Cont: Deeper than primes - Continuation 2

[jsfisher]

I find this gap as relevant because it is the result of forcing the finite on the infinite

Your claim is unsupported by anything but your claim.

...and I claim that we can't use the finite in order to conclude meaningful notion about 0.999...₁₀ that has |N| places.

This one is unsupported, as well. Mathematics, on the other hand, has a conceptually appealing way of imposing meaning on the infinite, and it's method is well-defined and consistent.

...
So my definition of limit is as follows:

Let X be a placeholder for that is summed.

If |X - L| = 0 then L is called the limit of X.

Until such time as you propose an effective method to compute this "placeholder for that is summed", your definition has no meaning.

Of course, too, if and when you propose an effective method to compute this "placeholder for that is summed", the need for your version of limit will vanish since it becomes possible to compute the infinite summation without involving limits.

 

[doronshadmi]

jsfisher, please reply in details to http://www.internationalskeptics.com/forums/showpost.php?p=10652261&postcount=540.

Thank you.

 

[doronshadmi]

Of course, too, if and when you propose an effective method to compute this "placeholder for that is summed", the need for your version of limit will vanish since it becomes possible to compute the infinite summation without involving limits.

Do not take X as a sequence of partial sums.

 

[jsfisher]

ETA: Discarding the definition for limit of a sequence on the basis it was not written specifically for sequences of partial sums screams of lack of understanding.

You are the one that wrote:

Exactly where does my version use partial sums? For your convenience, here's the definition again. Please just highlight were I make any mention of partial sums whatsoever.

A real-valued sequence (a₁, a₂, a₃, ...) has a limit, L, if and only if for all real numbers ε > 0 there exists a natural number N such that for all n > N, |a_n - L| < ε.​

isn't it?

Yes, and how does that support your bogus claim?

jsfisher, in case that you have missed it, the discussion is about the (standard) definition of convergence.

Doronshadmi, in case you have missed it, the discussions have been about the value of 0.999..., infinite series, sequences of partial sums, limits of sequences, the definitions in Mathematics that assure 0.999... is identical to 1, and the incredulity on your part that Mathematics might just be correct.

The definition for limit of a sequence you have most recently provided is slightly different from that in the main article for the Wikipedia section you pulled it from. Different, but equivalent, so if that's the version you'd prefer, so be it. It is substandard, though, since the norm is to present all the inequalities as strict.

 

[jsfisher]

Do not take X as a sequence of partial sums.

Wouldn't think of it. With that little detail dispensed with, now, what is an effective method to compute this "placeholder for that is summed"?

 

[doronshadmi]

Wouldn't think of it. With that little detail dispensed with, now, what is an effective method to compute this "placeholder for that is summed"?

I correct my previous reply.

Please do not take X as a sequence of partial sums (unless some term of it satisfies |X - L| = 0).

I am going to sleep, so good night for now.

 

[doronshadmi]

Yes, and how does that support your bogus claim?




Doronshadmi, in case you have missed it, the discussions have been about the value of 0.999..., infinite series, sequences of partial sums, limits of sequences, the definitions in Mathematics that assure 0.999... is identical to 1, and the incredulity on your part that Mathematics might just be correct.

The definition for limit of a sequence you have most recently provided is slightly different from that in the main article for the Wikipedia section you pulled it from. Different, but equivalent, so if that's the version you'd prefer, so be it. It is substandard, though, since the norm is to present all the inequalities as strict.

By using the (standard) definition of convergence you can't conclude that 0.999...₁₀ = 1 by using (0.9₁₀, 0.99₁₀, 0.999₁₀) as it trivially can be done in case of, for example, (1,0,0,0,...).

So, |X - L| = 0 is a must have term.

 

[jsfisher]

I correct my previous reply.

Please do not take X as a sequence of partial sums (unless some term of it satisfies |X - L| = 0).

I am going to sleep, so good night for now.

Wouldn't think of it. With that little detail dispensed with, now, what is an effective method to compute this "placeholder for that is summed"?

 

[jsfisher]

By using the (standard) definition of convergence you can't conclude that 0.999...₁₀ = 1 by using (0.9₁₀, 0.99₁₀, 0.999₁₀) as it trivially can be done in case of, for example, (1,0,0,0,...).

Sure it can. 0.999... by definition is the limit of the sequence (0.9, 0.99, 0.999, ...), and the limit of that sequence is easily proven to be 1.

(And before you ask, N = 1 + floor(log₁₀ ε), in terms of the "definition of convergence" you seem to prefer.)

