If the thought experiment is not possible to be done in physics then I guess there is no point discussing it...
So, now you understand why a point can not be divided any farther.
If the thought experiment is not possible to be done in physics then I guess there is no point discussing it...
But we are not talking about a point as defined by math. We are talking about reducing a ball so that it is the smallest it can be and yet still be greater than zero. We may be attempting to explore how that definition may need to be changed.So, now you understand why a point can not be divided any farther.
So basically Mathematics has given up on Achilles and stated that the paradox is indeed a paradox as the only way they can resolve it is to state that there is a limit to x/2.Hi rpenner.
From what I can observe of your x/2 line of reasoning, the axiom assumes that 'something' ('number') can be REPEATEDLY sub-divided into equal 'smaller and smaller' parts without limit. Is that what you wish to imply by that line of reasoning/axiom you use there?
But we are not talking about a point as defined by math. We are talking about reducing a ball so that it is the smallest it can be and yet still be greater than zero.
Mr. Q, You've mentioned that example many times, and it's completely wrong.
If we are talking about a mathematical ball, there is no smallest positive radius. This is readily seen, since if the radius is r > 0 then r/2 is also positive and strictly smaller than r.
If you're talking about a physical ball, then you'll have to tell me what a "ball" is, since at tiny scales of measurement all you have is quarks or strings or branes or whatever the latest physical theory is. There are no exact balls or spheres in the physical world at all. Not the earth, not a basketball, not a marble, not a "perfect" steel ball bearing. If it's physical, it's not a perfect sphere.
So if we're back to the world of mathematics, there is no such thing as a ball that is the "smallest it can be." You can always make it smaller, since there is no smallest positive real number.
What I am suggesting is that every possible co-ordinate location in 3 dimensional space has both 3 dimensional and zero dimensional attributes. The "zero point" both exists as a real 3 dimensional object and a zero dimensional object simultaneously. [ A zero point is not just a mere mental abstraction but is phenomenologically real and material because of it's 3 dimensional attributes]If you're talking about a physical ball, then you'll have to tell me what a "ball" is, since at tiny scales of measurement all you have is quarks or strings or branes or whatever the latest physical theory is. There are no exact balls or spheres in the physical world at all. Not the earth, not a basketball, not a marble, not a "perfect" steel ball bearing. If it's physical, it's not a perfect sphere.
So if we're back to the world of mathematics, there is no such thing as a ball that is the "smallest it can be." You can always make it smaller, since there is no smallest positive real number.
rpenner..."Except since the question is about the value of 0.999... as a real number",
just like saying π = 3.14159... is a statement about a real number and a representation with an infinite number of digits, then the proper domain is the field of real analysis which requires knowing what the rule of the game are.
You are perfectly free to ignore the real numbers, but then you lose any credibility you have on the topic of mathematics until you pick a set of rules and stick to them. So far rr6, your use of imperfectly understood Windows calculator tools instead of mathematics is not serving you well.
You're confusing smaller with smallest. What math is really saying is: given a real number r which is not zeroQuantum Quack said:So math is saying from what I can read of it,
1. There is always a smaller real number and
2. there are occasions where there is no smaller real number.
But they don't represent a process.rr6 said:irrespective of any label you want to attach to the value 0.999..., if those dots are to represent an infinite processs/procedure
Why cannot you agree with a simple statement. In logic, mathematics or philosophy one seeks to lay out a convincing argument than a conclusion follows from the premises. If you never state what your premises are then the best you can hope for is a cadre of sycophants mindless repeating phrases they have no basis to understand.This is expressing that unless you clearly assert what your principle assumptions are and develop those assumptions using logic and clear language, there is no possible meeting of the minds.Axioms are needed before a thought experiment is more than mental masturbation.
So what concept of number exists where 0.5 is a number but 0.25 isn't? If you claim there is a smallest positive number smaller than 1, then what is it and what mathematical property makes it impossible to divide it by two. Does this also mean there is a largest positive number in your philosophy? If there is a largest positive number then what happens when you add 1 to it? Does it conform to all the properties that every other number has?for the rational numbers or the algebraic numbers or the real numbers or the complex numbers [or the hyperreal numbers or the surreal numbers or the imaginary numbers], there is always a distinct third number halfway between any two distinct numbers. (The integers don't have this property, obviously.)
Links omitted, for if you had not shown interest in learning what was taught in one mathematical field, why would you care what was taught in another.This property of the real numbers has a geometric analogy in Euclid's Elements of Geometry, Book I, Postulate 1 and Proposition 10.
