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

The 0.999... = 1.0 argument might conceivably be an example of that inconsistency.

What "inconsistency" are you talking about? You have been given a precise explanation based on the way series function since their description by Taylor and McLaurin nearly 300 years ago.

I think that a few people here do have some rather crankish ideas.

This is a gross understatement. Look at this thread , for example.

But others of us are basically saying that what the 0.999... = 1.0 people are asserting seems counterintuitive and illogical.

For the cranks, yes. For the people who studied and understand math, no.

What's needed in these cases is a convincing account of why the counterintuitive ideas really do make sense.

But you received the explanation, so why do you persist in claiming that you haven't?

(3) poorly understand the common base-10 notational system.
IMHO this is the crux of the problem. We were taught the topic of different bases when I was 8 years old (1966 - for those keeping score). I intuitively grasped the concept and didn't understand why others didn't. Which is also why I find threads like this to be puzzling. The problem is a non-issue... It's an anomaly created by having 10 fingers.

I've actually seen introductory calculus textbooks that take a non-standard infinitesimals approach.

There's only one (Keisler). It came out in the 70's. It didn't catch on. And .999... = 1 is a theorem in nonstandard analysis, is it not? It would be a consequence of the transfer principle.

http://en.wikipedia.org/wiki/Transfer_principle

I'm not an expert on NSA so if I've got this wrong, please supply the correction.

The number represented by the infinite decimal $${0.a_1a_2a_3a_4\dots}_{\tiny 10}$$ is somewhere between 0 and 1 so we can write
$${0.a_1a_2a_3a_4\dots}_{\tiny 10} \in \left[0 , 1\right]$$ without using any of the information of the digits.
Using the information in the first digit, we can write:
$${0.a_1a_2a_3a_4\dots}_{\tiny 10} \in \left[\frac{a_1}{10}, \, \frac{a_1 + 1}{10}\right]$$
and likewise

$${0.a_1a_2a_3a_4\dots}_{\tiny 10} \in \left[\frac{a_1a_2}{100} , \, \frac{a_1a_2 + 1}{100}\right]$$
$${0.a_1a_2a_3a_4\dots}_{\tiny 10} \in \left[\frac{a_1a_2a_3}{1000} , \, \frac{a_1a_2a_3 + 1}{1000}\right]$$
$${0.a_1a_2a_3a_4\dots}_{\tiny 10} \in \left[\frac{a_1a_2a_3a_4}{10000} , \, \frac{a_1a_2a_3a_4 + 1}{10000}\right]$$

And all of these closed intervals form a chain of nested, non-empty subsets:
$$\left[0 , 1\right] \; \supseteq \; \left[\frac{a_1}{10}, \, \frac{a_1 + 1}{10}\right] \; \supseteq \; \left[\frac{a_1a_2}{100} , \, \frac{a_1a_2 + 1}{100}\right] \; \supseteq \; \left[\frac{a_1a_2a_3}{1000} , \, \frac{a_1a_2a_3 + 1}{1000}\right] \; \supseteq \; \left[\frac{a_1a_2a_3a_4}{10000} , \, \frac{a_1a_2a_3a_4 + 1}{10000}\right]$$
And so $${0.a_1a_2a_3a_4\dots}_{\tiny 10} \quad \in \quad \left[0 , 1\right] \; \cap \; \left[\frac{a_1}{10}, \, \frac{a_1 + 1}{10}\right] \; \cap \; \left[\frac{a_1a_2}{100} , \, \frac{a_1a_2 + 1}{100}\right] \; \cap \; \left[\frac{a_1a_2a_3}{1000} , \, \frac{a_1a_2a_3 + 1}{1000}\right] \; \cap \; \left[\frac{a_1a_2a_3a_4}{10000} , \, \frac{a_1a_2a_3a_4 + 1}{10000}\right] \; \cap \; \dots$$

