The meaning of infinity

[Jokodo]

Then why are you talking about performing arbitrary operations to 9/1 in a discussion of 1/9 as if there is any relationship?

Where am I?

You have a bunch of rounding tricks and nothing else.

You keep saying this, but can't point to a single one of them.

Of course it does. 0.04 in base 6, or 0.14 in base 12. The fact that it is not possible to express this value without somehow marking recursion in base 10 is a bug of base 10 notation, nothing more, nothing less. Definitely not a property of the number itself

1/9 is an operation. Not a number. You confuse operations with numbers.

And the operation never reaches a final value.

There is no final value to 1/9.

Of course there is, the question is only how to represent it in a given language such as the language of decimal notation. The decimal systems is not particularly well suited to precisely express fractions with a denominator that's a multiple of three because it's base, 10, does not include 3 as a prime factor. The duodecimal system is better suited for this purpose, but runs into similar problems with denominators that are multiple of five, and which are unproblematic under base 10. This doesn't say anything about the number itself. Your ability to cut a rod into nine sticks of equal length solely depends on the precision of your measuring and cutting tools, not on whether its initial length was one yard (yielding 9 sticks of exactly four inches each) or 1 metre (yielding nine sticks of 11.(1) cm).

It cannot be changed to some other base and retain all it's features. It a specific entity not any other entity.

You want to change it or perform rounding operations to it.

Not at all. I'm actively refusing to round it.

That is all you are doing.

Do you even know what "rounding" means?

Yes.

Rounding occurs when you pretend an infinite operation has completed.

Like when you pretend there is a final value to 9 * 0.1111...

That operation never finishes.

It finishes as soon as you've determined that the validity of the equation "1*10^n * 9 = 9*10^n" is independent of the value of n, which is a tautology. At that point, you know that, as long as the input contains no non-0, non-1 digits, the output has a "9" wherever there was a "1" in the input. As we defined ellipsis to mean "only '1's hereafter", the result follows per the definition of the input.

This method is no different in principle (rather, quite a bit more basic) from using long multiplication to derive the result of, say, 70 * 1100. Are you also claiming that the result 77,000 for this operation is valid only if I actually did at least 70 additions and record all intermediate values (and preferably 1100 -- who knows whether multiplication really is commutative)?

When you claim it does and by magic it somehow reaches 0.999... you have rounded off.

You have been taught to deal with infinities by pretending they are not there or by pretending they can complete. So you never see the irrationality in how you deal with them.

Do you understand that the decimal system is not the numbers

I am not talking about the decimal system.

I am talking about a specific entity. 0.999....

As defined it has no final value. There is no last 9.

The entity "0.999..." the way you are using it is a string. A string that could be a password on a server that doesn't require passwords to contain letters, that could be a partially masked credit card number, that could refer to "let's bomb the white house" in a code language you use to communicate with your co-conspirators, or any number of other things in any number of other actual or conceivable symbol systems aka languages.

In a limited (though still potentially infinite ) number of languages, in positional number systems with a base b > 9.0, 0.999... refers to a number. What number it refers to depends on which system you're looking at: For example 9/B₁₂ (=9/11₁₀) in duodecimal, 9/F₁₆=6/A₁₆=3/5₁₆ (=6/10₁₀=3/5₁₀) in hexadecimal (note how this actually has a final representation in decimal?), and 9/9=3/3=1/1 in decimal. More generally, it will always be 9/(b-1). Whenever (b-1) is divisible by 9 or 3 and the result of that division has no prime roots not also contained in the base, the result has a finite representation in that base*, otherwise it doesn't. However, that tells you something about that particular base (for decimal, that the base's prime roots are 2 and 5) and nothing about the number.

The entity referred to by the string "0.999..." in the arbitrary language that is the decimal notation is every bit as finite as the entity referred to by the same string in hexadecimal. The fact that 6/10 doesn't have a representation that can be explicitly enumerated in finite time in base 16 doesn't make the number more commonly known as 0.6 any more infinite than the fact that 1/9 cannot be in decimal makes that number infinite.

