How can limits be exchanged without proof of validity?

[matt grime]

In order that that other thread can get locked, here's something for JonF et al (sans organic detritus).

Firstly, 0.999... is the same as 9/10 +9/100 +9/1000+... that is what the decimal expansion means. This sum is equal to 1.

Now the other quesition involved taking the limit as n tends to infinity of (0.99...)^n now this is ne, but perhaps, JonF, you're switching limits, which you cannot do without proving it is valid. Meaning here, if x_m = 0.99999...9 with m nines that lim_n ( lim_m (x_m)^n) = 1, yet lim_m (lim _n (x_m)^n) =0

this is ok, and not contradictory if you remember that maths is just the manipulation of symbols to follow certain rules. One is that you cannot exchange limits whenever you feel like it. There are times you can times you can't.

Here's a couple of examples I gave in another thread today where it matters when you take limits.

for the interval [0,1] define f_n to be x^n the point wise limit of this os the function f where f(x) = lim_n f_n(x) which is the same as the function that is 0 for all x in[0,1) and 1 at 1. So limits don't commute with continuity.

the second is g_n defined on R as 1+x/n for x in[-n,0] 1-x/n for x in[0,n] 0 elsewhere. the pointwise limit is the zero function and the integral of that over R is zero, but the integral of each g_n is 1, so limits don't commute with integration.

 

[JonF]

Seemed to me that this would mean that .999... Is infinitely close to 1 but not actually 1.

But now I have figured out the error of my ways, by trying to make an example where: the infinitely small gap is multiplied an infinite many times. But, it turned out my example showed me the exact opposite of what I thought it would.

 

[matt grime]

The idea that it is infinitely close, but not the same, is what Abraham Robinson used in his non-standard analyisis. He defines infinitesimals like this. These ideas are not mainstream (they are intuitive, perhaps, to an untrained mathematician, but do not allow one to define the real numbers in the correct (correct means usual) way, and hence are unintuitive to a trained mathematician). They are of use in parts of economics. The difference is that ordinarily we use this idea of cauchy sequences where a real number is an equivalence class. That's uninterestingly technical, but it just means we say if a sequence of real numbers tends to zero, the limit is zero. In Robinson's version, we say the limit is an infinitesimal, and real, and therefore zero because zero is the only real infinitesimal. It is a subtle distinction that I don't think I've explained properly. But just remember that .99999... is 1 is a consequence of how we've defined the real numbers.

 

[Zurtex]

Would I be correct in saying that if infinitesimally small numbers existed within a number system then so would infinity. As infinity is not defined within real numbers then neither can infinitesimally small numbers.

 

[matt grime]

The infinitesimal numbers, of Robinsonian analysis, are not part of the real numbers. THe only real infitesimal is 0.

 

[Hurkyl]

Would I be correct in saying that if infinitesimally small numbers existed within a number system then so would infinity.

No, that would not be correct.

Now,

if the number system was ordered (so you can define infinite and infinitessimal)
if the number system was a field (aka division by nonzero is defined)
and
if there was a nonzero infinitessimal

then you can conclude there is an infinite number. In fact, you can conclude there are lots of infinite numbers; if ε is a (nonzero) infiniessimal, then, for instance, 1 / ε and 2 / ε are distinct infinite numbers.

 

[Organic]

The cardinal of the empty set = 0

The cardinal of a non-empty set is at least 1

So the meaning of small and smaller is meaningless when we compare them to emptiness.

Any transformation from a non-empty set to an empty set cannot be but a phase transition form cardinality 1 to cardinality 0.

Because of this reason we cannot define the smallest number.

A notation that can describe this beautiful situation is this:

[.000..., 1)

The meaning of this notation is this:

There are infinitely many empty levels of scales in the above singleton that cannot change the state of some non-empty set to an empty set.

It means that the cardinality of this half-open interval [.000..., 1) is at least 1

Therefore [.999..., 9) + [.000..., 1) = 1

By the way [.000..., 0) is for nonstandard analysis as [.000..., 1) is for standard analysis.

In both cases the result is a non-empty content of some set ( |{x}|=1 ).

 

[Zurtex]

Oh, O.K

Organic in terms of real numbers would [.999..., 9) + [.000..., 1) not be the same as 1 + 0?

 

[matt grime]

But that isn't the real number system you're talking about. It is not possible to claim there are infinitely many 0s and then a 1 and still be in the real numbers. What you're talking about has been formalized as non-standard analysis. It can easily be shown that it is not the real numbers because then you're implying there is a natural order preserving bijection to an uncountable well ordered set, which does not exist. Now, if you were to care about mathematics you might wish to learn about all the things I've just mentioned. You won't though will you?