But why closed intervals?
By definition an interval is a collection of all the numbers (for some concept of what number means) between the endpoints. A closed interval includes its endpoints while an open interval does not include them. It is not true that every infinite decimal corresponds with a number for concepts of number as coarse as the rational numbers, but by assumption we are using a concept of number which is fine enough that an infinite decimal always maps to one such number.
• If the intervals were open or half-open below 0.000... would have to be larger than 0 but smaller than any positive rational and we could not have any exact representations for rational numbers of the form $$\frac{n}{10^m}$$.
• And if the intervals were half-open above then this is again saying there are numbers that this decimal representation cannot express.
For example, 0.999... would have to differ from 1 by a number that would have to be larger than 0 but smaller than any positive rational, but since our intervals are half-open above, this number has no possible decimal representation.​
So by virtue of using closed intervals, we allow the possibility that every number has at least one representation in the infinite decimals and therefore at the cognitive expense of having some rational numbers that have more than one representation, we have a surjective mapping between infinite decimals and a particular concept of number. That concept is called the Real Numbers and has roots stretching back to antiquity -- long before the invention of the decimal number system.

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There is no "incompleteness". There is no "insufficiency".

The issue being discussed has nothing to do with physics, it is an elementary math problem. This has been explained to you countless times.

I am thrilled to see that you and MD are in agreement while sharing fringe misconceptions. Mainstream scientists, not so much. The math as we know it doesn't suffer from any issues and it is fully reliable, contrary to your persistent fringe dronings.

We need no such things, your theories belong in "Alternate Theories". Current axioms are firmly established.

Most of us can make perfect sense of the math, why can't you?

Whoa there, buddy! How can you sound so certain when even your own 'answers' tell us that axiomatically there are certain 'results' that can NOT bet 'determined' and so must remain 'undefined' because those axioms can NOT handle them consistent and complete, hence the undetermined/undefined excuses when the axioms hit the wall of their adequacy!

What is your agenda, Tach? Is it to claim that the mathematics and its axiomatic formulation is COMPLETE and CONSISTENT in EVERY way?

If so, then you should be avoiding giving 'explanations' whose 'axiomatic answers' are "undetermined", "undefined". Yes?

If you have nothing but repetition of 'textbook non-answers', then it would be best for you to leave these subtle searching discussions/points alone, and just observe the genuine mathematicians post reasonable arguments in context and in genuine attempt to progress the genuine discussions. Take a break from kneejerking-from-textbooks, and from defending patently inadequate axioms which lead to "undefined" and "undetermined" INCOMPLETENESS situations/excuses. OK? Thanks.

Hi gmilam.

IMHO this is the crux of the problem. We were taught the topic of different bases when I was 8 years old (1966 - for those keeping score). I intuitively grasped the concept and didn't understand why others didn't. Which is also why I find threads like this to be puzzling. The problem is a non-issue... It's an anomaly created by having 10 fingers.

It is the discussion itself that matters, not the cause of it, mate. During the discussions, many things about infinitesimals and non-zero '0' etc etc have been discussed which challenge the usual "undetermined" and "undefined" excuses for the axioms which lead to such incomplete and unsatisfactory 'results'.

As can be seen from all the discussions/points that have ensued, this old argument has progressed far from its initiation-stage 'quirk of decimal number system' aspects.

Thrill to the unusual twists and turns of the journey so far, and enjoy the ride, mate! Cheers!

Whoa there, buddy!

I am not your buddy, I am not buddies with the ones like you.

How can you sound so certain when even your own 'answers' tell us that axiomatically there are certain 'results' that can NOT bet 'determined' and so must remain 'undefined' because those axioms can NOT handle them consistent and complete, hence the undetermined/undefined excuses when the axioms hit the wall of their adequacy!

I'll have Italian with the above word salad of nonsense.

What is your agenda, Tach? Is it to claim that the mathematics and its axiomatic formulation is COMPLETE and CONSISTENT in EVERY way?