Except if you believe that the decimal notation -- and only the decimal notation -- has a special intimate relation to the numbers themselves.



* Assuming the arbitrary convention that recurring 0s need not be marked as such.

 

[Artemus]

This is off topic, but what the hell it is my thread. A sample problem.



An electric series resistor and capacitor driven by an AC source. What is the voltage across the resistor and capacitor vs time, and what is current vs time? It is a basic but not a contrived problem. It helps to understand electric circuits butbnot necessary. It is a simple differential equation.


Vc voltage across capacitor i = Cdv/dt Vc = integral(idt/C)
Vr voltage across resistor = i(t)*R
i(t) current
R resistor 1000
C capacitor 1*10^-6
Initial conditions i = 0, Vc = 0, Vr = 0.

Vs = 12 + 4*sin(2pi1000t) + 2*sin(2pi3000t)


Vs = Vr + Vc
Vs = 12 + 4*sin(2pi1000t) + 2*sin(2pi3000t) = i(t)*R + integral(idt/C)

Ironically you have written equations that depend entirely on concepts that you claim to reject throughout this thread. The value of the derivative is exactly that at the limit as dt goes to zero. The integral's exact value is the limit of a series as it goes to an infinite number of terms yet yields a finite value. Sinusoidal signals that extend for all time do not exist naturally. However, using Fourier analysis we can exactly represent any real signal as a spectrum of sinusoidal signals defined over all time and apply the circuit equations to the sinusoids individually, still getting the exact response when the inverse Fourier transform is performed. The Fourier transform is an integral operator defined when the limits of integration are extended to infinity, an extension of the infinite series, which yields an exact, finite value provided that the signal has finite energy (as all physically existing signals must) and the integral is therefore convergent. The spectrum of a physical, time-limited signal is continuous, so the inverse Fourier transform is adding an infinite number of sinusoidal terms, yet yielding a finite value. The (inverse) Fourier transform also depends on the complex exponential function, which is the analytic continuation of the real exponential function defined using the (convergent, and therefore exact) infinite series expansion.

None of this makes sense if the values obtained in the limit did not give the exact results of the mathematical operation, be it differentiation, series evaluation, or integration. As Maxwell's equations (since we are both electrical engineers) explicitly contain the differential operator (or equivalently, the integral operator when written in integral form), this is how reality works. It is not an approximation or a mathematical convenience.

 

[Jokodo]

I am talking about a specific entity. 0.999....

As defined it has no final value. There is no last 9.

I think I missed the elephant in the room.

The string "0.999...." very much does have a last "9", right there, for everyone to see, in position 4 (if you use 0-based indexing), directly before the second ".".

It is, within the language, defined to be synonymous with another string that doesn't have a last 9. But if you accept that to be meaningful, you might as well accept that the pair is also defined to be synonymous with 1.0 as all you need. Except that the latter synonymy relation is motivated: Without it, some basic properties of the language would have to change or else you'd run into contradictions, as it follows from them. The first synonymy relation, on the other hand, is entirely arbitrary, just a matter of convenience.

If, on the other hand, by the "specific entity. 0.999..." you mean the number it refers to in a particular dialect of base-10 positional notation*, it has been shown extensively that it equals 1.0, so whatever you claim about its properties will also have to apply to 1.0 (or, more likely, be plain false).

You are, of course, free to formalize a new language for referring to numbers where this doesn't hold. If it requires us to redefine multiplication and addition, it doubt it will be a useful addition, though.

* There are other dialects: most of Europe e. g. uses ',' rather than '.' to separate the integer part from the real part, and the recurrence property can be -- arguably more unambiguously -- expressed with a dot or arc above the recurring digit(s), or by paranthesising them.

 

[untermensche]

Where am I?



You keep saying this, but can't point to a single one of them.