So, |X - L| = 0 is a must have term.

So you have said. All you need is an effective method to compute this "placeholder for that is summed".

 

[doronshadmi]

Sure it can. 0.999... by definition is the limit of the sequence (0.9, 0.99, 0.999, ...), and the limit of that sequence is easily proven to be 1.

(And before you ask, N = 1 + floor(log₁₀ ε), in terms of the "definition of convergence" you seem to prefer.)

The fact that some a_n in some sequence is identical to the limit of that sequence (for example, in the case of sequence (1,0,0,0,...)), can't be extended to the case that no a_n in some sequence is identical to the limit of that sequence, in terms of the definition of convergence.

N = 1 + floor(log₁₀ ε) can't change that fact exactly because it is in terms of the definition of convergence.

So 0.999...₁₀ can't be easily proven to be 1 by the definition of convergence.

The best that the definition of convergence can do is to define (in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...)) that 0.999...₁₀ or 1 are greater than any term in sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), but it can't prove that 0.999...₁₀ is identical to 1 (the result in the case of sequence (1,0,0,0,...) can't be extended to the result in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), in terms of the definition of convergence).

So your "easily proven" has no basis in terms of the definition of convergence.

So you have said. All you need is an effective method to compute this "placeholder for that is summed".

Since your reasoning is completely closed under the notion of results that are derived from forcing the finite on the infinite (exactly as done by the definition of convergence), you simply missing the fact that actual mathematics already has a well-defined effective method to compute the exact value of infinitely many elements, exactly as done by define |N| (which is a transfinite number) as the number of elements in {1,2,3,...}.

I use this well-defined effective method in order to define 0.000...1₁₀ as the complement of number 0.999...₁₀ to number 1, as follows: 0.999...₁₀ + 0.000...1₁₀ = 1.

The reasoning at the basis of the construction of a number of the form 0.000...1₁₀, is derived from the different sizes of transfinite cardinalities, as follows:

0.999...₁₀ is not identical to 1, and the sequence of partial sums, even if it has |N| terms, does not establish -- rigorously -- that 0.999...₁₀ is identical to 1, simply because without using also|R| size, 0.999...₁₀ < 1 by exactly 0.000...1₁₀, where 0.000... is a place value keeper that the number of its places is in bijection with the number of places in 0.999... .

The number of places is at most at |N| size, so in order to actually close the gap between 0.999...₁₀ and 1, ...1₁₀ that is at |R| size is needed to actually do the job.

In other words: 0.999...₁₀ + 0.000...1₁₀ = 1.

More comprehensive view of such reasoning is given in http://www.internationalskeptics.com/forums/showpost.php?p=10652239&postcount=539.

------------------------------

My definition of limit (that is always under check, exactly as any definition has to be (which is a fundamental notion that is not in the scope of scholars like jsfisher)) is as follows:

Let X be a placeholder for that is summed.

If |X - L| = 0 then L is called the limit of X.

Here are two examples of X:

1) The sequence (1,0,0,0,...), such that |a_n - L| = 0.

2) The sum of the addition 0.999...₁₀ + 0.000...1₁₀ (where L=1), such that |(0.999...₁₀ + 0.000...1₁₀) - L| = 0.

 

[jsfisher]

Sure it can. 0.999... by definition is the limit of the sequence (0.9, 0.99, 0.999, ...), and the limit of that sequence is easily proven to be 1.

(And before you ask, N = 1 + floor(log₁₀ ε), in terms of the "definition of convergence" you seem to prefer.)

The fact that some a_n in some sequence is identical to the limit of that sequence (for example, in the case of sequence (1,0,0,0,...)), can't be extended to the case that no a_n in some sequence is identical to the limit of that sequence, in terms of the definition of convergence.

Be that as it may, that is irrelevant to the meaning of limit and of series. You are, again, objecting to things immaterial to the subject at hand.

For all ε, the value for N that I suggested meets the criterion for the limit (using the notation you provided). Were that not the case, you should be able to provide a value for ε and for n for which |a_n - L| fails to be less than ε.

...

So you have said. All you need is an effective method to compute this "placeholder for that is summed".

Since your reasoning is completely closed under the notion of results that are derived from forcing the finite on the infinite (exactly as done by the definition of convergence), you simply missing the fact that actual mathematics already has a well-defined effective method to compute the exact value of infinitely many elements, exactly as done by define |N| (which is a transfinite number) as the number of elements in {1,2,3,...}.

