?

No, it does not. I just don't think I should answer questions without first being sure they are relevant.

The issue is that because there can be no upper limit to numbers or infinities - there is no absolute infinite - there also can not be a lower limit --

If there is no "absinf" limit - there can be no "1/absinf" limit.

I think the biggest issue right now is that we appear to disagree on the meaning of the term "divisible". You appear to be using it to mean "able to be cut into smaller pieces using imagination". That's not how I use the term, and in my defense, that's not how people used it ever since Ancient Greeks. I am not talking about what can be imagined. I am talking about what can be done in reality. An apple can either be infinitely cut into smaller pieces or it cannot be. I don't care whether or not such can be imagined.

[obsrvr524]

that's not how people used it ever since Ancient Greeks.

That is not true.

I am not talking about what can be imagined.

Then you are not going to talk about maths.

I am talking about what can be done in reality.

Then you really need to learn physics.

An apple can either be infinitely cut into smaller pieces or it cannot be. I don't care whether or not such can be imagined.

Then nothing can be divided.

You can't cut a horse in half - much less "1/2 + 1/4 + 1/8 ..."

[Magnus Anderson]

That is not true.

Both Democritus (atomist) and Aristotle (anti-atomist) used it that way. They were talking about what's possible in reality. Democritus claimed that there exist physical objects that are inherently indivisible in the sense that it's impossible to cut them into smaller pieces regardless of what anyone did. Aristotle disagreed. Neither cared much about how many times a physical object can be cut in one's mind (a trivial matter.)

Then you are not going to talk about maths.

Maths isn't about what can be imagined, it's about what kind of relations exist between mathematical concepts.

"0.999..." means "The result of the infinite process of addition that is 0.9 + 0.09 + 0.009 + ..." If such a result doesn't exist because "..." means "no end" and "no end" means "it can't be completed", then "0.999..." is an oxymoron (since it represents the result of something that has no result.) But at the same time, if that is true, it is also necessarily true that an infinite number of moments cannot pass, for the same exact reason.

If you think that a second is infinitely divisible only in one's mind, then you do not really think that it is infinitely divisible, and you're really an atomist who's treating things as if they are infinitely divisible.

Then you really need to learn physics.

I am pretty sure I don't. That would be a massive overkill.

Then nothing can be divided.

In what sense? There is a difference between "I can't cut this thing because I don't have the means to do so" and "I can't cut this thing because it can't be cut any further regardless of what anyone did". There are personal limitations and then there are inherent limitations. Here, we don't care about personal limitations. We really only care about inherent limitations.

[obsrvr524]

-
Bullocks.

It seems to me that maths is almost entirely about imagined quantities and operations upon them - sometimes applicable to physical reality - sometimes not.

There is a point down around that Plank length where general relativity, quantum physics, and Zeno's paradox all come together coherently - without leaving gaping holes in logic or reality - but your attitude seems to be one of ignoring planetary and astronomical issues because you understand that the Greeks already discovered that Helios is responsible for dragging the Sun across the sky.

Arguing colors with a dog is particularly senseless when the dog doesn't even want to know.

[Magnus Anderson]

It seems to me that maths is almost entirely about imagined quantities and operations upon them - sometimes applicable to physical reality - sometimes not.

There are real quantities and there are imagined quantities. The number of people in the world is a real quantity. The number of elves in Tolkien's universe is an imaginary one. Maths is about neither. Though I can understand why someone would say it's about the latter. Not completely false but not completely true either. Maths is strictly about mathematical concepts and their relations. Quantities, whether real or imaginary, are not necessary in order to do math. Numerical concepts, however, are. Luckily, for both of us, this is not a relevant question.

The relevant point is that to say that time is infinitely divisible means that for any two real moments a and b there exists a real moment c that is located between the two moments a and b. It does NOT mean that for any two real moments another moment can be imagined to exist between them.

This entire thread is solely concerned with conceptual analysis. Everything I did so far has to do with concepts and what kind of relations exist between them. I never spoke about anything other than concepts. As an example, I never claimed that time is infinitely divisible or that time is finitely divisible. I merely claimed that if you say that a period of time we call "second" is infinitely divisible and that an infintie number of tasks cannot be completed, that it logically follows that a second cannot pass (for it would entail a passage of an infinite number of moments.)

There is a point down around that Plank length where general relativity, quantum physics, and Zeno's paradox all come together coherently - without leaving gaping holes in logic or reality - but your attitude seems to be one of ignoring planetary and astronomical issues because you understand that the Greeks already discovered that Helios is responsible for dragging the Sun across the sky.

Arguing colors with a dog is particularly senseless when the dog doesn't even want to know.

You have to prove that the direction you're suggesting we should go is the right direction. I don't think it is. Right now, what you're doing is complaining that the other person is not agreeing with you right away.