And you see the simple idea formally developed from the axioms of real numbers and set theory
"..." cannot represent an infinite process unless, like Zeno, you are doing the process the wrong way. 0.999... is an abbreviated representation (just a posh word for name) for an infinite series (a sum over an infinite sequence) and while no one can add up a column of unrelated numbers, "0.999..." has a repeating pattern that means it has very low information content. And since that sequence converges "0.999..." is just a representation for a number."..." dots represent infinite value and a infinite process/procedure and no amount of mind playing games with grammar or mathematics is will change that, in regards to 1.0 NOT equal to 0.999....
Incorrect use of the term of "mental masturbation" as the fields of logic, mathematics and philosophy exist as meaningful communications specifically contradicting the non-communicable virtues of "mental masturbation." That 1.000.. = 0.999.. is a consequence of defining the value of some infinite series in a consistent way. If you would prefer another way to do it, the burden is entirely on you to 1) point out that you subscribe to different axioms than those of the real numbers, 2) enumerate those axioms, 3) fairly and concisely explain just what the number 1.000... - 0.999... is and how your concept of number differs from the real numbers. To avoid your burden is to avoid differentiating your thoughts on the matter from "mental masturbation", got it?Those in the camp of 1.0 = 0.999... are playing mathematically illusionary, mental masturbation games. imho
Neither stopping nor rounding is in evidence in ANY of the proofs that 1.000... = 0.999... in the domain of real numbers. You simply didn't understand the demonstrations and probably did not understand what separates the concept of real numbers from rational numbers.This is simple conclusion and does require complex "mapping" math to understand but rather only rational, logical and common sense to understand.
To have 1.0 = 0.999... there are two simple processes that have to happen;
1) STOP the infinite process i.e. remove and exclude the dots from the value, number, equations( wiki ) whatever,
2) then round 0.9 higher to 1.0.
No it isn't. Even if what you said was true, which it is not, then at most you would have demonstrated that 1.000... = 0.999... has not yet been proven.This is rational, logical common sense and certainly a relatively simple explanatory guide to understanding that 1.0 does NOT equal 0.999....
The process of education consists largely of replacing common sense ideas with better approximations of truth. To embrace your particular notion of "common sense" is to deny that conversation on the topic is useful, so all of your posts can be replaced with: "I disagree. Common sense!!!11!!" and lose nothing in translation. Naturally, I disagree and think the beginning of the conversation requires agreement on what axioms we are asserting.No amount-- thousand of years --- of flawed thought process and flawed/invalid mathematics will ever make a false irrational nonsense a rational common sense fact/truth. imho
Repetitive and untrue. The process of arithmetic you speak of is not the whole of mathematics.RP, what label you want to a attach to the 0.999...is more mathematically illusionary, mental masturbation sidestepping of the primary issue, because, irrespective of any label you want to attach to the value 0.999..., if those dots are to represent an infinite processs/procedure,Except since the question is about the value of 0.999... as a real number,
Repetitive and untrue, for none of the demonstrations that 1 is not a higher value than 0.999... involve stopping at a finite number of 9's or rounding.then, the only way to arrive a the next higher value 1, is as follows;
1) STOP the infinite process/procedure, i.e. remov/drop the dots, ergo 0.9--- that is what the wiki link does, and then,
2) round to the next higher value, and in this specific case, the next highe value is 1.
Repetitive and untrue, for common sense requires that one not deem a field incorrect until one has understood it. To prejudge that 0.999... < 1.000... when one has only tools that work with finite number of digits is patently unwise and thus an an affront to common sense. Why would common sense with no experience of infinity have anything to say about the infinite?Common sense trumps invalid/incorrect mathematics, irrespective of extracurricular(?) lableing of stated finite and finite values
Super repetitive and untrue. Can you not see that you used this same baseless claim in this very same post? That's lazy even by the standards of professional bloviating know-nothings.The rule of the game, is that to convert an infinite value to a finite value, then we must;just like saying π = 3.14159... is a statement about a real number and a representation with an infinite number of digits, then the proper domain is the field of real analysis which requires knowing what the rule of the game are.
1) STOP the infinite process/procedure--- drop the dots ---, and then,
2) round to next higher value-- and that is is some manner ---what they actually do in the wiki link in the "infinite squence" section that your less complicated givens were almost word for word verbatim(?).