You have told us before this effective method exists, yet you are unable to display the actual method. I still anxiously await the opportunity to try this amazing method on the sequence (1, -1/2, 1/3, -1/4, 1/5, ...)

 

[doronshadmi]

Were that not the case, you should be able to provide a value for ε and for n for which |a_n - L| fails to be less than ε.

Nothing in there can be used in order to prove that, for example, 0.999...₁₀ is identical to 1.

You have told us before this effective method exists, yet you are unable to display the actual method.

I clearly showed its reasoning with some examples

I still anxiously await the opportunity to try this amazing method on the sequence (1, -1/2, 1/3, -1/4, 1/5, ...)

If no one of the terms of this sequence is identical to some number, the this sequence has no unique limit, exactly as seen in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...).

 

[jsfisher]

Nothing in there can be used in order to prove that, for example, 0.999...₁₀ is identical to 1.

Your incredulity notwithstanding, Mathematics has no trouble at all with that proof.

I clearly showed its reasoning with some examples

You have done nothing like that. Instead, you have simple restated that your assumed result is correct.

If no one of the terms of this sequence is identical to some number, the this sequence has no unique limit, exactly as seen in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...).

Once again, Mathematics presses on without regard to your lack of understanding. Limits of sequences continue to exist despite your objections, and infinite series reach finite sums irrespective of your wishes.

Mathematics shows you no respect whatsoever, doronshadmi.

 

[doronshadmi]

Your incredulity notwithstanding, Mathematics has no trouble at all with that proof.

First I wish to correct this part:

If no one of the terms of this sequence is identical to some number, the this sequence has no unique limit, exactly as seen in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...).

The right one is this:

If no one of the terms of this sequence is identical to some number, than this sequence has more than one unique value such that it is greater than all the terms of that sequence AND it is smaller than the limit, exactly as seen in the case of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...).

Let's take, for example, the limit of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...).

In that case L = 1 but 0.999...₁₀ > all the terms of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...) AND < 1 by 0.000...1₁₀.

So the problem of the current traditional mathematician is the lack of understanding of numbers like 0.999...₁₀ or 0.000...1₁₀, exactly because they have dogmatic view of mathematical definitions.

 

[jsfisher]

First I wish to correct this part:

The right one is this:

If no one of the terms of this sequence is identical to some number...

There is no such requirement for limits of sequences. Your corrected statement continues to be wrong.

Let's take, for example, the limit of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...).

In that case L = 1

So, are you finally agreeing that the limit of the sequence (0.9, 0.99, 0.999, ...) is 1? That's what you wrote means; is that what you mean?

...but 0.999...₁₀ > all the terms of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀,...)

Yes, that was never in dispute. 1 (=0.999...) is greater than any element of the sequence (0.9, 0.99, 0.999, ...).

This trivial fact has no bearing on the meaning of limit, nor infinite series, nor the meaning of 0.999... specifically. Why do you continue to wallow in irrelevant details?

 

[doronshadmi]

So, are you finally agreeing that the limit of the sequence (0.9, 0.99, 0.999, ...) is 1? That's what you wrote means; is that what you mean?

1 is considered as the limit of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) not because |a_n - 1| < ε, but because |(0.999...₁₀ + 0.000...1₁₀) - 1| = 0

Again, your definition forces the finite on the infinite in order to get the result, and this attitude prevents numbers like 0.999...₁₀ or 0.000...1₁₀.

My definition does not force the finite on the infinite in order to get the result, and this attitude enables numbers like 0.999...₁₀ or 0.000...1₁₀.

Moreover, by my definition L can also be 0.999...₁₀ (it is not restricted to numbers that can be defined by finitely many symbols) and in this case |0.999...₁₀ - 0.999...₁₀| = 0

In case that you have missed it 0.111...₂ < 0.222...₃ < 0.333...₄ < ... < 0.999...₁₀ < ... < 1, where all of them are numbers that are greater than all the terms of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) and each one of them can be considered as its alternative limit.

It is trivially understood that no positive factional number > 1, but it does not mean that given an infinite sequence of such numbers, 1 is the only possible limit.

Once again, dogmatic approach about definitions (and in this case we are talking about the limit concept as defined by the definition of convergence "for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε") prevent new notions (and in this case, new notions about limit) that can be used in order to develop Math.

 

[jsfisher]

1 is considered as the limit of the infinite sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) not because |a_n - 1| < ε

If you were less sloppy with your notation and bothered to declare the meaning of a_i and include that exists an N part and so on, then, yes, that would be exactly why 1 is the limit (not is considered the limit, but is the limit) of the sequence (0.9, 0.99, 0.999, ...).

