Cont: Deeper than primes - Continuation 2

[doronshadmi]

Edit 2: what number do you get when you divide 1 into 3 equal parts?

1/3 (which is > 0.333...₁₀ by 0.000...1₁₀, if the relative observation between |N| and |R| is not ignored (and it is definitely ignored in case that 0.333...₁₀ = 1/3)).

 

[doronshadmi]

Let me know when you are done editing your messages.

Done.

 

[jsfisher]

The fallacy of 0 < |a_n - L| AND 0.999...₁₀ = 1 is far from being precise.

Let me know when you stop trying to disprove definitions.

 

[doronshadmi]

Let me know when you stop trying to disprove definitions.

You actually do not wish to know anything that does not fit to your philosophical invariant view of definitions, that stays invariant even if it is logically false, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=10630295&postcount=408.

 

[jsfisher]

You actually do not wish to know anything that does not fit to your philosophical invariant view of definitions

Definitely not true. I would be delighted to see what you have in mind as an alternate to the definition of infinite series and of limit. You refuse to provide that. Instead, you complain that the established definitions are wrong; definitions cannot be wrong.

...that stays invariant even if it is logically false, as clearly seen in http://www.internationalskeptics.com/forums/showpost.php?p=10630295&postcount=408.

Your posts are seldom clear, and they are never worth revisiting, even after your extensive editing of them.


Tell us, Doronshadmi, how would you define 'infinite series'? If it has any reliance on the concept of limit, then how would you define 'limit'?

 

[doronshadmi]

Definitely not true ...
definitions cannot be wrong.

jsfisher's logical reasoning is (Definitely not true (which means:"definitions are not invariant")) AND (definitions cannot be wrong).

Tell us, Doronshadmi, how would you define 'infinite series'?

The absolute value of the sum of at least infinitely many terms.

 

[Little 10 Toes]

Tell us, Doronshadmi, how would you define 'infinite series'?

The absolute value of the sum of at least infinitely many terms.

So an infinite series is a sum. Sheeze. What's the value of {o,t,t,f,f,s,s,e,n,t,e,t,t,f,f,s,s,e,n,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,t,f,f,f,f,f,f,f,f,f,f,f ...}?

If you haven't figured it out, the series is the first letter of each positive number written out as a word.

By the way, what is the decimal representation of 1 divided by 3? When you multiply the result by three, how come it does not equal 1?

 

[doronshadmi]

If you haven't figured it out, the series is the first letter of each positive number written out as a word.

Sum of elements and series is the same thing.

So the series of {o,t,t,f,f,s,s,e,n,t,e,t,t,f,f,s,s,e,n,t,t,t,t,t,t ,t,t,t,t,t,t,t,t,t,t,t,t,t,t,f,f,f,f,f,f,f,f,f,f,f ...} elements is the sum of at most countable |N| elements.

By the way, what is the decimal representation of 1 divided by 3?

1/3 does not have decimal representation since 0.333...₁₀ is a number of its own < 1/3 exactly by 0.000...1/3₁₀, and the needed details are found in http://www.internationalskeptics.com/forums/showpost.php?p=10630361&postcount=416. Yet you do not grasp them, and repeat again and again on your representation argument.

 

[doronshadmi]

Tell us,

Since when you became the representative of mathematics in this forum?

 

[jsfisher]

Tell us, Doronshadmi, how would you define 'infinite series'?

The absolute value of the sum of at least infinitely many terms.

Absolute value? Seriously? Well, that nonsense aside, how, exactly does one find the value of "the sum of at least infinitely many terms"?

We all know how to add two values, and that is easily extended to any finite number of values. How do you make the leap to an infinite number?

 

[doronshadmi]

Absolute value? Seriously? Well, that nonsense aside, how, exactly does one find the value of "the sum of at least infinitely many terms"?

By add at least |N| absolute values of them in one step, where in the case of, for example, the sum over |N| terms of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) < 1 by 0.000...1₁₀, where this number is explained in http://www.internationalskeptics.com/forums/showpost.php?p=10630295&postcount=408 and more comprehensive explanation is given in http://www.internationalskeptics.com/forums/showpost.php?p=10622312&postcount=358.

How do you make the leap to an infinite number?

How mathematics makes the leap from finite cardinality to some transfinite cardinality, and how there are leaps (defined as "strictly greater than ...") among transfinite cardinals?

Mathematics has no problem with such leaps, so what is your point?

 

[jsfisher]

By add at least |N| absolute values of them in one step

Absolute values, still, eh? Two blunders, and you aren't even through the first sentence.

How, exactly do you propose we "add at least |N| absolute values of them in one step"? As I wrote before, adding two values is understood, and adding any finite number of values is a simple extension of that. I.e.,

(a₁ + a₂ + ... + a_n + a_n+1) = (a₁ + a₂ + ... + a_n) + a_n+1​

Adding an infinite number of values is not a simple extension of that. In fact, that is one of the purposes of limit, to give meaning to infinite series.