Of course it does. 0.04 in base 6, or 0.14 in base 12. The fact that it is not possible to express this value without somehow marking recursion in base 10 is a bug of base 10 notation, nothing more, nothing less. Definitely not a property of the number itself

1/9 is an operation. Not a number. You confuse operations with numbers.

And the operation never reaches a final value.

There is no final value to 1/9.

Of course there is, the question is only how to represent it in a given language such as the language of decimal notation.

It is an operation not a final value.

You arrive at the decimal simply by performing the operation.

Not by thinking about where you want to end up. Or by transforming the fraction into something else.

It is like I said "go left".

And you claimed that meant the same thing as "arriving at the restaurant" because you know that is where you want to go.

You are not making rational sense.

The decimal systems is not particularly well suited to precisely express fractions with a denominator that's a multiple of three because it's base, 10, does not include 3 as a prime factor.

Yes.

You end up with infinite strings that cannot possibly end.

The fraction cannot be faithfully represented by the decimal. The decimal is always a little off. Thus the string has no end.

The fraction and the decimal are not the same thing.

Just like "go left" does not mean "eat your dessert". The operation (1/9) "divide one by nine" is not the same thing as the result (0.111...). The operation is what produces the result.

You see operations as results because you know where you want to go. You want your dessert.

The entity referred to by the string "0.999..." in the arbitrary language that is the decimal notation is every bit as finite as the entity referred to by the same string in hexadecimal.

0.999... does not refer to something.

It is something.

 

[DBT]

AT ANY PRESENT MOMENT ALL the events in the past have completed.

NO more events in the past will take place AT THAT PRESENT MOMENT.

No more events can take place in the past at a present moment.

They are ALL completed.

They could not have been infinite.

Time in the past could not have been infinite.

This still doesn't relate to what I said. You are still making up stuff that nobody has claimed and calling it absurd. You appear to be arguing with yourself.

Address my arguments or move along.

I was. I am. I am pointing out that you are making up terms and conditions that nobody has claimed, calling your self constructed terms and conditions absurd and arguing against them, therefore, to all appearances, arguing with yourself.

Why are you doing this? Nobody is saying that past events are not completed - ie, finished, no longer happening, except for possible causation - so why imply that this is an issue for your opponents?

 

[untermensche]

Address my arguments or move along.

I was. I am. I am pointing out that you are making up terms and conditions that nobody has claimed, calling your self constructed terms and conditions absurd and arguing against them, therefore, to all appearances, arguing with yourself.

Why are you doing this? Nobody is saying that past events are not completed - ie, finished, no longer happening, except for possible causation - so why imply that this is an issue for your opponents?

You see those words you quoted before you did not address them in any way?

That is part of my argument.

You have not addressed it in any way.

Address my argument.

You have no argument beyond we can pretend an infinity exists in some way therefore it is possible.

You having some tangential argument does not address mine in any way.

AT ANY PRESENT MOMENT ALL the events in the past have completed.

NO more events in the past will take place AT THAT PRESENT MOMENT.

No more events can take place in the past at a present moment.

They are ALL completed.

They could not have been infinite.

Time in the past could not have been infinite.

 

[DBT]

You see those words you quoted before you did not address them in any way?

That is part of my argument.

You have not addressed it in any way.

Address my argument.

You have no argument beyond we can pretend an infinity exists in some way therefore it is possible.

You having some tangential argument does not address mine in any way.

I am addressing your words and your arguments by pointing out that you appear to be arguing with yourself.

I am pointing out that nobody, as far as I know, is contesting statements such as ''NO more events in the past will take place AT THAT PRESENT MOMENT'' and its variations.

I am saying that events are finite, that events have a beginning, a middle and an ending with a related causal effect, causal ripples that bring changes to other objects and their relationships over time.

 

[steve_bank]

This is off topic, but what the hell it is my thread. A sample problem.



