Cont: Deeper than primes - Continuation 2

[doronshadmi]

You'll be able to quote the specific part where you gave that definition, then?

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.

 

[doronshadmi]

Actually, it was assessing your capabilities,

Assessing my capabilities can't be done by beliefs, but since you have a dogmatic approach about the current discussed subject, you are not aware of this simple fact.

 

[doronshadmi]

as x goes to infinty.

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.

Now, please define "x goes to infinity".

 

[jsfisher]

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.

How does that in any way relate to L = limit(x -> infinity) F(x)?

How would you apply that to the limit of:

F(x) = 1 + 1/x², when x is not an integer, and
F(x) = 1 + 1/x, otherwise​

as x goes to infinty.

 

[doronshadmi]

How does that in any way relate to L = limit(x -> infinity) F(x)?

It does not related to L = limit(x -> infinity) F(x) since infinity is not actually involved in your case, and as a result you force the finite on the infinite, which is no more than hands waving.

If you disagree with me then please define the exact meaning of "x goes to infinity" (also notated as "x -> infinity").

EDIT:

Also please answer to the following question:

What is the limit of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) according to the definition of convergence?

 

[jsfisher]

It does not related to L = limit(x -> infinity) F(x) since infinity is not actually involved in your case, and as a result you force the finite on the infinite, which is no more than hands waving.

So, you have no definition for limit(x -> infinity) F(x). That leaves a big gap. Sequence convergence uses such a limit in its definition, but you don't have one.

What, then, is your definition for sequence convergence?

 

[doronshadmi]

So, you have no definition for limit(x -> infinity) F(x). That leaves a big gap.

You have missed it, your definition leaves a big gap exactly because x does not actually reaches infinity.

On the contrary, in my definition x actually reaches infinity (in case that infinity is involved) and therefore there is no gap.

You simply ignore http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 and the contents of all of the chain of links within it, which very clearly define, explain and provide concrete examples of my reasoning about the discussed subject.

Yours' not so much, for example, you still did not support your claim that, for example, 0.999₁₀... = 1 by using your reasoning.

This time please answer to the following question:

What is the limit of the sequence (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) according to the definition of convergence?

In case that you are going to avoid the answer it will be clear to any poster here that you can't support you claims in http://www.internationalskeptics.com/forums/showpost.php?p=10669440&postcount=625 about this sequence.

 

[jsfisher]

You have missed it, your definition leaves a big gap exactly because x does not actually reaches infinity.

Not only does it not actually reach infinity, it cannot. That's the whole point. It is not meaningful to ask what is a function's value at infinity (of any flavor) when the function has the reals as its domain. It is, however, perfectly meaningful to ask how does a function behave for ever-increasing values.

That's what limit(x -> infinity) F(x) deals with.

F(x) could be a function over the positive integers, and then it is equivalent to a sequence. The sequence converges if and only if F(x) has a limit as x -> infinity.

No gaps.

On the contrary, in my definition x actually reaches infinity (in case that infinity is involved) and therefore there is no gap.

Your "definition" includes no x and no mention of infinity. It has a capital X and an underscored capital X. The latter is described as "placeholder for that is summed," whatever that was meant to mean, and the former is left undefined. With so much undefined, your "definition" has no applicability.

So, whatever your "definition" does, it has nothing to do with the limit of some function as it's parameter approaches infinity. You need one, though, because the definition of convergent sequence requires it. Either that, or you will need your own private definition for convergent sequence.

 

[doronshadmi]

Not only does it not actually reach infinity, it cannot. That's the whole point. It is not meaningful to ask what is a function's value at infinity (of any flavor) when the function has the reals as its domain. It is, however, perfectly meaningful to ask how does a function behave for ever-increasing values.

That's what limit(x -> infinity) F(x) deals with.

F(x) could be a function over the positive integers, and then it is equivalent to a sequence. The sequence converges if and only if F(x) has a limit as x -> infinity.

No gaps.

That's the whole point. There are gaps, exactly because it cannot reach infinity. A given infinite sequence has a limit exactly because by my definition |N| and |R| are involved.

Your "definition" includes no x and no mention of infinity. It has a capital X and an underscored capital X. The latter is described as "placeholder for that is summed," whatever that was meant to mean, and the former is left undefined. With so much undefined, your "definition" has no applicability.

It is clear that you simply do your best in order to not understand http://www.internationalskeptics.com/forums/showpost.php?p=10666379&postcount=612 and the contents of all of the chain of links within it, which very clearly define, explain and provide concrete examples of my reasoning about the discussed subject.

