1=0.999... infinities and box of chocolates..Phliosophy of Math...

You're referring to this:

As stated by me and others several times, mathematics is not physical. Physics is physical. If you want to equate real with physical and claim that mathematics is unreal, that's your prerogative, but mathematics "really" exists. Where is it? Where do numbers come from and why do we bother to use them as labels for real physical objects?

Why do we do it then? Why do we use "unreal" numbers to label things? Why would anyone say they own "one car"?

But it is complete, Euclidean geometry is axiomatically closed; points, lines, and planes along with arbitrary coordinate systems describe reality very well. Let me quote someone I have a bit more faith in than you:

But numbers are abstractions, mathematics is abstract because that's what it deals with--abstract entities. Are you going to rework the whole of mathematics since Euclid, and what are you going to use instead of numbers?

But you intend to change that? I don't think you will, I don't think you can get around the fact that mathematics and physics are not, and never will be, the same thing.

Sorry, I don't really understand any of that. If that's what your "ToE" will look like, I'd say not many other people will understand it either.

As for your reply to James R, and your claim that 1/3 is a division that "hasn't even started", can you do long division? Do you know what dividing 1 by 3 implies when you do it?

You know, you say "3 goes into 1.0 0.3 times with 0.1 remainder, then 3 goes into 0.10 0.03 times, etc".
So the implication is that you will get an unending string of 3s after the decimal point. Also that you won't need to actually keep dividing to conclude that 1/3 is equal to 0.333...

These are the only logical conclusions you can make; there is no room for "not yet done", or "incomplete". That is mathematical reality (logically consistent and no errors).
You would need to be quite unintelligent, or intentionally stupid, to see otherwise.

 

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Hi arfa.

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I was just logging out and checking for typos when I caught your prompt reply. Just quickly then.

Why do you advocate keeping maths and physics separate from the starting premises of each? Why not make both consistent, starting the axioms and the postulates from self-evident truths observable/testable in reality as the 'starting premises', rather than obviously philosophical UNrealities which FOREVER DOOM maths to NOT ever becoming consistent with the physically observable reality? Are you ALLERGIC to the notion of IMPROVING the mathematical axioms/construct so that it WILL be PHYSICALLY CONTEXTUAL and not forever UNREAL contextual fantasy that outputs "undefined" etc absurdities from its starting axioms?

Where does any imaginary/abstract 'world construct' come from. The human imagination, of course; such abstractions don't actually have 'independent objective existence' outside the abstract world construct itself. However, any imaginary constructs CAN be useful, but let's not run away with or imaginations and fall 'down the rabbit hole' of fantasy as if it is 'real' in any sense that is meaningful in any physically extant sense. OK? Take a cold shower and get a grip on those fantasies.

Consider: We ALSO 'label' PEOPLE/THINGS with NAMES; are YOU suggesting even for a moment that the NAME 'Barbara" is in any sense 'real objective extant thing' as the LIVING BEING IDENTITY you 'label' Barbara"?

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Complete fantasy, and consistent only with its own starting fantasy premise. So what? Imagination and Philosophy produces many 'worlds' consistent with 'themselves and their starting imaginary/philosophical premises' having no relation to any reality.

And who asked you to practice 'faith' in anything/anyone? I just got though recommending you think everything out for yourself 'from scratch reality' starting premises, and you come back with innuendo that I demand 'faith' from you in any way shape or form? Is this how you 'win' debates, by misattribution and misrepresentation and ignoring the objective/substantive point made already? Not a good sign for your consistency or understanding of what it is you are 'defending' or what it is your are being shown objectively without need for belief, faith or fantasy 'authority' acceptance or denial. "Just the [reality] facts, m'am", as the detective used to say in the "Dragnet" TV series.

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Yes, it will be easy to re-jig the whole construct from real staring self-evident premise for axioms (and consistent with real physics postulates too!). Numbers, including ZERO will be Axiomatically re-defined according to the reality aspects I already pointed to that are superior to current axioms/definitions, such that the new construct will be 'reality-referential' from the word go, so that "undefined" and "dimensionless point infinities", "singularities" etc etc outputs/concepts are SUPERSEDED by superior stuff; and present 'unreal things' in maths will no longer have any role to play, and no longer will they gum up the works so much as they have done by creating such fantasies upon fantasies that has led so inexorably to incompleteness and impasse for both maths and physics.