[origami]

Posted this last week, but it had to be approved. Reposting it now.

When you try to move from point A to point B using Zeno's algorithm, the total amount of distance that you will cross is equal to "1/2 + 1/4 + 1/8 + ..." times the distance between A and B. And "1/2 + 1/4 + 1/8 + ..." isn't equal to "1" for the same reasons that "0.999..." isn't equal to "1".

It doesn't quite work that way. Let's postulate, instead of the set you represented, 0.525125..., 5 being a representation for 1/2, 2.5 for 1/4 and 1.25 for 1/8. In this number, the quantity is never going to be above 0.6. After we determine that, we are just trying to find out how much of 6 is there, 6 and however much. But less than 6, in any case. So on for every digit in the number. In the case of 0.999..., the quantity is clearly above 0.9. How much above? The next digit asks the same question infinitely.

What I am trying to say is that the number 0.999... doesn't postulate that 1 is made of infinite 9's. 9 here is arbitrary due to the decimal system, in another base it would be whatever single digit of that base gets closest to 1, to the next order. The number 0.999... is simply a notation for 1. What matters isn't the number, as has been stated, the number is imaginary. What matters is the quantity.

What matters about postulating that 9 is repeated infinitely is not a postulation of an actual progression of infinite 9's, but a matter of notation. Whereas it can be postulated that 9 is repeated infinitely, it cannot be postulated that the digit 1 would follow an infinite progression of digits 0. There is no actual residue from 0.999....

[Meno_]

Got it , i think. At some point, qualification if notation overrides quantitative analysus, since there us some minute d is solution fir every next repetition of differences.

That explains 6 as being some kind of virtual sign.

[Meno_]

Got it , i think. At some point, qualification if notation overrides quantitative analysus, since there us some minute d is solution fir every next repetition of differences.

That explains 6 as being some kind of virtual sign.

and guessing an expansion into further differentials suffer the same fate until even virtual signs become useless

again a conjecture with apologies.

[Meno_]

Found it:



gamma wave or gamma Rhythm is a pattern of neural oscillation in humans with a frequency between 25 and 140 Hz, the 40-Hz point being of particular interest.[1] Gamma rhythms are correlated with large scale brain network activity and cognitive phenomena such as working memory, attention, and perceptual grouping, and can be increased in amplitude via meditation[2] or neurostimulation.[1][3] Altered gamma activity has been observed in many mood and cognitive disorders such as Alzheimer's disease,[4] epilepsy,[5] and schizophrenia.[6]

[Motor Daddy]

It doesn't quite work that way. Let's postulate, instead of the set you represented, 0.525125..., 5 being a representation for 1/2, 2.5 for 1/4 and 1.25 for 1/8. In this number, the quantity is never going to be above 0.6.

It doesn't work that way!

Let's start at the start line, and the finish line is 1.0 mile down the road.

If you first went 1/2, then went 1/4 more... right there you are at the .75 mile marker, so your statement that it is never going to be above .6 is BS! Do you agree so far?

You represented 1/2 + 1/4 as "0.525"??? That is 525/1000, or 525 Thousandths. But you know that 1/2 + 1/4 = .75 (750/1000) right??

1/2 = .5
1/2 + 1/4 = .75
1/2 + 1/4 + 1/8 = .875

Almost to the 1 mile marker, right? No, and you will never get there using that strategy of traveling .5 the remaining distance.

.5 + .25 + .125 + .0625 + .03125 + .015625 + .0078125 + .00390625 + .001953125 + .0009765625 + .00048828125 = 0.99951171875

See where this is going? The strategy of always traveling 1/2 the remaining distance will NEVER get you there, even if stepping infinitely.

You want to get to the 1 mile marker?

Take 1760 steps of 3 feet each. 1760 x 3 = 5280 feet! 1 mile = 5280 Feet, or 1760 Yards.

Heck, if trying to get there by traveling 1/2 the remaining distance, then why not go 1 step forward and 2 steps backwards, and see if THAT gets you there??? LOL

[origami]

Yes, you are correct. In constructing the number, I assumed 1/4 of 1/2 instead of of the original 1 as the original equation described. I made a mistake there.

[origami]

In any case, we are discussing an ever increasing number, so a series of numbers, instead of a single number. 0.999... is a single number. We are trying to determine if the quantity described by 0.999... is equal to the quantity described by 1. There is no following number, no series, only the one number 0.999...

Being base 10, you write the highest number at the order following 1 and replace 1 with 0. That number is 9. Because 9 does not reach 1, it needs an additional description in the next order to determine how much more than 9 is needed to describe the original 1. Because it is the maximum amount, you seek the highest number in base 10, which is 9, so you write 9. Because the amount will always have to be the maximum, it will always prompt a further order containing 9. In fact, because the process is projected infinitely, there is no residue. No residue can ever happen.