An electric series resistor and capacitor driven by an AC source. What is the voltage across the resistor and capacitor vs time, and what is current vs time? It is a basic but not a contrived problem. It helps to understand electric circuits butbnot necessary. It is a simple differential equation.


Vc voltage across capacitor i = Cdv/dt Vc = integral(idt/C)
Vr voltage across resistor = i(t)*R
i(t) current
R resistor 1000
C capacitor 1*10^-6
Initial conditions i = 0, Vc = 0, Vr = 0.

Vs = 12 + 4*sin(2pi1000t) + 2*sin(2pi3000t)


Vs = Vr + Vc
Vs = 12 + 4*sin(2pi1000t) + 2*sin(2pi3000t) = i(t)*R + integral(idt/C)

Ironically you have written equations that depend entirely on concepts that you claim to reject throughout this thread. The value of the derivative is exactly that at the limit as dt goes to zero. The integral's exact value is the limit of a series as it goes to an infinite number of terms yet yields a finite value. Sinusoidal signals that extend for all time do not exist naturally. However, using Fourier analysis we can exactly represent any real signal as a spectrum of sinusoidal signals defined over all time and apply the circuit equations to the sinusoids individually, still getting the exact response when the inverse Fourier transform is performed. The Fourier transform is an integral operator defined when the limits of integration are extended to infinity, an extension of the infinite series, which yields an exact, finite value provided that the signal has finite energy (as all physically existing signals must) and the integral is therefore convergent. The spectrum of a physical, time-limited signal is continuous, so the inverse Fourier transform is adding an infinite number of sinusoidal terms, yet yielding a finite value. The (inverse) Fourier transform also depends on the complex exponential function, which is the analytic continuation of the real exponential function defined using the (convergent, and therefore exact) infinite series expansion.

None of this makes sense if the values obtained in the limit did not give the exact results of the mathematical operation, be it differentiation, series evaluation, or integration. As Maxwell's equations (since we are both electrical engineers) explicitly contain the differential operator (or equivalently, the integral operator when written in integral form), this is how reality works. It is not an approximation or a mathematical convenience.

WOW, That is a lot to digest I really do not see what you are saying. The solution is obvious if you have had diffeq.

lim x -> inf y = 1/[2 | 1/x] = 0.5 verbally expressed as x GOES TO inf y GOES TO 0.5. It never gets to exactly 0.5 but the error is infinitesimal. A convergence test taken as an infinite limit is not exact.

If you want to discuss calculus and differential equations that would be another thread. There is Maxwell's equations as written as continuous variables and then there ais solving them in practice. Beyond simple problems they are solved numerically.

Faraday'S Law V = B(dphi/dt)

All of science and engineering reduce to the same basic math such as curl and divergence. Water in a pipe or an EM field descriptions all boil down to multivariable calculus and differential equations. They are all expressions of conservation. Curl and divergence tell where a volume has a net output, input, or zero. Water flow or an EM field.

 

[steve_bank]

I was probing to see if anyone had any computational math experience. I knew it wasn't directly on the net. Complex variables and Fourier Transforms are too complicated beyond simple problems. Differential equations are commonly solved numerically.

It is very simple. Euler's Method. A common method for solving differential equations.

https://en.wikipedia.org/wiki/Euler_method


Differentials from calc 1 i = c dv/dt dv = (i/C)dt where dt is the timestep.

Vc = 0 initial conditions

for n = 1 to 1000
V = Vc
i = [12v-Vc]/r
Vc = Vc + (i/C)dt
next n

V is the cap voltage vs time.

 

[Juma]

I was probing to see if anyone had any computational math experience. I knew it wasn't directly on the net. Complex variables and Fourier Transforms are too complicated beyond simple problems. Differential equations are commonly solved numerically.

It is very simple. Euler's Method. A common method for solving differential equations.

https://en.wikipedia.org/wiki/Euler_method


Differentials from calc 1 i = c dv/dt dv = (i/C)dt where dt is the timestep.