It takes courage and willing to work/think to rework/advance the maths/science from what it was. This has happened before; only some rework/advances have proven disastrous because they introduced fantasy as reality and proceeded to create a fantasy math/physics modeling constructs/interpretations base on 'unreal logics chain' of abstraction upon abstraction.

And yet you pretend to "understand" fundamentally untestable/unfalsifiable 10-150 'dimensional' string fantasies; and Big Bang 'beginnings Hypotheses" that don't cover the actual supposed t=0 beginning state?

Seems like any old mere fantasies of maths and physics 'respected authority figures' whom you have "faith in" are preferable to actually doing the hard yakka necessary to rework everything to AVOID having to 'believe' and 'have faith in' such fantasies in order to defend/maintain your illusions? Better to prefer and have faith in you yourself getting to work and start thinking things out for yourself starting from scratch REALITY, hey?

Just because you "don't understand" what I have only hinted at, it does not make your "understandings" in current fantasies of your own 'preference' any more palatable/valid. You'll have to wait for my complete and consistent (including Gravity, Inertia etc mechanism/entities etc) ToE IN FULL before you can make any real informed judgement. It will be readily comprehensible and reality referable from your own intellect/testing. That is because it starts from reality and ends in reality....'no faith or fantasies or purveyors thereof need apply', as it were.

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You missed all the points/subtleties involved. The 'trivial construction' of a symbol "1/3" is NOT the action itself. As agreed with Trippy when he finally introduced REALITY into the discussion of what 1/3 actually entails in fact, not in "assumption of things done that have yet to be done". If you don't understand the meaning and import of any of that in the context/discussion so far on 1/3 (and also unitary 3/3, 9/9 etc etc) trivial constructions/symbols, which no amount of equally trivial 'decomposition/recomposition 'proof' exercises makes any more 'real' or 'already done' fait accompli 'statements', then you've missed some crucial exchanges over these discussions; so you may have to wait for my full publication of math/physics ToE to get the whole reality picture/understanding straight.

Until then, keep at least a few 'objectively processing' brain cells away from the current training stream of consciousness being inculcated into you by the current maths system/teachers/texts. When you finally read my stuff, it will be less of a shock to your system if your mind hadn't been unanimously convinced to be/remain 'faithful' to whatever 'fantasies/personalities of the moment' you are currently 'following' without objective scrutiny 'as from outside looking in'. Better a realistic heretic than a captive of fantasy.

Anyhow, gotta log out again. Bye till next time, arfa, everyone.

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Physics is about what we observe, and how that gets explained. Mathematics is about logic. What starting premises did you have in mind?

And yet Euclidean geometry remains a mathematical theory which coincides very well with what we observe. This is, I believe, why Roger Penrose labels Euclidean geometry a "superb" physical theory. Are you calling an emeritus professor of mathematics a fantasist?

Numbers are abstract, and are defined axiomatically, so I don't think you're going to achieve much. Furthermore, distances are abstract labels for real things. You can't buy 5 metres, you have to buy 5 metres of "something".

You think I don't know that writing "1/3" doesn't mean I need to start doing long division? Long division equates to proving something, once you know that 1/3 = 0.333... you don't have to prove it every time you write down the equation.

You're coming across as someone who thinks there is something wrong with mathematics, rather than your understanding of it. Have you considered that the latter option might be true rather than the former? I know, it takes courage and a bit of willingness to admit you could be wrong, and then it takes a bit of work "undoing" the misunderstanding you have.

I've just completed a graduate diploma in mathematics (/shines awesome badge), and golly, there were times when I was just wrong about things pertaining to numbers and logic (but not always, otherwise they've made a big mistake conferring the diploma, yes?).

 

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Hi arfa.

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I've taken a little break from my other work, so....

We construct theoretical physical models using abstract math/geometry axiomatic concepts/logics. Isn't that what you actually meant to say? If so, the maths and physics are inextricably linked when it comes to analysis, prediction and communication of the science. Why do you seem to want the two (physics and maths) to never be in a position to be consistent and complementary aspects of the reality-based construct of complete and consistent ToE we are all working towards? Seems like you want to 'specialize' in abstractions and don't want anyone else to 'sully' your abstract maths system by actually making it real and directly relevant to physics in a true marriage of disciplines for the benefit of both. Strange.