I have no agenda, you on the other hand.....have serious delusions of being a researcher that challenges the existent paradigms.

If so, then you should be avoiding giving 'explanations' whose 'axiomatic answers' are "undetermined", "undefined". Yes?

Nope, you fail. Again. The "undefines" are a direct resultants of the existent theorems.

If the shoe fits, wear it.

If you have nothing but repetition of 'textbook non-answers',

Trouble is, the textbooks do GIVE the appropriate answers, you should try studying them for a change.

then it would be best for you to leave these subtle searching discussions/points alone, and just observe the genuine mathematicians

Stop deluding yourself, you are no mathematician.

uhm as I mentioned earlier but appear to have been totally ignored. The issue, to me, is simply:
Does 1/infinity = 0 ?
According to a PHD, in Pure Mathematics 1/infinity = infinitesimal and not zero. [An infinitesimal is NOT a number I believe]
this whole issue about 0.999... = 1 is a no brainer when you clarify the position about what 1-0.999... = ?

What is wrong with stating that 0.999...= 1 is only valid because it is a defined result using a process of limitations?
To say that 0.999... = 1 is a "defined" result appears to be axiomatically correct where as to say that it is a calculated result generates all sorts of issues.

I am not your buddy, I am not buddies with the ones like you.

I'll have Italian with the above word salad of nonsense.

I have no agenda, you on the other hand.....have serious delusions of being a researcher that challenges the existent paradigms.

Nope, you fail. Again. The "undefines" are a direct resultants of the existent theorems.

If the shoe fits, wear it.

Trouble is, the textbooks do GIVE the appropriate answers, you should try studying them for a change.

Stop deluding yourself, you are no mathematician.

Ummm, ok. Since 'professional/trained' mathematicians don't seem to be getting anywhere further from the "undetermined" and "undefined" etc state of affairs that has obtained for so long now, perhaps it may be left to non-professional mathematicians who work at it for the LOVE of maths and science (and not the 'status/ego' etc) to make the next leap forward towards completion of both maths and science. Especially if 'professional' mathematicians are all (I certainly hope not!) as clueless as you still seem to be about the incompleteness status of the current maths/axioms which output such absurdities as "undefined" and "undetermined" etc. Good luck with that while the dialogue and advances pass you by in your in-status-quo-belief-without-question way of 'discussing-by-insulting'.

Ummm, ok. Since 'professional/trained' mathematicians don't seem to be getting anywhere further from the "undetermined" and "undefined" etc state of affairs that has obtained for so long now, perhaps it may be left to non-professional mathematicians who work at it for the LOVE of maths and science

Rubbish. First of all, there is no need for your intervention, second off, you don't know where to start, third off, you have no love of science, all you are doing is spamming the forums.

to make the next leap forward towards completion of both maths and science.

...like in you Theory of Everything that no one has ever seen? Still weaving the clothes of the emperor, RC?

QQ,
Infinity, as I'm almost sure someone has addressed, is a concept, not a number. So 1/infinity is at best a concept.

Since infinity is larger than any finite concept of magnitude, it follows that 1/infinity, to such an extent the concept is rigorous, must be smaller than any finite concept of magnitude.

Further if your number system (like the hyperreals or surreal number system) admits one transfinite number, then it admits an infinite number of them, and infinity conceptional is larger than all of the admitted transfinite numbers. Therefore it follows that 1/infinity must be smaller than any positive infinitesimal.

Therefore, one is tempted to declare that 1/infinity = 0. But this does not promote the concept of infinity to the status of number. Nor does it admit 0 times infinity equals one.

The Riemann sphere conceptionally adds just one transfinite number, called $$\infty$$, to either the real numbers or the complex plane, but in doing so it it forces to strip infinity of the property of being larger than all real numbers, for it is also smaller than all real numbers and $$-\infty = \infty = \sqrt{-1} \infty$$.
http://en.wikipedia.org/wiki/Riemann_sphere#Arithmetic_operations

1/(any transfinite number) is a specific non-zero infinitesimal in systems that permit such objects. This is surely what your PhD authority in Pure Mathematics was trying to convey. [Citation of that purported authority requested, please.]