Vc = 0 initial conditions

for n = 1 to 1000
V = Vc
i = [12v-Vc]/r
Vc = Vc + (i/C)dt
next n

V is the cap voltage vs time.

Answer this post instead of changing the subject:
https://talkfreethought.org/showthr...ng-of-infinity&p=572462&viewfull=1#post572462

 

[steve_bank]

https://www.quora.com/Why-is-0-999-ldots-equal-to-1?share=1&srid=X68a

https://en.wikipedia.org/wiki/0.999...

You're welcome.

Already looked at it, does not affect my arguments. You are assuming when you see 0.999... = 1 via geometric series on the web it infers a finite 1. and not 1/1 as a fractional approximation. Googling is not a substitute for reasoning.

If its an aporoximation then you should be able to say how big the error is.
I know that this error = 0.
Can you show that it must be > 0? As you claim?

In an infinite limit that converges on a finite vale the error is infinitesimally small. Infinity can never be reached.

y = 1/(2 + 1/x) x can never go to zinfinity in reality, 1/x can never go to zero.. Computationally in practice 0.5 would be taken as exact.

In control systems the feedback equation is 1/( 1/A + F) where A is the forward gain and F is the feedback gain. In physical systems A > 100,000 iand the term 1/A is taken as zero and F controls the system. Yes, I understand error analysis. In harfer problems I used Monte Carlo techniques.

 

[Juma]

If its an aporoximation then you should be able to say how big the error is.
I know that this error = 0.
Can you show that it must be > 0? As you claim?

In an infinite limit that converges on a finite vale the error is infinitesimally small. Infinity can never be reached.

That wont do. The answer must be a real number. If you cant come up with a real value then the error is EQUAL to 0. That is the definition of equal for reals.

 

[untermensche]

You see those words you quoted before you did not address them in any way?

That is part of my argument.

You have not addressed it in any way.

Address my argument.

You have no argument beyond we can pretend an infinity exists in some way therefore it is possible.

You having some tangential argument does not address mine in any way.

I am addressing your words and your arguments by pointing out that you appear to be arguing with yourself.

I am pointing out that nobody, as far as I know, is contesting statements such as ''NO more events in the past will take place AT THAT PRESENT MOMENT'' and its variations.

I am saying that events are finite, that events have a beginning, a middle and an ending with a related causal effect, causal ripples that bring changes to other objects and their relationships over time.

Of course nobody can contest what I am saying.

It is a series of truisms.

It seems we have nothing left to discuss.

 

[Jokodo]

Where am I?



You keep saying this, but can't point to a single one of them.



Of course there is, the question is only how to represent it in a given language such as the language of decimal notation.

It is an operation not a final value.

You arrive at the decimal simply by performing the operation.

Not by thinking about where you want to end up. Or by transforming the fraction into something else.

It is like I said "go left".

And you claimed that meant the same thing as "arriving at the restaurant" because you know that is where you want to go.

You are not making rational sense.

The decimal systems is not particularly well suited to precisely express fractions with a denominator that's a multiple of three because it's base, 10, does not include 3 as a prime factor.

Yes.

You end up with infinite strings that cannot possibly end.

The fraction cannot be faithfully represented by the decimal. The decimal is always a little off. Thus the string has no end.

The fraction and the decimal are not the same thing.

Of course they're not. One is a number, the other one is a string, that in a particular language refers to that number. Just like the strings "Australia" and "Asia" refer to continents in the English language. What you're doing is not unlike proclaiming that Australia is more than twice as big as Asia because you did a character count.

Just like "go left" does not mean "eat your dessert". The operation (1/9) "divide one by nine" is not the same thing as the result (0.111...). The operation is what produces the result.

You see operations as results because you know where you want to go. You want your dessert.

The entity referred to by the string "0.999..." in the arbitrary language that is the decimal notation is every bit as finite as the entity referred to by the same string in hexadecimal.

