Cont: Deeper than primes - Continuation 2

[doronshadmi]

Look at the definition for 'limit' as the domain variable tends to "infinity". No where does "infinity" of any form find its way into the definition, and no where does "infinity" of any form find its way into the proof.

Dear jsfisher are there finitely or infinitely many elements in sequence (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) that all of them are used in order to construct 0.9₁₀+0.09₁₀+0.009₁₀+ ... = 0.999...₁₀ (such that 0.999...₁₀ is not less than the sum over all (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) elements) ?

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Once again:

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

So |a_n - L| must be = 0 in order to conclude that the sum over a_n = L.

It must be stressed that if infinity is not involved here, then |a_n - L| > 0 and we can't conclude that the sum over a_n = L.

For example, where is the rigorous proof that the sum over all (0.9₁₀, 0.09₁₀, 0.009₁₀,...) = L = 1?

(My solution is given in http://www.internationalskeptics.com/forums/showpost.php?p=10617467&postcount=316).

 

[jsfisher]

Dear jsfisher are there finitely or infinitely many elements in sequence (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) that all of them are used in order to construct 0.9₁₀+0.09₁₀+0.009₁₀+ ... = 0.999...₁₀ (such that 0.999...₁₀ is not less than the sum over all (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) elements) ?

Nowhere does that enter into the proof. Why to you harp on the irrelevant?

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

See? By your own words, no infinity anywhere to be found.

So |a_n - L| must be = 0 in order to conclude that the sum over a_n = L.

No, this does not follow at all. The requirement is |a_n - L| < ε where ε > 0. It is perfectly acceptable for 0 < |a_n - L|.

It must be stressed that if infinity is not involved here, then |a_n - L| > 0 and we can't conclude that the sum over a_n = L.

In fact, we can conclude that. It is a matter of definitions for those things you don't understand: sequences and limits.

 

[doronshadmi]

It is perfectly acceptable for 0 < |a_n - L|.

In that case it is perfectly acceptable that 0.999...₁₀ < 1 by |a_n - L| > 0, or in other words

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

is not a rigorous proof that, for example, 0.999...₁₀ = 1.

Moreover, you actually ignore this (http://en.wikipedia.org/wiki/Sequence#Finite_and_infinite):

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence, when disambiguation is necessary.

Furthermore, since 0.999...₁₀ is the sum over one-sided (infinite) sequence like (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) where the number of its terms = |N|, 0.999...₁₀ alone can't be equal to 1 and the solution is given in http://www.internationalskeptics.com/forums/showpost.php?p=10617467&postcount=316.

 

[jsfisher]

It is perfectly acceptable for 0 < |a_n - L|.

In that case it is perfectly acceptable that 0.999...₁₀ < 1 by |a_n - L| > 0, or in other words

The two are unrelated. My observation is supported by the very definition of limit. Yours is not supported by anything.

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

is not a rigorous proof that, for example, 0.999...₁₀ = 1.

Of course not. The "for all..." is merely a part of the formal definition of limit. However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1.

Moreover, you actually ignore this (http://en.wikipedia.org/wiki/Sequence#Finite_and_infinite):

Normally, the term infinite sequence refers to a sequence which is infinite in one direction, and finite in the other—the sequence has a first element, but no final element, it is called a singly infinite, or one-sided (infinite) sequence, when disambiguation is necessary.

I did not ignore it at all.

Furthermore, since 0.999...₁₀ is the sum over one-sided (infinite) sequence like (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) where the number of its terms = |N|, 0.999...₁₀ alone can't be equal to 1 and the solution is given in http://www.internationalskeptics.com/forums/showpost.php?p=10617467&postcount=316.

And just how is "the sum over one-sided [sic] (infinite) sequence" determined? By definition, it is the limit of the sequence of partial sums.

Determining the limit of the sequence of partial sums does not involve infinity at any step.

 

[doronshadmi]

Determining the limit of the sequence of partial sums does not involve infinity at any step.

And how many partial sums there are in (s₁, s₂, s₃, ...) sequence, finitely many or infinitely many?

 

[doronshadmi]

And just how is "the sum over one-sided [sic] (infinite) sequence" determined?

Exactly as given in http://www.internationalskeptics.com/forums/showpost.php?p=10617467&postcount=316.

 

[doronshadmi]

My observation is supported by the very definition of limit.

The very definition of limit, if it is limited to finite sequences, does not support the assertion that the sum over all terms of sequence (0.9₁₀, 0.09₁₀, 0.009₁₀) actually = 1.

Moreover, the sum of |N| terms of that sequence < 1 , and we need a gap closer at |R| size in order to actually define 1 as the limit of 0.999...₁₀.

The "for all..." is merely a part of the formal definition of limit. However, that definition comes into play in showing the limit of a particular sequence of partial sums is unity, and that establishes the value for a particular series, and that, in turn, establishes -- rigorously -- that 0.999... is identical to 1.

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 generally, the sum over a_n where a_n has at most |N| terms < L by sum at |R| size.

 

[doronshadmi]

Let's clarify it by a given example that can be used without loss of generality:

The sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, 0.001₁₀) = 1

The sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, 0.0009₁₀) < 1

Please pay attention that in this finite example the sum = 1 if we are using the smallest value 0.001₁₀ that is at higher level than value 0.0009₁₀.

In the case of the sum over infinity many values like (0.9₁₀, 0.09₁₀, 0.009₁₀, 0.0009₁₀, ...) the smallest value > 0 (notated as 0.000...1₁₀) is determined as the sum at |R| size (notates here as ...1₁₀), which is at higher level than the sum at |N| size (notate here as 0.000..., and 0.000...1₁₀ is taken as a unique value exactly as 0.001₁₀ is taken as a unique value.