Grossly untrue.You refuse to answer my questions in regards to that specific wiki section of "infinite sequences" becuase you know that what I've stated, is for the most part, exactly what is happeing in that "infinite sequence" section where the dots "..." are mysterious just dropped, with no explanation, as to why their dropped.
You have given no explanations -- you simply have not had anything useful to say about infinite decimals, or the real numbers. You can't even point to specific illegitimate steps on Wikipedia or in my demonstrations. This speaks to me of your inability to understand the demonstrations. Your repeated invocations of your purported common sense speaks to me of your inability to articulate your thinking and the arrogance to believe that yours is the principled approach.I can tell you why there dropped, because that is the ONLY way to get to ifinite 0.999... = finite 1.0.. Common sense and no complex "mapping", or "infinite sequence" or "real" vs non-real, natural vs unatural, etc....processes and labeling will ever trump the common sense and valid explanations I've repeated given.
1.000... has the same number of digits as 0.999...I am not ignoring finite 1.0 and infinite 0.999...You are perfectly free to ignore the real numbers, but then you lose any credibility you have on the topic of mathematics until you pick a set of rules and stick to them. So far rr6, your use of imperfectly understood Windows calculator tools instead of mathematics is not serving you well.
You need a theory of number to coherently communicate what you believe those expressions represent. The theory of real numbers say they represent the same number.and what those two represent.
More like nakedly asserted.Your are ignoring the rational, logical, and common sense explanations I've repeated stated,
None of which I recall because your misconceptions make your questions uninteresting and your lack of a clearly expressed theory of number leaves nothing of your viewpoint as a communicated message.and the questions I've asked you specifically in regards to this issue,
Citation requested. I certainly don't remember your post, my response or any wikipedia page that resembles closely what I wrote.except for the one place you made an effort to address my givens as stated, and that was you eventually, put the very complex "mapping" aside, and offered us a more simple alledged "proof" that I later found to be almost word for word, verbatim from the wiki link regarding 0.999...
What givens?Rp, please come back and talk to me, when your want to address my given rational, logic common sense explanations as stated and references to the wiki links "infinite sequences given formual equations, for starters.
If you cannot do that, then all I see you doing is a continual sidstepping of the majority of my givens.
arfa brane.."But they don't represent a process.
They represent numerals, an infinite string of 9s in this case.
The "..." does not mean "unless you write down an infinite string there will be trouble".
I believe I am not mistaken, so which one of us is mistaken? Why does an infinite string have to be the result of a process? Does something happen if you assume there is no process?rr6 said:Arfa, I believe you are mistaken, on what appears to me, an obvious case of inherently infinite process/procedure. of creating and infinite set of 9'' if even if only[/B--- and the only way --- in an metaphysically conceptually, abstract sense.
Из вашего сообщения не очень понятно, чего вы в итоге хотите добиться… Просто повысить самооценку или найти цель вашей жизни или еще что-то? Хотя, соглашусь, семинар о мотивации – не самый худший вариант. Попробуйте, это точно не помешает. Расскажете и нам, что да как
arfa brane..."I believe I am not mistaken, so which one of us is mistaken?
Why does an infinite string have to be the result of a process? Does something happen if you assume there is no process?
In formal language theory, strings are just arbitrary compositions of characters or letters (i.e. abstract symbols), so there is a set containing all the finite length strings for some alphabet. But this set must contain an infinite number of strings if there is no restriction on their length, other than it must be finite.
This follows from the fact that, in the natural numbers, there is no largest number; there will always be strings with greater length than some n.
Strings can be considered as letters under concatenation, and that concatenation is a process. However, given a string composed from some alphabet, can you decide how it was composed? It seems that the process or operation of concatenation, if there is one, has no inverse. You can't "unconcatenate" an arbitrary length string because you don't know if the last character was the result of a concatenation--there is no information about the order the characters were added to the string.
Arfa, I believe you are mistaken, on what appears to me, an obvious case of inherently infinite process/procedure. of creating and infinite set of 9'' if even if only[/B--- and the only way --- in an metaphysically conceptually, abstract sense.
Does that mean it will take me forever to cut a pie into thirds?
Don't make it harder than it needs to be. It's an anomaly inherent in base 10 - nothing more.GM, when you want address my comments as stated, then please come back and talk to me. Otherwise, no need to attempt conversation.
Finite 1.0 does not equal finite 0.999... process/procedure. Common sense not neccessarily relevant to a common pie.
r6:
Because you don't have any "givens" that I or so many others here can understand.rr6 said:Once again, and as so many time with yourserlf or various others here, you have not and do not directly address my givens