0.999... does not refer to something.

It is something.

It is something indeed: a string of length eight, with four digits and four punctuation characters.

In a certain set of languages, the positional notation systems, it is defined to be synonymous with another string, an infinite one, and to refer to a rational number. The fact that it has an infinite alias doesn't make the string infinite any more than me defining an unending sequence of "u"s as an alias for "untermensche's mother" makes it impossible to refer to your mother. The existence of such a label doesn't make the number infinite anymore than my alias makes yo mamma infinitely massive.

What number 0.999... refers to depends on the language: in duodecimal, it's 9/11 (decimal 0.(81)), in hexadecimal 9/15 and in decimal 9/9. All of them rational numbers. The last also happens to be a natural number.

 

[Artemus]

WOW, That is a lot to digest I really do not see what you are saying. The solution is obvious if you have had diffeq.

The point is that the solution to the differential equation has meaning only under the precise definitions of the mathematical operations used. You deny the validity of those definitions and yet claim that the solutions have meaning. You can't have it both ways.

 

[Jokodo]

Another illustration: All of the following are equivalent labels that can be used to refer to you:

"untermensche"

"the person posting as untermensche at talkfreethought.org"

"untermensche@TFT"

"untermensche, child of <your mothers name> and <your father's name>"

"untermensche, child of <your mother's name> (daughter of <your maternal grandmother's name> and <your maternal grandfather's name>) and <your father's name> (son of <your paternal grandmother's name> and <your paternal grandfather's name>)"

The last can be trivially expanded by including your great-grandparents, and their parents, etc. So there trivially exists a label for you that explicates your genealogy back 32 generations, roughly 1000 years. Pronouncing that label at one second per parent-child relation takes 2 ^ 32 seconds, or 136 years, longer than any human has ever lived. There even exists a label for you that explicates your genealogy back 64 generations, roughly 2000 years. Pronouncing that label would take roughly 40 times the estimated time since the big bang.

By the logic that let's you conclude that the number referred to by 0.(9) "doesn't have a final value" because there exists a label that cannot be actualised, I herewith derive that you don't have a final value, aka don't really exist.

It's either that, or your logic is flawed.

 

[steve_bank]

WOW, That is a lot to digest I really do not see what you are saying. The solution is obvious if you have had diffeq.

The point is that the solution to the differential equation has meaning only under the precise definitions of the mathematical operations used. You deny the validity of those definitions and yet claim that the solutions have meaning. You can't have it both ways.

How so? A Reimann integration is a limut as dx goes to zero. It is an approximation.

A convergence test on a series is a limit.

0.000... is a definition, fotrver there are only 5's.

A geometric series used to describe it is not the same as 0.999...

As to definitions finite and infinite are mutually exclusive, do you wish to argue that? If you accept the two as mutually exclusive then any algoriym that equates 0.999... to a finite 1 must eiter be wrong, or the conclusion is based on misunderstanding of the operation and intent of the algorithm.

Tp me equating 0.999.. 10 1 is like someone claiming they have a perpetual motion machine or a process that outputs more energy than in.

I can write the differential equation for a parallel inductor capacitor circuit, convolve it with an impulse and it will mathematically oscillate forever. It can never be constructed.

The point is mathematical consistency does not mean a real condition. Geometric Series applied to repeating decimals provides approximate fractional equivalents. It does not serve as a proof 0.999... = 1. The only difference between 0.333.. and 0.999.. is that for the later the equation cam only produce 1/1.

If in real calculation on material objects 0.999.. can never equal 1. That would be analogous to violating conservation.

 

[steve_bank]

If its an aporoximation then you should be able to say how big the error is.
I know that this error = 0.
Can you show that it must be > 0? As you claim?

In an infinite limit that converges on a finite vale the error is infinitesimally small. Infinity can never be reached.

That wont do. The answer must be a real number. If you cant come up with a real value then the error is EQUAL to 0. That is the definition of equal for reals.