The new reality-based starting premises(replacing the current UNreality-based maths axioms and physics postulates) will be published in the overall maths-physics ToE. Patience. Like I said, I have let too many cats out of the bag as it is. Plagiarism is an ever present danger to original works. History cautions.

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"Superb"? An abstract theory rooted in unreal philosophical dimensionless concepts and notions is now considered "a superb physical theory"? Pull the other one!

Anyhow, I have great respect for Sir Roger. I have often quoted his courage in admitting the old cop-out retort used by him and his peers against anyone asking the legitimate question: "What came before the Big Bang?" was just that, a cop-out. He now admits that for the Big Bang scenario to have any meaning, there must be a 'before', else the concept of Big Bang is just another UNreal concept. So please don't ever suggest for one moment I disrespect one of my favorite mainstream thinkers, Sir Roger. Thanks.

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Everything said by anyone must be taken and treated IN CONTEXT, not as some 'chopped item' for making some inane innuendo of your own about what I think of Sir Roger. The context OBVIOUSLY is that ALL PROFESSIONAL physical theories to date have associated DOMAINS OF APPLICABILITY beyond which and out of the context of which you are not allowed to push it like some 'generic barrow' of your own 'attributions/assumptions' about what he meant. Be careful not to do that anymore, as it does nothing for your 'arguments'.

In any case, any theory based on fantasies like 'infinities of dimensionless points' etc etc deserves to be called fantasy, by self-definition from the starting axioms/postulates being rooted in fantasy as I already pointed out, and which you cannot deny or refute so far when reality is the final arbiter of everything that purports NOT to BE 'fantasy'. Yes? Fantasies are useful, but not the final word. The next step is full reality; with the maths/physics abstract fantasies of our 'science childhood' needing to be finally put aside, like all 'fairy-tales and toy-models' must be IF we and theory are to finally reach 'all grown up' status at last. Yes?

Yes, that's what I said; only, as I have also been saying, the current axioms are PATENTLY inadequate to the demands of reality, as the limitations of the current 'outputs' and 'models' demonstrate by holding us back with PARTIAL picture "near enough is good enough" incomplete and inconsistent and unreal 'infinities', 'undefineds', 'singularities' etc. No, not real, and not good enough. We must move on with the math/physics axioms/postulates. Hark!....REALITY beckons, arfa! Stay and play with the fantasies and toys, mathematician; but some of us physicists are growing out of such childish things, useful and fun as they may have been until now.

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You again missed the original point made in the original context. The point was that just constructing/writing down a symbol FOR long division process is insufficient to identify 1/3 AS a specific number. It is NOT anything except NOTATION CONVENTION. Neither PROCESS or ACTION or VALUE or actual NUMBER of any real meaning UNTIL one does carry out that IMPLIED long division process....and if one does start the 1/3 long division process, that process hits the brick wall of NEVER COMPLETING to arrive at a specific number/value; and making a further notation convention of writing it out as 0.333... is a trivial cop out that does nothing to PROOF the assumptions behind the notational EXPECTATION that a long division 1/3 will end and output a finite number by the further assumptions of limits arguments which are not proof of anything except that limits convention, not real action/result in any real exercise of long division actually under way. Just saying 1/3 or 0.333... are "equal to each other" does not proof it unless the action is completed AS the ONLY proof which can be mathematically justified above and beyond mere conventional notation of something 'assumed to have been done but not actually yet done in reality' at all.

If you still don't get that, then you obviously missed some of the discussion, and so will have to wait for your full insight to arrive when you read my math/physics ToE. Sorry.

Not at all. I (and others) have pointed to the problems/incompleteness/inadequacies. It's just you haven't been listening objectively, but with 'defensive kneejerking' drowning out your own intellects attempts at listening/understanding without bias and self interest as a professional mathematician. That is a well known human trait/tendency; as history shows amply whenever an advance has been finally made despite all that reactionary professional self-interest etc. You're obviously only human, arfa. Not everyone may be willing and/or able to remove themselves from the problem and just treat the reality as coldly and calculatingly as I have, with scrupulously impartial self-directing reality approach that takes me and all other personalities and their flaws out of the theorizing process.