1/(the point on the Riemann sphere identified by the symbol $$\infty$$) = 0, but this does not mean $$0 \times \infty = 1$$ or even $$\infty - \infty = 0$$ so some violence to our concept of number has occurred.

But if we agree to work with real numbers and their completeness axiom, then it is required that the only possible "completion" of the unending sequence 0, 0.9, 0.99, 0.999, 0.9999, ... is 1. This claim doesn't rest on which form of the completeness axiom you choose, it's the same if you use Cauchy sequences and limits, supremum of the infinite set, topological continuity, etc.

I disagree with the claim that "0.999... = 1" is a no-brainer. You need practice in applying formal logic to know that if formal logic decides this, then deciding it once decides it for all time, and that one's choice of definition for number matters in important ways. This is what [post=3127165]Hamming meant when he wrote "In mathematics we do not appeal to authority, but rather you are responsible for what you believe."[/post] As someone with a physics background, I have a geometry background, which means my concept of number is closely related to my concept of length and I can conceive of lengths in the ratio $$1\, :\, 2$$, $$1\, :\, \frac{2}{5}$$, $$1\, :\, \frac{1}{3}$$, $$1\, :\, \sqrt{2}$$, and $$1\, :\, \pi$$. In geometry, you can't have two points next to each other with no room between them to add a third distinct point. So the concept of magnitude derived from geometry, the real numbers, cannot admit a difference between 0.999... and 1. They are two names for the same point on the real number line.

To calculate 0.999... ab initio one needs a general mechanism for evaluating an infinite sum of ever smaller pieces. That mechanism, both in analysis and non-standard analysis, is the limiting value of the sequence of finite partial sums as the number of terms in the partial sums grows without limit. (Thus the concept of infinity is used without requiring that we describe infinity as a number.) And that limiting value in this particular case is 1.
If x is a number different than 1 and y is a number smaller than | 1 - x |, then at most only a finite number of the partial sums will be in the y-radius neighborhood of x. Thus a number other than 1 cannot be the limiting value of the sequence of partial sums.

Rubbish. First of all, there is no need for your intervention, second off, you don't know where to start, third off, you have no love of science, all you are doing is spamming the forums.

"Rubbish?" "No need for intervention?"

Where have you been? Haven't you been here reading the discussion? If you had, then your 'answer' to MD about something being "undetermined" and "undefined" etc FROM THE CURRENT AXIOMS has made it clear that even YOU must by now realize that the current axioms are INCOMPLETE and INADEQUATE.....hence the absurd loophole 'excuses' of "undefined" and "undetermined".

Don't you even know what you write or what the implications of what you wrote to MD were when it comes to the NEED for intervention sooner or later to eliminate such "undetermined" and "undefined" outputs from the mathematics/physics modeling so as to allow the next step towards consistent completion of BOTH?

What use are you, except to keep yelling insults at people and pretending to have some authority which you only repeat from textbooks while missing the NEW points being made which will eventually CHANGE those textbooks?

...like in you Theory of Everything that no one has ever seen? Still weaving the clothes of the emperor, RC?

Since you brought it up: it is to demonstrate the current emperor's lack of clothes that I will publish the maths and physics TOE as a complete and consistent real treatment from BOTH mathematical and physical axioms/postulates. Maybe you should wait until then before making more uninformed insults and assumptions based on your current incomplete knowledge in both. That might help to curb your ego-tripping tendencies to a tolerable level. Good luck!

PS: Tach, you have (against the rules and against recent admin/mod directives to you) referred to me as RC (and hence referred to my past RealityCheck personna). This blatant trolling/baiting post cannot be ignored, as it flies against the admin/mod and site rules without any excuses. Hence you are reported. Good luck and do better next time, Tach.