 

[doronshadmi]

Absolute values, still, eh?]

Yes, for example, the |N| terms of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...)

How, exactly do you propose we "add at least |N| absolute values of them in one step"?

Since time is not involved (or any kind of process), it is done in one step.

Adding an infinite number of values is not a simple extension of that. In fact, that is one of the purposes of limit, to give meaning to infinite series.

Infinite series does not need limit in order to get its meaning.

All is needed is the relative knowledge of transfinite cardinality, or the absolute knowledge of actual infinity (which is strictly greater than all transfinite cardinalities) such that |the sum of infinite sequence - L| = 0.

It is already explained in http://www.internationalskeptics.com/forums/showpost.php?p=10633086&postcount=431.

0 < |a_n - L| AND (for example) 0.999...₁₀ = 1, is a logical fallacy by both following cases:

1) If a_n is a finite series, it can't be used in order to conclude anything about the series of an infinite sequence with |N| terms like (0.9₁₀, 0.09₁₀, 0.009₁₀, ...).

2) If a_n is an infinite series such that 0 < |a_n - L|, then this series is definitely < 1 by |a_n - L|.

For example: no amount of finitely many partial sums like, for example, (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) can conclude anything about the series of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...), and you, jsfisher, use finite sums in order to conclude something about an infinite sum, so you are the one that actually use this arbitrary and artificial extension from the finite to the infinite.

 

[jsfisher]

Since time is not involved (or any kind of process), it is done in one step.

Yes, you have said that before. The question is how. Without an effective method for this, then anything you say about 0.9 + 0.09 + 0.009 + ... is simply an assertion on your part.

Mathematics gives meaning to infinite series by defining it to be the limit of the sequence of partial sums. It is a convenient, conceptually sound, and consistent definition. You define (by virtue of a complete lack of any such definition) infinite (doron-)series as whatever value you say it is.

Mathematics by fiat isn't how its done.

 

[doronshadmi]

Mathematics gives meaning to infinite series by defining it to be the limit of the sequence of partial sums.

Again, no amount of finitely many partial sums like, for example, (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) can be used in order to conclude anything about the series of (0.9₁₀, 0.09₁₀, 0.009₁₀, ...), and you, jsfisher, use finite sums like (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) terms in order to conclude something about an infinite sum over (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), so you are the one that actually use an arbitrary and artificial extension from the finite to the infinite.

 

[doronshadmi]

Without an effective method for this, then anything you say about 0.9 + 0.09 + 0.009 + ... is simply an assertion on your part.

I use an effective method for this, as clearly seen in:

http://www.internationalskeptics.com/forums/showpost.php?p=10625610&postcount=379

http://www.internationalskeptics.com/forums/showpost.php?p=10633086&postcount=431

http://www.internationalskeptics.com/forums/showpost.php?p=10622312&postcount=358

 

[doronshadmi]

Yes, you have said that before. The question is how.

Exactly by the same reasoning that defines |N| (which is a transfinite number) as the number of elements of the infinite set {1,2,3, ...}, known as the set of all natural numbers.

|N| value is defined by one step exactly because {1,2,3, ...} is taken as a whole.

 

[doronshadmi]

Some correction of http://www.internationalskeptics.com/forums/showpost.php?p=10637029&postcount=435.

Instead of

... and you, jsfisher, use finite sums like (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) terms in order to conclude something about an infinite sum over (0.9₁₀, 0.99₁₀, 0.999₁₀, ...), so you are the one that actually use an arbitrary and artificial extension from the finite to the infinite.

it has to be

... and you, jsfisher, use finite sums like (0.9₁₀, 0.99₁₀, 0.999₁₀, ...) terms in order to conclude something about an infinite sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, ...), so you are the one that actually use an arbitrary and artificial extension from the finite to the infinite.

 

[doronshadmi]

In addition to http://www.internationalskeptics.com/forums/showpost.php?p=10630394&postcount=421 I wish to stress that, for example:

0.1₃ = 1/3, where 0.333...₁₀ < 1/3 by 0.000...1/3₁₀, so 0.333...₁₀ < 0.1₃ by |0.333...₁₀ - 0.1₃| = 0.000...1/3₁₀ > 0.

 

[Little 10 Toes]

In addition to http://www.internationalskeptics.com/forums/showpost.php?p=10630394&postcount=421 I wish to stress that, for example:

0.1₃ = 1/3, where 0.333...₁₀ < 1/3 by 0.000...1/3₁₀, so 0.333...₁₀ < 0.1₃ by |0.333...₁₀ - 0.1₃| = 0.000...1/3₁₀ > 0.

Sheeze. Again.

What is the decimal representation of one divided by three? Please write down what the decimal representation of .01/3?

Edit: And why even bring up base 3 numbers? Why not use base 17?