Sincerely, congratulations. But perhaps that explains your zealous defense of the fantasy-based maths construct, against increasingly mounting evidence/arguments about that which you've just 'mastered' is an abstraction from start to finish as it stands (for reasons of notional philosophical point etc already pointed to). Maybe though, when you stop polishing that 'awesome [Dan Dare?] badge', you are still youthful enough to maybe learn NEW tricks based on REALITY. Then you will have FULLY grown up, that 'shiny new badge' can be replaced by some 'shiny new insights' necessary to understanding the WHOLE complete and consistent universal picture, and not the 'fractured fantasies' we still receive from the current professional theories? Good luck in your chosen profession, arfa. But never stop learning from REALITY, hear? Cheers.

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We construct physical theories to explain observations, which are themselves physical.

Yes, I agree, equations can "communicate" more than ordinary words, and a physical theory cannot be "math-free", for reasons that we probably don't really understand.

Why do you think that's what I want? I've consistently stated that mathematics is not physics, they are not the same thing.

No, what's strange is that you can form this opinion based on what I've actually said in this thread. Mathematics is "real and directly relevant" to physics, and it always has been, or at least it's something we've had to accept since we started counting things, like camels or goats, baskets of wheat, people, what have you . . .

Have you ever tried counting sheep in a paddock? If you had to do this, would you use "unreal philosophical concepts" like the numbers 1,2,3, . . . which are dimensionless? Or have you thought up some alternative to using "whole" numbers?

But anytime you count things, like sheep, you are assigning dimensionless objects to each real, physical animal. When you finish counting the sheep, is the total then based on a fantasy?

Notation is exactly how we identify specific numbers, so that's a load of bollocks, my son. As for "process", we perform long division, it's an algorithm. Mathematics, being unphysical, does not define process or algorithm which are physical (like us). Numbers don't "do" things, they just are what they are.

I already told you how to prove it, what you're displaying here is a misunderstanding of long division and the mathematical induction that follows when you get a repeating pattern of digits in the dividend.

I don't think you've pointed to any problems or inadequacies, seriously. You've stated a few times that Euclid's notion of a dimensionless point is a problem or a "fantasy", but haven't provided any supporting evidence. What about Newton's idea of a centre of mass (an abstract Euclidean point), what are you going to do about that?

I know numbers are abstractions (from start to finish), you intend to "redefine" numbers somehow because you perceive a problem, I don't.
Whenever you count things, you're using abstract dimensionless objects (much like Euclid's points), despite all the physical things you're counting the numbers remain abstract, dimensionless, a "fantasy". You want to address this "problem", good on you, but there really isn't a problem.

 

Consider it as real number, including infinitesimal and infinity. So, real numbers with hyper-real numbers or extended real numbers can be considered.

After making number 1, this 1 can make up number 2 or any other number.

The rule is same which guarantees, there are infinite points in the line segment corresponding to number 1.

If you see the number 0.999... as the sum of its parts, it will become as:

0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + ... ;

Here all the terms are having the last digit 9 and the sequence of 0's are increasing gradually. So, one of the terms will be having infinite sequence of 0's with the last digit 9. This term can be written as 0.000...9. So, the number 0.000...1 also is possible.

Where you are not finding the sense?

LHS number is in the decimal form whereas the RHS number is in the fraction form and both the numbers are same.

 

So it can be any real number.

What about before it makes the number 1? Can you make up 2, then subtract 1 instead?

What about the line segment corresponding to the number 2? Or 2.1? Or any number at all?

But there is no last digit, if there was the sequence would be finite instead of infinite. So can you see how that means there is a contradiction?

In any of your posts, that's where.

 

No question should be "off limits" but certainly some just reflect lack of understanding.

I suspect you think time has some existence independent of matter and energy fluxes. It does not. Time is a parameter in many useful equations that allows two material process to be compared. Much more convenient than direct comparison of two, like saying the standard candle burns down 1.2cm while that dripping faucet leaks 1 Liter of water. I. e. Time ONLY expresses how much of one physical process completes in terms of another, like the number of oscillations in a digital watch's crystal. Without matter changing (or energy) to relate to other matter changing (or energy) the concept of time is meaningless. As I understand it there was no matter or energy before the BB, so no meaningful question about time "before" the BB exist.