Incidentally, why is 0 not the smallest (non-negative) number? You omitted to deal with that. And your analysis doesn't hold up as there's nothing there which doesn't apply to the natural numbers, so you've just "proven" that 1 is not the smallest natural number.

To elaborate on that, at no point do you use what the symbol 0.999... means ie that it is a decimal expansion of a real number. We could equally be dealing with hexadecimal expansions and all of what you said remains just as "true" and you've just concluded something even less true.

Moreover, the object you write as [.00000000...,1) you call a half open interval. Well do you mean [0,1)? and you're saying that has cardinality is at least 1? Well, it is c, which I suppose is greater than 1. I'm only hesitating because you don't use the correct definition of cardinality.

If you don't mean that half open interval, I'm not sure which one you mean. The last sentence is also equally puzzling as the only interpretation of that is not correct, mathematicall.

Also a supposedly singleton set that is a half open interval of real numbers does not make much sense, as the set [x,x) means all the y greater than or equal to x and srtictly less than x, which is empty - there are no real y satisfying that criterion.

Apart from all those mistakes you've also introduced yet more undefined terms and concepts.

 

[Organic]

Zurtex,

Be aware that what I am doing here is a totally new point of view on Math language.

If you are a student than my ideas can make you a lot of troubles with your teachers, at this stage.

 

[Zurtex]

You can't be serious? I was confusing my teachers with the whole infinitesimally small numbers at GCSE level when I had much less of an understanding of mathematics.

 

[matt grime]

Just because it is a totally new point of view doesn't stop it being totally wrong and inconsistent and not suitable for posting in this thread. Please stick to theory development as requested.

 

[JonF]

Ok, I get now why .999... = 1

Previously I thought $.\bar{0}1 + .\bar{9} = 1$

Which it does, but that is because: .0000….1 = 0

The infinitely small gap I thought existed is actually 0

 

[Organic]

It is not possible to claim there are infinitely many 0s and then a 1

You did not understand [.000...,1) notation.

Here I am using the idea of the half-open interval on a representation of a single R member, which is a non-standard way of using.

By this notation I mean that there are infinitely many empty levels of scales in the above singleton that cannot change the state of some non-empty set, to a state of an empty set.

You can say: "But number 0 is not an empty set content, but a non-empty set content, for example {0}, so what are you talking about?"

My answer is: "Please don't interfere between the notation and the meaning behind the notation, which are two different things, and I am talking about the meaning behind the notations, which are:

If our singleton's notation is not 0, we are talking about a non-empty set, and if our singleton's notation is 0, we are talking about an empty set.

More than that, in general I am talking about the relative relations between two different representations of to different numbers, where their self unique values are being kept exactly as |{}| and |{0}| cannot have the same cardinality."

I am not talking about 1 notation as some level that we can reach.

1 notation is here to tell us that it does not matter how many levels notated by 0 we have, still it stays a content of a non empty set with cardinality 1 ( |{x}|=1 ).

By the way, by [.000..., 0) notation we get more general representation of the above idea, which says that our singleton cannot get cardinality 0 ( |{}|=0 ) no matter what notation we are using.

therefore the gap between the two numbers remains open(>0), in both standard and nonstandard analysis.

 

[ahrkron]

Organic,

When explaining or elaborating on your own version of math, please do so in the TD forum. It can confuse people that are learning, especially because the definitions you use for many terms are different from the standard ones. It is not enough to state that yours is a "new point of view".

 

[Organic]

Hi ahrkron,

Is General Math is a special forum for students?

 

[quantumdude]

Hi ahrkron,

Is General Math is a special forum for students?

No, it's a forum for math!

 

[Organic]

Hi Tom Mattson,

Is there any problem to discuss about non-standard ideas in General Math forum?

 

[Zurtex]

So Organic what exactly do you think you have proved? Because to me it just looks like you are trying to prove something by defining it to be so.

 

[matt grime]

Ok, I get now why .999... = 1

Previously I thought $.\bar{0}1 + .\bar{9} = 1$

Which it does, but that is because: .0000….1 = 0

The infinitely small gap I thought existed is actually 0

but 0.00000...1 does not have any meaning as a rea number! Infinitely many zeroes then a 1? Sorry.

 

[matt grime]

You did not understand [.000...,1) notation.

But yo'uve not explained it.

Here I am using the idea of the half-open interval on a representation of a single R member, which is a non-standard way of using.

Which you've not explained

By this notation I mean that there are infinitely many empty levels of scales in the above singleton that cannot change the state of some non-empty set, to a state of an empty set.

Makes no sense

You can say: "But number 0 is not an empty set content, but a non-empty set content, for example {0}, so what are you talking about?"

I could do, yes.