...but because |(0.999...₁₀ + 0.000...1₁₀) - 1| = 0

This, on the other hand is just meaninglessness you concocted to cover your incredulity and ignorance.

Again, your definition

It is not my definition. It is the definition. Your unwillingness to accept it disturbs Mathematics not at all.

...forces the finite on the infinite in order to get the result, and this attitude prevents numbers like 0.999...₁₀

Nonsense. 0.999... is a perfectly fine number. It happens to be identical to 1.

...or 0.000...1₁₀.

Nonsense. 0.000...1 is not a number at all, but merely a bit of symbolic gibberish you've invented to cover your ignorance.

My definition does not force the finite on the infinite in order to get the result, and this attitude enables numbers like 0.999...₁₀ or 0.000...1₁₀.

Your definition fails. Until such time as you are able to show how your magic X would be calculated, your definition is stuck in nonsense.


Your misunderstanding of Mathematics provides you not leverage to declare it wrong or limiting. On the contrary, Mathematics is wide open to new realms under alternate foundations. There is a set theory without the axiom of infinity; that does not, however, invalidate ZFC.

As I have invited you to do many times in the past, feel free to build whatever mathematical world you like upon whatever foundation you like, but don't even pretend you've disproved the rest. And given your demonstrated inability to define anything, don't be too surprised if you fail trying to articulate your foundation in any meaningful way.

 

[doronshadmi]

If you were less sloppy with your notation and bothered to declare the meaning of a_i and include that exists an N part and so on, then, yes, that would be exactly why 1 is the limit (not is considered the limit, but is the limit) of the sequence (0.9, 0.99, 0.999, ...).

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε

So, by this definition you can't conclude that 0.999...₁₀ is identical to 1, or that 1 is the only possible limit of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), as very simply shown in http://www.internationalskeptics.com/forums/showpost.php?p=10653953&postcount=556.

It is not my definition. It is the definition. Your unwillingness to accept it disturbs Mathematics not at all.

This is a concrete example of your dogmatic approach about mathematical definitions.

Nonsense. 0.999... is a perfectly fine number. It happens to be identical to 1.

0.999... is undefined unless some base > 9 is provided.

Moreover is the number is, for example, 0.999...₁₀, then what is called the definition of convergence does not prove that 0.999...₁₀ is identical to 1 exactly because it is based on the wrong notion of forcing the finite on the infinite (as seen in http://www.internationalskeptics.com/forums/showpost.php?p=10652939&postcount=550).

Nonsense. 0.000...1 is not a number at all, but merely a bit of symbolic gibberish you've invented to cover your ignorance.

0.000...1 is undefined unless some base > 1 is used.

Your definition fails. Until such time as you are able to show how your magic X would be calculated, your definition is stuck in nonsense.

I do not have to show any magic, all is needed is to use a test that shows that it diverges, converges, or between them, and then, if it is converges, then it does have more than one alternative as its limit, exactly because on one of its terms is identical to that limit (as can be seen, for example, in sequence (1,0,0,0,...), where one of its terms is identical to its limit).

Your misunderstanding of Mathematics provides you not leverage to declare it wrong or limiting.

Mathematics is not a privet property of any authority, but your dogmatic approach about it does not allow you to be aware of it.

As I have invited you to do many times in the past, feel free to build whatever mathematical world you like upon whatever foundation you like, but don't even pretend you've disproved the rest. And given your demonstrated inability to define anything, don't be too surprised if you fail trying to articulate your foundation in any meaningful way.

Your dogmatic approach about definitions, and your illusion that Mathematics is a privet property of some authority, clearly seen in this quote of yours.

 

[jsfisher]

for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε

So, by this definition

...of the limit L of the sequence (a₁, a₂, a₃, ...)...

you can't conclude that 0.999...₁₀ is identical to 1

Well, not without a few additional details--those pesky things that cause you so much angst--but once all pieced together, sure 0.999... is identical to 1.

...or that 1 is the only possible limit of sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...)

No, that's not any more difficult. Your prior attempts to show otherwise fell victim to your lack of attention to details coupled with a few irrelevancies you added for no good reason.

 

[Little 10 Toes]

I always like how Doronshadmi likes to drag in other number systems.

0.999... is undefined unless some base > 9 is provided.

and

0.000...1 is undefined unless some base > 1 is used.

Really? There's a base 1 system?