BTW, the trash content of this thread got so high I delisted it, but when I happen to notice arfa brane was still wasting his time posting here, I clicked on it again before continuing down the list of all forums to B&E to see if, as the mod there, I needed to do something. He seems to not have learned that some will not learn.

 

However did you get that idea?! I have argued and supported my arguments scientifically, maintaining the EXACT OPPOSITE of what you just attributed to me, mate. So no further comment on the rest is necessary. OK?

Your missing of some context results in such mistaken situations when 'responding' out of the full context. I was merely referencing Sir Roger Penrose's change of mind regarding the generic question of what came before the BB. That was an example of changing his mind about the old 'cop out' responses to that generic question in the past, but that he no longer uses. That was all.

No worries, though, as long as we're straight on that now? Cheers.

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The abstractions you have been steeped in for so long have made you miss the fundamental reality to which all things must have reference to ultimately if they are to be able to represent that reality in our modeling.

Consider my last clue I gave you: The first/last "infinitesimal quantum of physical effectiveness" as the first/last-step quantum of effectiveness before change occurs/transitions from one state/extent to the next IN REALITY energy-space PROCESS.

That includes NUMBERS, because at base, the ONLY UNIT CONCEPT/QUANTUM is that 'first/last infinitesimal of physical effectiveness' from which all REAL concepts of NUMBERS arise as COMPOSITIONS/DECOMPOSITIONS from/back to such quantum energy-space real INDIVISIBLE states which we may abstractly associate in our analyses constructs with NUMBERS/VALUES which ultimately must refer to that energy-space quanta in all its variation states/extents from minimal to maximal effectiveness.

Your concerns about the 'problem' of things like Newtonian centers of mass, locations, etc is not a 'real' concern, nerely an abstract notion in itself. You see, any 'energy-space feature/object' in reality has a REAL centre comprising its center of mass/location etc, and it is the 'last infinitesimal quantum of physical effectiveness' energy-space COMPONENT AT THAT center/location with respect to the objects overall. Whether that central quantum is 'energy-space' in its 'background' GROUND STATE or whether in its MATTER form energy-space state, is immaterial to reality.

See? Your abstract math/philosophy notions, associating 'dimensionless point' UNREAL 'properties' TO whatever in reality IS there AT that center/location of real thing, have no bearing on what IS there: ie, the relevant infinitesimal of effectiveness physical quantum that cannot be further 'treated' mathematically because it cannot be 'divided' any further and still remain its minimal effectiveness action/state.

The universal quantum energy-state expanse/process knows where it is, even if you abstractly cannot actually center/locate any bit of it using 'dimensionless point' abstract analysis. Process happens, locations/centers arise and change and subside.

The whole point NOW is to make the mathematical analysis/model MORE REAL. And to do that requires reviewing axioms and redefining 'points', zero, infinity, singularity, infinitesimal etc etc so that they will give those otherwise philosophical/imaginary 'things' REAL PROPERTIES that can represent the REAL PHYSICS we want to model.

No big deal, once you know how to go about it. No drama. No 'imagined problems' with center/location etc concepts/applications etc. No panic! Anyway, I will leave it at that because I don't want to reveal any more about the full and complete and consistent math-physics ToE to be published. Sorry, but you'll have to be patient to have the rest explained.

Have run out of time for posting again, Will just be reading only again from tomorrow for a while. Good luck starting out in your chosen career for which you studied and graduated, mate! Seriously, congratulations and well done! Bye for now, arfa brane, everyone.

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YES.

Number 2 is greater than number 1. If the constituents add up, it will make the smaller numbers first, then the greater numbers.

Can you tell me how many points are there in the line segment corresponding to number 2 or number 2.1 or any number?

Consider, 0.999... = 0.9 + 0.09 + 0.009 + 0.0009 + 0.00009 + ...... upto infinite terms.

Can you tell me, which term in the RHS above is not having the last digit as 9?

 

Yes: the last one, simply because there is no "last one".

 

By the mere FACT that when you create the number 0.999... you start typing 0 first then . then 9 then 9 then 9 then . then . then . already proves 0.999... will never equal 1, unless you round it off.