If you postulate 0.000..., you cannot add 1 to the order that follows the progression, because the progression is infinite. At no point in the 0.999... progression can you write 1, because 1 is a smaller number than 9, and would describe a lower overall number. Any number but the highest allowed by the base indicates a quantity known to be less than the entirety of whatever order that number appears in, making the entire progression known to be less than 1, so invalid.

[Magnus Anderson]

If you postulate 0.000..., you cannot add 1 to the order that follows the progression, because the progression is infinite.

That's what I've been asking obsrvr524.

More abstractly, the question is:

Is a sequence of numbers such as \((1, 2, 3, \dotso, 0)\) a contradiction in terms?

If he says "Yes" then it logically follows that it's impossible to cross a line that is truly infinitely divisible (and not merely in our heads.)

Personally, I don't think it's a contradiction in terms which is why I think it's logically possible to cross an infinitely divisible line. But that's obviously not what obsrvr524 thinks.

[origami]

Is a sequence of numbers such as (1,2,3,…,0) a contradiction in terms?

Well it's not a contradiction in terms in the sense that 0 has no quantity, and can be said to follow an infinite progression or be inserted anywhere along the progression. I am assuming I think correctly that ... here represents an infinite progression.

The sequence (1,2,3,...,anynumberover0) is invalid. At any point of the progression, a specific number will be there that is different from an arbitrarily different order of the progression, so a single arbitrary number cannot be inserted anywhere either as addition, subtraction or replacement, and no quantity can be postulated after the progression either because there is no after the progression, it is projected without end.

I have been following your discussion to do with infinite division with great interest. I do believe the sideline about imaginary or real division is a distraction. It makes no difference to the mathematics, which you did point out, mathematics being a system of accounting quantity.

My contribution for the moment, I believe, is to make the distinction between a process that is infinite, and a quantity that is infinitely divisible. An infinite process can describe a finite division, as is the case of pi, which is a finite quantity. The quantity represented by pi is a proportion rather than a number, and I doubt it can be said it exists as a number, but as a proportion it is decidedly finite.

[Magnus Anderson]

Well it's not a contradiction in terms in the sense that 0 has no quantity, and can be said to follow an infinite progression or be inserted anywhere along the progression. I am assuming I think correctly that ... here represents an infinite progression.

We're talking about a sequence of symbols. We're not talking about a sequence of quantities of things. In other words, \((1, 2, 3, ..., 0)\) does not stand for (x, x,x, x,x,x, ... _) where "_" represents "no element". Rather, it stands for a sequence of symbols with "0" also being a symbol. So it's not "nothing" or "no element" or "absence of element". It's a real element. A real nice numeral that is "0" (:

The sequence (1,2,3,...,anynumberover0) is invalid. At any point of the progression, a specific number will be there that is different from an arbitrarily different order of the progression, so a single arbitrary number cannot be inserted anywhere either as addition, subtraction or replacement, and no quantity can be postulated after the progression either because there is no after the progression, it is projected without end.

What that indicates is that you think that \((1, 2, 3, \dotso, 0)\) is a contradiction in terms because "there is no after the progression" and the reason you believe that is because "it is projected without end".

The three dots do mean "no end". But what does that mean? It means "no last element". And "no last element" means "no element at the position with the highest index". And what that in turn means is "there is no position with the highest index". That's what "endless" means. And it refers to the subsequence that is \((1, 2, 3, \dotso)\) and not to the entire sequence that is \((1, 2, 3, \dotso, 0)\). It means there is no position with the highest index in the subsequence that is \((1, 2, 3, \dotso)\). The sequence containing that subsequence may or may not have such a position. Now, if you think that a sequence that has an end cannot contain a sequence that has no end, I am all ears as to why that's the case.

But the bottom line is that, if you're right, and the above sequence is indeed a contradiction in terms, then it's not possible to cross an infinitely divisible line.

[origami]

Yes, they are symbols, and each existing symbol contains information, so not absolute nothingness. But the information contained in the symbol "0" is that there is no quantity.

What that indicates is that you think that (1,2,3,…,0) is a contradiction in terms because "there is no after the progression" and the reason you believe that is because "it is projected without end".

What is being projected without end is a sequence of quantities. 0, having no quantity, does not alter the progression. At any order, the following order can be said to have 0 added to it without altering the progression, even if we decide that the order is the last order, regardless of the fact that that last order will never be reached, because it can be postulated without contradiction.

When I say order, I mean the same thing you mean when you say index.

I understand you, though, with the use of the word subsequence, to be postulating an infinite series of numbers, a series of numbers where there is no last order, within a finite quantity. The numbers could then continue in sequence, without ending, so long as they do not surpass the finite quantity that contains them, and other numbers or sequences can follow that sequence.

You are correct.