"Rubbish?" "No need for intervention?"

Where have you been? Haven't you been here reading the discussion? If you had, then your 'answer' to MD about something being "undetermined" and "undefined" etc FROM THE CURRENT AXIOMS has made it clear that even YOU must by now realize that the current axioms are INCOMPLETE and INADEQUATE....

'''in your mind. In the minds of the mainstream scientists, there is no such issue.

hence the absurd loophole 'excuses' of "undefined" and "undetermined".

Well, this is your opinion. No one cares.

Don't you even know what you write or what the implications of what you wrote to MD were when it comes to the NEED for intervention sooner or later to eliminate such "undetermined" and "undefined" outputs from the mathematics/physics modeling so as to allow the next step towards consistent completion of BOTH?

There are no "implications" we are doing very well without your constant droning about any "NEED for intervention".

What use are you, except to keep yelling insults at people and pretending to have some authority which you only repeat from textbooks while missing the NEW points being made which will eventually CHANGE those textbooks?

The textbooks are fine as they are, you should try understanding them instead of your constant droning about changing them.

Since you brought it up: it is to demonstrate the current emperor's lack of clothes that I will publish the maths and physics TOE as a complete and consistent real treatment from BOTH mathematical and physical axioms/postulates. Maybe you should wait until then before making more uninformed insults and assumptions based on your current incomplete knowledge in both. That might help to curb your ego-tripping tendencies to a tolerable level.

Stop deluding yourself, your "TOE" will never get published. For the mere reason that it is just a figment of your imagination.

QQ,
Infinity, as I'm almost sure someone has addressed, is a concept, not a number. So 1/infinity is at best a concept.

Since infinity is larger than any finite concept of magnitude, it follows that 1/infinity, to such an extent the concept is rigorous, must be smaller than any finite concept of magnitude.

Further if your number system (like the hyperreals or surreal number system) admits one transfinite number, then it admits an infinite number of them, and infinity conceptional is larger than all of the admitted transfinite numbers. Therefore it follows that 1/infinity must be smaller than any positive infinitesimal.

Therefore, one is tempted to declare that 1/infinity = 0. But this does not promote the concept of infinity to the status of number. Nor does it admit 0 times infinity equals one.

Hmmm. rpenner, thanks for your polite and genuine contributions to the discourse so far. Much appreciated.

About the above, I recall recently where you invoked the notion of "reasonableness" when approaching such things as are being discussed here. I therefore ask the question: Would it be more reasonable to base maths axioms on reality than pure abstraction?

I ask this because when it comes to zero, there is an obvious (already exhaustively pointe out and explained) difference between the 'dimensionless point' NOTION on which all the 'dimensional' properties/logics etc are based in subsequent mathematics treatments)...and the PHYSICAL 'location' where SOMETHING must exist that has some dimension in energy-space terms.

The former 'point' is abstract and dimensionless NOTHING, whereas the latter 'location' must, in reality energy-space terms (even before any other notions/abstractions are made) be associated with SOMETHING.

As the discussions have elicited, the 'mathematical point' notion is NOTHING ZERO, while the 'physical location' notion is based on the fact that SOMETHING EXISTS in energy-space terms no matter how much we 'abstract' that something in our maths/models?

See? That is the 'reasonableness' test which ALL axioms and postulates must be based on if they are ever to be complete and consistent in both maths and physics.

In the physical universe, there is always SOME THING that exists; whereas currently, mathematics assumes from the dimensionless point axiom (based on abstract notion only) that there can exist NOTHING (ie, no energy-space or dimensions in any context whatsoever....ie, evidence the 'dimensionless point abstraction of maths).

Anyhow, that's where I and others wish to take the reasonableness into, the REALITY as well as just the maths in isolation from reality which leads to incomplete/unreal axioms which in turn lead to "undetermined" and "undefined" etc results where maths is effectively made impotent to further help us elucidate the complete physical reality.