1 = 1.000...
0.999... = 0.999...
1.000... - 0.999... = 0.000...1

If 0.999... = 1 then 0.888... = ? and 0.777... = ? etc...

Theres no last one... but we know what all of them are. In the case of 1 = 0.999... they are 9's and not 0's for 1.000...

 

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Hansda, consider \(0.999... = \sum_{k\geq 1} \frac{9}{10^k}\)
If you agree that the left side is a number, then the right side must also be just a number -- that's what the equals sign means.
So what is the number? I'll pretend I don't know so I'll call it \(S\).
\(0.999... = S = \sum_{k\geq 1} \frac{9}{10^k}\)
That didn't do much but first it saves some typing and it lets me write the family related quantities \(S_n = \sum_{k=1}^{n} \frac{9}{10^k}\).
Now every n specifies how many terms we use before we stop the sum. \(S_1 = \frac{9}{10}, \; S_2 = \frac{9}{10} + \frac{9}{100}, \; S_3 = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000}, \; \dots\)
And for every n greater than or equal to 1, we can prove \(S_{n+1} = S_n + \frac{9}{10^{n+1}}\) and \(\frac{9}{10^{n+1}} \gt 0\) so \(S_{n+1} \gt S_n\).
So presumably \(S_1 \lt S_2 \lt S_3 \lt \dots \lt S_{n+5}\lt S_{n+6}\lt S_{n+7} \lt \dots \lt S\) and indeed this is provable from the rules of inequalities.
So any member of the family of \(S_n\) is less than \(S\), in other words each member is a lower bound on what the value of \(S\) could be. But none is the greatest possible lower bound on what \(S\) could be.


\(S_n\) gets ugly to compute the hard way -- imagine the pain if n = 1000. And yet most of all counting numbers are actually larger than 1000. So I want to prove that:

\(S_n = \frac{9}{10} \times \frac{1 - 10^{-n}}{1 - 10^{-1}}\) for all counting numbers, n.​

First let me prove that \(S_1 = \frac{9}{10} \times \frac{1 - 10^{-1}}{1 - 10^{-1}}\). The proof writes itself: \(S_1 = \frac{9}{10} = \frac{9}{10} \times 1 = \frac{9}{10} \times \frac{1 - 10^{-1}}{1 - 10^{-1}}\).
Next, I want to prove that if for a particular counting number k, that \(S_k = \frac{9}{10} \times \frac{1 - 10^{-k}}{1 - 10^{-1}}\) is true, that \(S_{k+1} = \frac{9}{10} \times \frac{1 - 10^{-(k+1)}}{1 - 10^{-1}}\) follows as a result. So from the definition of the family, I know that \(S_{k+1} = S_k + \frac{9}{10^{k+1}}\) and from assumption: \(S_{k+1} = \frac{9}{10} \times \frac{1 - 10^{-k}}{1 - 10^{-1}} \, + \frac{9}{10^{k+1}}\), So we have \(S_{k+1} = \frac{9}{10} \times \frac{1 - 10^{-k}}{1 - 10^{-1}} \, + \; \frac{9}{10} \times \frac{1}{10^k} = \frac{9}{10} \times \left( \frac{1 - 10^{-k}}{1 - 10^{-1}} + 10^{-k} \right) = \frac{9}{10} \times \left( \frac{1 - 10^{-k}}{1 - 10^{-1}} + \frac{10^{-k} - 10^{-k-1}}{1 - 10^{-1}} \right) = \frac{9}{10} \times \frac{1 - 10^{-k} + 10^{-k} - 10^{-(k+1)}}{1 - 10^{-1}} = \frac{9}{10} \times \frac{1 - 10^{-(k+1)}}{1 - 10^{-1}}\) which is what I wanted to prove.
Using the principle of finite induction, I now know that \(S_n = \frac{9}{10} \times \frac{1 - 10^{-n}}{1 - 10^{-1}}\) is true for all counting numbers, n.
But because \(\frac{9}{10} = 1 - 10^{-1}\) this also proves that \(S_n = 1 - 10^{-n}\) for all counting numbers, \(n\).