 

[doronshadmi]

According to the example above, one can understand that a sum over sequence with |R| terms (which can't be symbolized since it is uncountable) needs ...1 at |P(R)| size in order to be identical to 1, ... etc. ad infinitum, so number systems (whether they are countable or uncountable) are determined by different levels of resolutions, where the higher resolution is taken as one unique value (notated as ...1) from the point of view of the lower resolution, and it closes the gap between the limit value and the sum of the lower resolution.

On top of all this there is the non-composed highest level that is beyond all resolutions, and it closes the gap to 1 for all resolutions.

In other words, the non-composed highest level is independent of any point of view, in order to be used to close the gap between the sum of a given resolution, and 1.

So, that is beyond transfinite cardinality is indeed actual infinity, and it is invariant and objective to all levels of collections, that each one of them or all of them together are not invariant and objective as the non-composed level is.

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If we use the point of view of Physics on this case, then Newtonian physics is useful even if it is not accurate as GRT, or in other words, by using this analogy, the relative different points of view of different resolutions are still useful in order to close the gap between some sum at smaller transfinite cardinality to some limit value, by using a given sum of bigger transfinite cardinality (which is equivalent here to Newtonian physics).

Yet the total solution is provided only by using the non-composed highest level that is beyond all resolutions (which is equivalent here to GRT, in this analogy).

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Definitions like "for all ε > 0, there exists a natural number N in N such that for all n ≥ N, |a_n - L| < ε", simply block the ability to understand the relative view of different levels of cardinality (or the concept of relative levels, in general), and the absolute view of the non-composed level beyond all transfinite cardinalities, and how the relative or the absolute views are used in order to solve the limit problem.

 

[jsfisher]

Let's clarify it by a given example that can be used without loss of generality...

Let's not. You continue to show complete disregard for and lack of understanding of how things are defined. Despite your objections, the series for an infinite sequence is, by definition the limit of the sequence of partial sums. Determination of that limit does not at any point involve infinity, in any of its possible forms.

If you are unwilling to accept and comply with fundamental mathematical definition, then you are not doing Mathematics.

You, doronshadmi, are not doing Mathematics.

 

[doronshadmi]

You, doronshadmi, are not doing Mathematics.

You, jsfisher, put arbitrary restrictions to mathematical development, and use them in a dogmatic way.

If you are unwilling to accept and comply with fundamental mathematical definition, then you are not doing Mathematics.

If you are unwilling to re-examine fundamental mathematical definitions, then you are not doing Mathematics.

Despite your objections, the series for an infinite sequence is, by definition the limit of the sequence of partial sums. Determination of that limit does not at any point involve infinity, in any of its possible forms.

This time please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10621174&postcount=345.

 

[jsfisher]

You, jsfisher, put arbitrary restrictions to mathematical development, and use them in a dogmatic way.

As you have been invited many times in the past, if you want to explore new areas of Mathematics, do so. However, you are not allowed to redefine established terminology and declare prior results wrong as a consequence.

This time please reply to http://www.internationalskeptics.com/forums/showpost.php?p=10621174&postcount=345.

It is an irrelevant question and has no bearing on the valuation of a series.

 

[doronshadmi]

However, you are not allowed to redefine established terminology and declare prior results wrong as a consequence.

I can re-examine or re-define your so called "established terminology", and this is a perfectly valid way of mathematical development.

The difference between you and me is that since you are now a moderator, you can shut me up because I do not follow after your so called "established terminology".

But before you going to shut me up, be aware of the fact that this thread is a part of "Religion and Philosophy" forum, which somehow you became one of it moderators.

 

[doronshadmi]

It is an irrelevant question and has no bearing on the valuation of a series.

It is relevant, but you choose to ignore this question because it does not fit to your so called "established terminology".

Again, if you are unwilling to re-examine fundamental mathematical definitions, then you are not doing Mathematics.

 

[jsfisher]

Again, if you are unwilling to re-examine fundamental mathematical definitions, then you are not doing Mathematics.

You are not examining them; to are totally disregarding them and substituting your own.

 

[doronshadmi]

You are not examining them; to are totally disregarding them and substituting your own.

I do not follow after your arbitrary restriction of finitism on limits, which is no more than your philosophical approach about this subject, so?

 

[jsfisher]

I do not follow after your arbitrary restriction of finitism on limits, which is no more than your philosophical approach about this subject, so?

Knock yourself out: Define 'doron-limit' any way you like, but leave 'limit' alone.

So, how would you like to define limit? Be specific.

 

[doronshadmi]

Be specific.

I am very specific in:

http://www.internationalskeptics.com/forums/showpost.php?p=10621388&postcount=347

http://www.internationalskeptics.com/forums/showpost.php?p=10621469&postcount=348

http://www.internationalskeptics.com/forums/showpost.php?p=10621635&postcount=349

where limit is not restricted only to finitism, and relative or absolute observations are used.

In all cases (depending on relative or absolute observations) a sum over a given sequence reaches the limit (and not just tends to, as defined by your finite-only observation).

 

[jsfisher]

I am very specific in:

http://www.internationalskeptics.com/forums/showpost.php?p=10621388&postcount=347

http://www.internationalskeptics.com/forums/showpost.php?p=10621469&postcount=348

http://www.internationalskeptics.com/forums/showpost.php?p=10621635&postcount=349

where doron-limit is not restricted only to finitism, and relative or absolute observations are used..

[Highlighted text added.]
Great. So now all that is left for you to provide is an actual definition.

 

[Little 10 Toes]

Please pay attention that in this finite example the sum = 1 if we are using the smallest value 0.001₁₀ that is at higher level than value 0.0009₁₀.

define Doron-level please.