[origami]

Importantly, that a finite quantity contains an infinite, or a series of infinite sequences, does not require that the quantity be infinitely divisible. It only requires that a process that does not end can be applied to it.

[Meno_]

Removed for any sense of an intended duplicity having some constructed rationale as its object

[Meno_]

The idea of rationality/ irrationality of sequential numbers comes up, where the roll back from approximated values toward the sequences which 'aim at' informing the 'symbols' with increasing rational numbers, even numbers which can not contradict unending sequences because if the limiting possibility as to their 'endlessness'
That possibility makes irrational numbers which are other then endless


Pi is endless in a sense fir there are no possible gaps in the sequence which can be de-differentiated, but then the possible limit does consist in the absolute limit of such integration, always minus the absolutely shrinking possible limited particle/segment.

That possible segment is both: even and odd approaching the null that serves both purposes , near the critical point of it's progression.

Please, please do not comment on a topic almost near identical but yet possibly miles apart ( a part)
The amateurish attempt to replace the count of incessant sheep, as motive, shows it's sheepish potential.
The point is trying, of the analogue having processing 'draw' within it's symbolic content, to integrate and form ' rings-circles around it's self, as if making sense of the need to become binary, hence rationally objective within it's own frame of reference.

So the symbolic value diminishes the real number's potential to fill in within it's own quantitative/qualitative differential value.

Not at all sure of the import if this, but let it stand anyway
What's the worst that can happen other then hitting some refreshers in the am.


But i think the real import was in conjunction to the network of differing sequencing where 0 could perform both functions at variance and or simultaneously as it approaches more or less uncertainty within some assigned form of continuity. Something along that line where the less certain will indicate more possible contraindication within it's own system within other systems.

I prefer probability functions to incurred more uncertainty as where system error is firmed by objective criteria and the the opposite where conditional criteria becomes the function's specialized 'role'? The counter positioning happens when the probability reduces toward lesser deviance from some model.

These two ways of programming may become the hidden objective with which to view the dufferentially 'diagnosed' semblance of what can be purported in the reasonable-unreasonable continuum.

[origami]

Let's take an interesting case, using obsrvr524's series, that will also show the equality of 0.999... and 1. In his series, we add 1/2 + 1/4 + 1/8 and so on infinitely. This is an infinite series with a boundary of 1, meaning it infinitely approaches but never reaches one.

Eventually, the amount of 9's, of orders containing the number 9, will increase and increase before a non-9 number is reached, or an order with a number lesser than 9. It will increase infinitely. However, there will always be some order down the line of 9's that is a number lower than 9, meaning that the entire number is less than 1. It infinitely approaches but never reaches 0.999.... Since the boundary is 1, we can realize that 0.999... is 1.

[Meno_]

Ok. That one is the boundary, and as the sequential. numbers reduce from .999999999 they descend from a last .9 toward the absolute where the number of nines reach a point with a differential from 1 of .000000000000000000000000toward less then absolute ( boundary) - 1 an equal number of 0's minus that 1 unit.

That presents the problem of the eventual curvature of that extended line from it's beginning toward it's absolute limit.

That curvature becomes a necessary function of it's inescapably impossible formation into an perfectly ideal horizontal continuum.
The varience between that absolute ( horizontal measurement and the number of singular units comprising it, is the metaphysical roll back of an ideal circularity toward it's absolutely divisible parts.

PIe's functionality represents the differential function of that idea(l) right?

[Meno_]

Let's take an interesting case, using obsrvr524's series, that will also show the equality of 0.999... and 1. In his series, we add 1/2 + 1/4 + 1/8 and so on infinitely. This is an infinite series with a boundary of 1, meaning it infinitely approaches but never reaches one.

Eventually, the amount of 9's, of orders containing the number 9, will increase and increase before a non-9 number is reached, or an order with a number lesser than 9. It will increase infinitely. However, there will always be some order down the line of 9's that is a number lower than 9, meaning that the entire number is less than 1. It infinitely approaches but never reaches 0.999.... Since the boundary is 1, we can realize that 0.999... is 1.

The difference between a series of halving sums and. the sequance of subtracting. 1/ 10 the of every successive derivation is formally uses the identical functional pattern, but. the differential all entail a two different functional derivatives, , while the ideal limit if the boundary will remain the same.

However, there will always remain a virtual remainder, between the real ( the accuration of the slope. of that with the nominally unchanged function of the other assymtote


The rotation of which will form the added circular dimension .
The functional sphere varience will then reform the level of linear uncertainty into the infinite cuts of the firmed circularities .

But the 1 unit is always an approximation of the ideal into it's virtually reduced , specified reality. Hence there are never 'ideal representations'

Leibnitz should have avoided saying that " This is the best possible world. There may be other innumerable planets in the universe adhering to much higher degrees of specificity