My answer is: "Please don't interfere between the notation and the meaning behind the notation, which are two different things, and I am talking about the meaning behind the notations, which are:

If our singleton's notation is not 0, we are talking about a non-empty set, and if our singleton's notation is 0, we are talking about an empty set.

Just a label, I could define it differently, and?

More than that, in general I am talking about the relative relations between two different representations of to different numbers, where their self unique values are being kept exactly as |{}| and |{0}| cannot have the same cardinality."

makes no sense again.

I am not talking about 1 notation as some level that we can reach.

who is? I don't know because again that is meaningless

1 notation is here to tell us that it does not matter how many levels notated by 0 we have, still it stays a content of a non empty set with cardinality 1 ( |{x}|=1 ).

By the way, by [.000..., 0) notation we get more general representation of the above idea, which says that our singleton cannot get cardinality 0 ( |{}|=0 ) no matter what notation we are using.

therefore the gap between the two numbers remains open(>0), in both standard and nonstandard analysis.

Please, take heed of the requests of more important people than me. You are not talking about the real numbers. It is evident that you do not understand the ideas of mathematics, please stop possibly confusing people unnecessarily because you're not making any sense.

 

[Hurkyl]

Is there any problem to discuss about non-standard ideas in General Math forum?

The (current) problem is that you are using others' threads as a vehicle to promote your own ideas.

$$.\bar{0}1$$

The thing about decimals is that the number of digits is very well specified. There is one digit for every positive integer; no more, no less. (yes, I'm just talking about the ones to the right of the decimal point)

So the problem with something like $0.\bar{0}1$ is that this sequence of digits cannot fit the definition of a decimal. By definition of decimal, the '1' has to occur in the n-th place for some positive integer n, which would mean that there aren't an infinite number of zeroes to its left.

 

[NateTG]

Which it does, but that is because: .0000….1 = 0

Without some extra explanation, $$0.\bar{0}1$$ is not a real number -- it's not covered by the typical 'bar' notation. Depending on how you extend 'bar' notation there is a variety of exotic interpretations available in addition to just ignoring everything on the right of the bar.

 

[Organic]

Sorry Hurkyl,

So the problem with something like is that this sequence of digits cannot fit the definition of a decimal. By definition of decimal, the '1' has to occur in the n-th place for some positive integer n, which would mean that there aren't an infinite number of zeroes to its left.

You missed the point.

In [.000..., 1) or [.000..., 0) the " x)" notation cannot be reached by "[.000..." and this is exactly the meaning of the open interval idea.

Therefore no infinitely many 0 notations in infinitely many n-th places can close the gap between
[.000..., x) and 0.

Again you look on the notations themselves and forget the meaning the they are representing, which in this case representing a structure of a fractal upon infinitely many scales, where its structural property depends on the base value which we choose to use.

For example, please see an example of these fractals when bases 2,3 or 4 are used:
http://www.geocities.com/complementarytheory/ComplexTree.pdf

For more detailed information on this structural/quantitative point of view on numbers, please look at:
http://www.geocities.com/complementarytheory/Complex.pdf

[.000..., 2) > [.000..., 1) > [.000..., 0) (the last one is for nonstandard analysis).

Again, the ", x)" notation stands for an unreachable value (permanent-next state, which is like an unreachable shadow or a mirage effect of the current examined number).

Thank you,

Organic

Again you look on the notations themselves and forget the meaning the they are representing, which in this case representing a structure of a fractal upon infinitely many scales, where its structural property depends on the base value which we choose to use.

But this is about the real numbers, not whatever you choose to make it. Why are you posting this? You haven't said anything with any bearing on the original topic

Hi Matt,

I am talking about R members, but from a richer point of view.

Your point of view is based on the shadow of my point of view, therefore you don't understand that we are talking about the same thing, but from different levels.

So you're saying that in the real numbers, because of this richer structure that you have observed, 0.999... and 1 are distinct (as decimal representations)?

Firstly that is wrong as the definition (description) of the real numbers shows, secondly, as your "rucher structure" is entirely dependent on the decimal (or other base) representation, I don't think It qualifies as richer in the slightest.

Matt,

By structural/quantitative point of view, any given quantity that is not= 0, has more then one structural representation (and not just decimal).

If you ignore the structural property of a given number and look only on its quantity, then this point of view is not more abstract then the structural/quantitative point of view, but more trivial then the structural/quantitative point of view.

We don't have to be happy if there is a way to take two complex systems like, for example, two persons, and then to say that we have two objects.

This is a trivial point of view of these persons, and by my number system we can choose if we want to ignore or not to ignore their internal complexity.

Therefore the result of this kind of view is richer then the standard quantitative-only point of view, which has no choice but to use its trivial point of view and define that .999... = 1

 

[Hurkyl]

Last warning, Organic. The only reason I'm letting this post stay is so that you get the last word in this topic on your theories.