You are not listening, an approximate real number.

Acrealmworld example.

https://www.electronics-tutorials.ws/opamp/opamp_8.html

Reference the non inverting circuit. The opamp is the traingle symbol.

The op amp amplifier the voltage difference across the + - inputs.. This is called the open loop gain or aol. The resistors are the feedback function f.

Vout = vin x 1/[1/aol + f] In the limit as aol -> inf it reduces to vout = vin x 1+rf/rin a finite value., Infinite gain is impossible nd is typically> 100,000. for an f of 2 in the limit as aol -> inf for 1 volt in there is exactly 2 volts out. In reality as infinity is not reachable the output will be less than 2. You end up with real numbers but applied yo reality limits are approximations.

Likewise limit x -> inf 1/[2 + 1/x] approaches 2 but never gets there. We take it as 2 in a calculation.

Infinite limits are a tool for analysis, but are nor real.

 

[Jokodo]

WOW, That is a lot to digest I really do not see what you are saying. The solution is obvious if you have had diffeq.

The point is that the solution to the differential equation has meaning only under the precise definitions of the mathematical operations used. You deny the validity of those definitions and yet claim that the solutions have meaning. You can't have it both ways.

How so? A Reimann integration is a limut as dx goes to zero. It is an approximation.

A convergence test on a series is a limit.

0.000... is a definition, fotrver there are only 5's.

A geometric series used to describe it is not the same as 0.999...

As to definitions finite and infinite are mutually exclusive, do you wish to argue that? If you accept the two as mutually exclusive then any algoriym that equates 0.999... to a finite 1 must eiter be wrong, or the conclusion is based on misunderstanding of the operation and intent of the algorithm.

And that conclusion is itself based on a misunderstanding of what 0.999... is. The number 0.999... is finite every bit as much as 1.000... The symbols "0.999..." and "1.000..." are also finite (I'm counting eight characters for both). They're both shorthand for an infinite symbol though. The only difference between those symbols is that the conventions of the language allow us to elide recurring 0s but not recurring 9s without marking that ellipsis. So we get a "1" that doesn't signal an elided series of infinitely recurring 0s, and a "0.(9)" or "0,999..." that does signal its elided nines.

Tp me equating 0.999.. 10 1 is like someone claiming they have a perpetual motion machine or a process that outputs more energy than in.

You're assuming your conclusion: You think you know that 0.999... is smaller than 1.000..., therefore turning one into the other would be creating something out of nothing. But, if as numbers (not as the strings) they're the same to start with? Your entire argument breaks down when you don't assume your conclusion. It's circular.

I can write the differential equation for a parallel inductor capacitor circuit, convolve it with an impulse and it will mathematically oscillate forever. It can never be constructed.

The point is mathematical consistency does not mean a real condition. Geometric Series applied to repeating decimals provides approximate fractional equivalents. It does not serve as a proof 0.999... = 1. The only difference between 0.333.. and 0.999.. is that for the later the equation cam only produce 1/1.

If in real calculation on material objects 0.999.. can never equal 1. That would be analogous to violating conservation.

To the contrary, even if 0.999... and 1 were distinct mathematically, they'd have to be the same in any "real calculation on material objects" because material objects have a finite number of constituent particles.

 

[Kharakov]

In an infinite limit that converges on a finite vale the error is infinitesimally small. Infinity can never be reached.

y = 1/(2 + 1/x) x can never go to zinfinity in reality, 1/x can never go to zero.. Computationally in practice 0.5 would be taken as exact.

1/x is 0 from certain points of view. If you divide a line into an infinite amount of pieces, each piece is a point. It shifts dimensions from 1d to 0d. You can do it with volumes of spacetime as well (shifting down a dimension by dividing infinitely).

The fact that it can be done mentally means that it is done by nature.... but that's a rabbithole of an argument if I've ever seen one. And I haven't. I've read and herd them.