So, whatever your "definition" does, it has nothing to do with the limit of some function as it's parameter approaches infinity. You need one, though, because the definition of convergent sequence requires it. Either that, or you will need your own private definition for convergent sequence.

Since my definition enables to use countably infinite cardinality and uncountably infinite cardinality, it is not restricted to the reasoning of the definition of convergence (for example, http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587).

Once again, the definition of convergence is some particular case (with a loss of generality) of my definition of limits.

 

[jsfisher]

That's the whole point. There are gaps, exactly because it cannot reach infinity. A given infinite sequence has a limit exactly because by my definition |N| and |R| are involved.

No. A infinite sequence can have a limit exactly because 'limit' is a well-defined construct in Mathematics.

So far, you have not produced a substitute definition for the limit of an infinite sequence. (That would be of the form L = limit(x -> infinity) F(x), where the function domain would be the positive integers.)

 

[doronshadmi]

No. A infinite sequence can have a limit exactly because 'limit' is a well-defined construct in Mathematics.

An infinite sequence can have more than one limit, as very simply explained in http://www.internationalskeptics.com/forums/showpost.php?p=10653953&postcount=556 (where this explanation is based on my definition of limits, that enables the use of countably infinite cardinality and uncountably infinite cardinality).

What you call 'limit' is based on the trivial notion (without loss of generality) that no positive fractional number > number 1.

So far, you have not produced a substitute definition for the limit of an infinite sequence.

So far your dogmatism about the definition of convergence (and about definitions, in general) prevents any notion of limits that is not based on the trivial notion mentioned above.

 

[jsfisher]

No. A infinite sequence can have a limit exactly because 'limit' is a well-defined construct in Mathematics.

An infinite sequence can have more than one limit

That's not how limits work in Mathematics.

...as very simply explained in http://www.internationalskeptics.com/forums/showpost.php?p=10653953&postcount=556 (where this explanation is based on my definition of limits, that enables the use of countably infinite cardinality and uncountably infinite cardinality).

We are still anxiously awaiting a serviceable version of your definition. It won't change how Mathematics deals with limits in the main, but it could be an interesting notion for a side-branch. So far, though, a vague reference to "placeholder that is summed" doesn't get you to a serviceable definition.

What you call 'limit' is based on the trivial notion (without loss of generality) that no positive fractional number > number 1.

Positive fractional number? Do you mean a number between 0 and 1? If so, then, wouldn't it be unreasonable to expect a number between 0 and 1 to be greater than 1?

That aside, though, no where is the Mathematical notion of limit based what you claim.

So far, you have not produced a substitute definition for the limit of an infinite sequence.

So far your dogmatism about the definition of convergence (and about definitions, in general) prevents any notion of limits that is not based on the trivial notion mentioned above.

You are evading instead of addressing the criticism. You have provided no serviceable definition for your notion of limit. Had you, then we could have applied your definition to the examples I've provided.

 

[doronshadmi]

That's not how limits work in Mathematics.

That's not how limits work in Mathematics that does not use countably infinite cardinality and uncountably infinite cardinality (as done in my system).

We are still anxiously awaiting a serviceable version of your definition.

Your "We" is a concrete example of how Mathematics becomes some kind of religion that has fixed dogmas. So no, these "We" are not anxiously awaiting a serviceable version of my definition, unless it fits to "We" limit's dogma.

Positive fractional number? Do you mean a number between 0 and 1? If so, then, wouldn't it be unreasonable to expect a number between 0 and 1 to be greater than 1?

That aside, though, no where is the Mathematical notion of limit based what you claim.

"We" are not aware of their dogmatic approach about Mathematics.

You are evading instead of addressing the criticism. You have provided no serviceable definition for your notion of limit. Had you, then we could have applied your definition to the examples I've provided.

I have provided serviceable definition for my notion of limits, including examples and detailed explanations, where one of its cases is the trivial case that is based on the trivial notion (without loss of generality) that no positive fractional number > number 1.

Since my definition enables to use countably infinite cardinality and uncountably infinite cardinality, it is not restricted to the reasoning of the definition of convergence (for example, http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587).

Once again, the definition of convergence is some particular case (with a loss of generality) of my definition of limits, but "We" are not aware of it, because the definition of convergence is one of "We" dogmas.

 

[realpaladin]

Good! I see I can whack your ears again with limits and how they work! Fun!

 

[Little 10 Toes]

I'm still here Doronshadmi. I just find it funny that you still define limit. "that which is summed". Freakin' funny!