And if 1/infinity is a concept like any other abstract concept, then just as there can be concept of "imaginary number" then there can be a concept logically consistent with what MD pointed out to Tach (using Tach's own examples) of 'infinitesimal number' which has its basis in the 1/infinity conceptual treatment of SOMETHING (and not 'no thing') REAL such that when it 'disappears' into INEFFECTIVENESS (physical and mathematical) then it is the smallest 'imaginary' step/number which completes the process and transitions the term from one DOMAIN of effectiveness/ineffectiveness to the SUPERPOSED/ADJACENT DOMAIN of phenomenological state/dynamics etc.

A reasonable approach/goal, yes?

Let's also take a moment and consider:

Whenever the LIMITS argument/concept is used for such expressions as 0.999..., we automatically and EFFECTIVELY INVOKE a 'last infinitesimal' of that 'infinity' concept used in that limits notation/treatment, in order to bring the expression from a never ending undetermined state to a determined state of "1"...by effectively appending an (IMAGINARY NUMBER?) INFINITESIMAL non-zero 'step' value we call '0' but which CAN NOT be 'nothing' in reality?

Yes?

Thanks again for your contributions in so professional and courteous and erudite a manner (something which certain wannabe's could do worse than try to emulate!). Kudos, rpenner!

'''in your mind. In the minds of the mainstream scientists, there is no such issue.

Well, this is your opinion. No one cares.

There are no "implications" we are doing very well without your constant droning about any "NEED for intervention".

The textbooks are fine as they are, you should try understanding them instead of your constant droning about changing them.

Stop deluding yourself, your "TOE" will never get published. For the mere reason that it is just a figment of your imagination.

Quoted for the record. Reported for opinionating and personal insults and evading the very issues which Tach claims "don't exist".

... What is wrong with stating that 0.999...= 1 is only valid because it is a defined result using a process of limitations? ...
That "limiting process" is not the ONLY way to show 1 = 0.999... here that is done without speaking of limiting:
{post 301 in part}...The proof I and others have given that 1 = 9/9 = 0.99999.... is just particular case of the fact ALL rational fractions like a/b or a/9 (both a & b being integers and a < b) are equal to an infinitely repeating decimal (if they are not a finite decimal when b is a factor or product of factors of the base).

For example, the general proof of this goes like:
Rational Decimal, RD = 0.abcdefg abcdefg abcdefg .... Where each letter is one from the set (0,1,2...8,9) and the spaces are just to make it easier to see the repeat length in this case is 7.
Now for this repeat length 7 case, multiplying RD by 10,000,000 moves the decimal point 7 spaces to the right. I. e. 10,000,000 RD = a,bcd,efg . abcdefg abcdefg ... Is a 2nd equation with comas for easy reading the integer part.

Now, after noting (10,000,000 - 1) = 9,999,999 and subtracting the first equation from the second, we have the integer:
a,bcd,efg = 9,999,999 x RD. Note there are no infinitely long numbers here and 9,999,999 certainly is not zero so we can divide by it to get: RD = a,bcd,efg / 9,999,999 the rational fraction of integers exactly equal to the infinitely long repeating (with repeat length =7) decimal, RD.

Now lets become less general and consider just one of the repeat length = 7 cases. I. e. have a=b=c=d=e=f=g = 9 and recall RD was DEFINED as 0.abcdefg...so is now in this less general RD = 0.9,999,999,... and from green part of line above, RD = 9,999,999 / 9,999,999, which reduces to the fraction 1/1 which is unity as the numerator is identical with the non-zero denominator. I.e. the least numerator rational fraction equal to 0.999,999... is 1/1. ...
Someguy1 objected to multiplying an infinite decimal string. I noted in my reply that "time 10" was only descriptive of what moving decimal point achieves. It is really built into the meaning of the notation using a decimal point. I.e. first place to left of the decimal point tell how many unit second how many 10s, third how many 100s etc. So 5.00 is five and 50.0 is fifty. That times 10 per shift of point one space more left is part of the meaning of the notation, not an operation o multiplying.