Thus our lower bounds on what \(S\) can be looks like this:
\(1 - 10^{-1} \, \lt \, 1 - 10^{-2} \, \lt \, 1 - 10^{-3} \, \lt \dots \lt \,1 - 10^{-(n+5)} \, \lt \, 1 - 10^{-(n+6)} \, \lt \, 1 - 10^{-(n+7)} \, \lt \dots \lt S\)

Now I want to prove that if \(X\) is less than 1 then there are members of the family \(S_n\) which are larger than it.
Say \(X \lt 1\). Then \(0 \lt 1 - X\). Then \(1 - X\) is a positive number. Then there is a real number \(\epsilon\) such that \(e^{-\epsilon} = 1 - X\) because \(f(x) = e^x\) is a continuous function takes on all positive real values. Also \(f(x \ln 10) = e^{x \ln 10} = 10^x\). So \(X = 1 - e^{-\epsilon} = 1 - 10^{- \frac{\epsilon}{\ln 10}} \). So the question of if there are members of the family \(S_n\) which are greater than \(X\) reduces to the question of asking if there are counting numbers which are larger than the real number \(- \frac{\epsilon}{\ln 10} = - \frac{\ln ( 1 - X) }{\ln 10}\) and the answer is yes, most of them.
So every number less than 1 is less than some (or all!) members of the family \(S_n\) and thus less than \(S\).

Thus 1 is greatest lower bound on what S could be, because all numbers less than 1 fail to qualify.

One of the biggest principles of the real numbers is that if a non-empty set of numbers has a upper (or lower!) bound then it has a least upper bound (or a greatest lower bound). By this, we see that the set of positive rational numbers whose square is less than 2 has greatest lower bound of 0 and a least upper bound of √2. 0 is not positive and √2 is not rational, but they are the minimal bounds of that particular set of positive rationals. That's why \(S\) which is the least upper bound of the family of \(S_n\) need not be a member of the family and can be 1.

Likewise \(\lim_{n \to \infty} S_n = \lim_{n \to \infty} (1 - 10^{-n}) = 1 - \lim_{n \to \infty} 10^{-n} = 1\) provides a concise line of reasoning for what S is.

Claims that \(1 - S\) are some sort of positive infinitesimal have been asserted but not reasoned from first principles. For one, there are no rules for manipulating infinitesimals in conjunction with the real numbers, so there are no first principles under discussion.

Secondly, positive infinitesimals make no sense within the context of other numbers we have experience with. Let's assume that near 0 there is the collection of all positive infinitesimals that differ qualitatively from finite rational and real numbers, \(P\). 0 is the greatest lower bound of \(P\) and let \(p\) be the least upper bound of \(P\). So \(p\) is either finite or infinitesimal. But because \(p \gt 0\) then \(p + p > p\), both \(2 p\) and \(10 p\) must therefore be finite and positive while \(p/2\) must be infinitesimal. But then there is always some counting number \(n\) such that \(p \leq 10^{-n} \leq 10 p\). Then \(p \gt 10^{-(n+2)}\) so there must be some members of \(P\) that are between \(10^{-(n+2)}\) and \(p\). But that contradicts the assumption that infinitesimals are qualitatively different than finite numbers. Something must be wrong in the assumptions being made, but the easiest assumption to deny is that there are any positive infinitesimals. And indeed there aren't in the real numbers.

 

Well. that all looks impressive and all, but what about counting sheep or camels?
Can we say a camel is "an infintesimal of physical effectiveness", or "a real indivisible state"?

I mean, humans started using numbers by counting whole things and keeping records, thereby "inventing" number systems. The notational conventions of the day were sufficient or fit for their purpose, and they survived as long as they remained so (viz the persistence of the Roman number system and its eventual replacement with the Arabic system).

Now you propose to "update" our concept of numbers, but haven't really pointed out why they are no longer fit for the purpose. We have computers that seem to do quite a good job handling numbers (as digital on/off states), I just can't see why, or even how, this needs revising. Computers of course, can't handle infinities because they all have an "overflow problem". Infinity is a concept though; computers don't "handle" concepts.

 

I just want to ask, does a particle having half-life decay completely?

 

When k = infinity(consider it(k) as hyper-real or extended real number), how do you think the infinity-th term would be in the RHS of the above equation?