 

[realpaladin]

That's not how limits work in Mathematics that does not use countably infinite cardinality and uncountably infinite cardinality (as done in my system).

Is that a convoluted way of saying "Discrete and continuous mathematics"?

 

[doronshadmi]

Here is a quote taken from http://en.wikipedia.org/wiki/Georg_Cantor#Set_theory :

The beginning of set theory as a branch of mathematics is often marked by the publication of Cantor's 1874 article,[31] "Über eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen" ("On a Property of the Collection of All Real Algebraic Numbers").[40] This article was the first to provide a rigorous proof that there was more than one kind of infinity. Previously, all infinite collections had been implicitly assumed to be equinumerous (that is, of "the same size" or having the same number of elements).[41] Cantor proved that the collection of real numbers and the collection of positive integers are not equinumerous. In other words, the real numbers are not countable.

Here is a quote taken from http://en.wikipedia.org/wiki/Karl_Weierstrass#Soundness_of_calculus :

Weierstrass was interested in the soundness of calculus, and at the time, there were somewhat ambiguous definitions regarding the foundations of calculus, and hence important theorems could not be proven with sufficient rigour. While Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many had only vague definitions of limits and continuity of functions.

Delta-epsilon proofs are first found in the works of Cauchy in the 1820s.[3][4] Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement interpreted as being incorrect by many scholars. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus.

So when Weierstrass formalized the definition of limits (around 1838) as used today, he did not know that there was more than one kind of infinity (first formalized by Cantor only in 1874).

My work (please look at http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587 including all its chain of links) uses Cnator's work on different kinds of infinities in order to define numbers like 0.999...₁₀ or 0.000...1₁₀, such that 1 - 0.999...₁₀ = 0.000...1₁₀

 

[jsfisher]

So when Weierstrass formalized the definition of limits (around 1838) as used today, he did not know that there was more than one kind of infinity (first formalized by Cantor only in 1874).

Nor did he need to. Infinity, as used in limits, is not as a cardinal number. Limits answer a question about the behavior of functions for ever-larger arguments. The "as x goes to infinity" is figurative in this regard.

The cardinality of the domain set is immaterial.

As an aside, too, were one to extend basic arithmetic to include "infinity" as a value, the infinity so introduced would be the same for arithmetic over the real numbers as it would be over the integers.

Again, the cardinality of the domain set is immaterial.

 

[doronshadmi]

Limits answer a question about the behavior of functions for ever-larger arguments.

If you speck about the behavior of some function such that it tends from 1 to 0 trough 1/2, 1/3, 1/4, 1/5, ..., then also in this case |a_n - L| must be > 0, exactly as explained in http://www.internationalskeptics.com/forums/showpost.php?p=10649134&postcount=501.

So your definition is limited to trivial cases like (0,0,0,...), (1,1,1,...), (0.34₁₀, 0.34₁₀, 0.34₁₀,...), etc.

I have no idea what you are going on about now, but it doesn't follow anything under discussion in any obvious way.

Be that as it may, (1, 0, 1/2, 0, 1/3, 0, 1/4, 0, 1/5, ...) is a sequence; the sequence has a limit; and half the elements of the sequence have the same value as the limit.

Another, even more trivial example would be (1, 1, 1, 1, 1, ...).

Zigzaging, jsfisher, is immaterial to the discussed subject.

What is essential to the discussed subject is the behavior of functions, and my definition extends the behavior by using infinite countable and uncountable cardinal numbers, exactly as defined and explained in details in http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587 including all its chain of links.

Limits answer a question about the behavior of functions for ever-larger arguments. The "as x goes to infinity" is figurative in this regard.

You simply force the finite on the infinite, and "x goes to infinity" is the result of such forcing.

The behavior of functions in my system is not based on forcing the finite on the infinite, exactly because I use infinity as defined by Cantor's transfinite cardinality.

 

[doronshadmi]

the infinity so introduced would be the same for arithmetic over the real numbers as it would be over the integers

In other words your system does not distinguish between the countably infinite and the uncountably infinite.

Please define such infinity by your Traditional Mathematics.

EDIT:

Also be aware that I logically have found such infinity, which is beyond the notion of collections (known also as absolute or actual infinity, where the relativity among infinite collections is the result of the reflection principle of absolute infinitely among transfinite cardinality (please see http://en.wikipedia.org/wiki/Absolute_Infinite) such that larger transfinite cardinality is observed as a single element from smaller transfinite cardinality , as briefly explained in http://www.internationalskeptics.com/forums/showpost.php?p=10652239&postcount=539.