That "limiting process" is not the ONLY way to show 1 = 0.999... here that is done without speaking of limiting:

Someguy1 objected to multiplying an infinite decimal string. I noted in my reply that "time 10" was only descriptive of what moving decimal point achieves. It is really built into the meaning of the notation using a decimal point. I.e. first place to left of the decimal point tell how many unit second how many 10s, third how many 100s etc. So 5.00 is five and 50.0 is fifty. That times 10 per shift of point one space more left is part of the meaning of the notation, not an operation o multiplying.

Hi Billy T.

I have already mentioned where trivial non-actions (like using 1/1, 9/9 'constructions') as a 'step' in 'proofs' is defeating the purpose of fractional treatment first and foremost without introducing self-serving circuitous constructions which assume a-priori what the 'fraction' will be (because you already have the abstraction of 1/1, 9/9 and unity as somehow being relevant to the FRACTIONAL states being discussed).

Any unity construction using trivial notations like 9/9 etc do not really 'explain' anything fundamental; because, of course if you divide like with like you essentially are ALREADY at the UNITY state of abstraction.

Hence your setup is self-determining already, and hence not a valid way to 'proof' of anything but that 1/1=1 and 9/9 =1 etc.

It's not such trivial already-self-determined 'unity states' we are exploring, but the FRACTIONAL state, and how to get from that to UNITY via a last step (as suggested by the reasonable concept of an 'imaginary' infinitesimal non-nothing zero value/step 'number'?).

If imaginary numbers can 'exist', then so can an imaginary infinitesimal concept number. Yes?

Thanks for your time and trouble, Billy T. Your contribution to discourse here and elsewhere is much appreciated, I assure you. Cheers.

If imaginary numbers can 'exist', then so can an imaginary infinitesimal concept number, Yes?

No.

Calling numbers "imaginary" is an unfortunate historical accident. But imaginary (and more generally, complex) numbers are easily constructed from the rules of set theory. If you believe in the real numbers, and if you believe in pairs (x,y) of real numbers, then you can define addition and multiplication of pairs in such a way that the resulting system is exactly what we mean by the complex and imaginary numbers.

No such construction is available to construct infinitesimals, except for nonstandard analysis, in which .999... = 1 is still a theorem. The vague, magic 1/infinity infinitesimals referred to repeatedly in this thread simply do not have any mathematical existence at all. "1/infinity" is not defined for the simple reason that there's no mathematically sensible way to define it.

No.

Calling numbers "imaginary" is an unfortunate historical accident. But imaginary (and more generally, complex) numbers are easily constructed from the rules of set theory. If you believe in the real numbers, and if you believe in pairs (x,y) of real numbers, then you can define addition and multiplication of pairs in such a way that the resulting system is exactly what we mean by the complex and imaginary numbers.

No such construction is available to construct infinitesimals, except for nonstandard analysis, in which .999... = 1 is still a theorem. The vague, magic 1/infinity infinitesimals referred to repeatedly in this thread simply do not have any mathematical existence at all. "1/infinity" is not defined for the simple reason that there's no mathematically sensible way to define it.

Yes, I realize that as currently axiomatically treated/defined, things are as you say. But I was exploring the changes to that axiomatic set such that the infinitesimal CAN then be treated consistent and complete within the new axiom? That was the thrust of my using the 'imaginary' label for infinitesimal, to highlight how axiomatic treatments depend on said axioms, but IF the axioms change, then the imaginary status is available as a conceptual tool for thinking about an 'final straw' like infinitesimal of effectiveness step/change (ie, some 'NON-nothing zero' point/value) in reality that transitions the string/state from one maths/physical domain to the adjacent domain (in either direction).

Thanks for that clarification of 'imaginary' usage/history, mate!