Cont: Deeper than primes - Continuation 2

[doronshadmi]

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

What I wrote is much more rigorous than your fantasy of "the series for an infinite sequence [that] is, by definition the limit of the sequence of partial sums".

Your definition is a fantasy (relatively or absolutely) exactly because

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

so your call for definition does not hold water.

 

[jsfisher]

What I wrote is much more rigorous than your fantasy of "the series for an infinite sequence [that] is, by definition the limit of the sequence of partial sums" exactly because your definition, by your own words,

Curious you think the word you inserted belongs there. It is further evidence you don't understand the meaning of 'limit' nor 'sequence'. And, no, you didn't provide anything to discredit what I have written. You merely complained about something that did not even appear in the proof.

Nevertheless, you may use your own doron-limit if you wish. You just need to define it. Why has defining things proven so difficult for you?

 

[doronshadmi]

define Doron-level please.

There is no doron-level, but only the different levels of place value method, for example:

1 in 0.001₁₀ is in higher level than 9 in 0.0009₁₀

 

[doronshadmi]

Curious you think the word you inserted belongs there. It is further evidence you don't understand the meaning of 'limit' nor 'sequence'.

Please support your argument about me (that I "don't understand the meaning of 'limit' nor 'sequence'" because I'v added the word [that]) in details.

Only if you do that rigorously, then and only then you can start to talk about the validity of your, so called, definition.

 

[jsfisher]

Please support your argument about me (that I "don't understand the meaning of 'limit' nor 'sequence'" because I'v added the word [that]) in details.

Not 'because'. The curious word addition you made was additional evidence. My statement was correct without the addition, and becomes a non-sentence with the addition.

Only if you do that rigorously, then and only then you can start to talk about the validity of your, so called, definition.

Validity of a definition? I see you are back to trying to disprove meaning. It doesn't work that way. Definitions stand as is, whether you understand them or like them or would prefer something else.

 

[doronshadmi]

you didn't provide anything to discredit what I have written.

Since infinity is not involved in your definition of limits, no sum of any given sequence is identical to a given limit, accept in your fantasy.

 

[doronshadmi]

EDIT:

Validity of a definition?

Exactly, since infinity is not involved in your definition of limits. As a result it is no more than your fantasy, simply because no sum of any given sequence is identical to a given limit if infinity is not involved.

Once again your dogmatic philosophical (and I would even add religious) approach about definitions, is exposed.

 

[jsfisher]

Since infinity is not involved in your definition of limits, no sum of any given sequence is identical to a given limit, accept in your fantasy.

The definition of series makes your statement incorrect. Please stop trying to disprove definitions. They are not subject to that sort of thing.

 

[doronshadmi]

The definition of series makes your statement incorrect. Please stop trying to disprove definitions. They are not subject to that sort of thing.

Your definition of limits is incorrect exactly because infinity is not involved in it.

Its validity it is no more than your fantasy, exactly because no sum of any given sequence is identical to a given limit if infinity (in its relative or absolute forms) is not involved in the definition.

 

[doronshadmi]

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

In that case it is perfectly acceptable that, for example, 0.999...₁₀ < 1

 

[jsfisher]

Your definition of limits is incorrect exactly because infinity is not involved in it.

Still, despite your protestations, the definition stands unphased.

Now, it may be its utility you challenge. Challenge away, but the best way to do what would be to define for us your new and improved version of doron-limit. Who knows what great insight your alternate might provide. You need only define doron-limit so we can proceed to the next step.

 

[jsfisher]

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

In that case it is perfectly acceptable that, for example, 0.999...₁₀ < 1

My statement is consistent with the meaning of 'limit'. Your statement is not.

 

[doronshadmi]

My statement is consistent with the meaning of 'limit'.

Your statement is consistent with the current agreement about the meaning of 'limit', which is wrong exactly because infinity is not involved.

As a result 0 < |a_n - L| is perfectly acceptable and so is 0.999...₁₀ < 1, simply because they are inseparable of each other.

 

[jsfisher]

Your statement is consistent with the current agreement about the meaning of 'limit'

Current? Well, I suppose. These things don't tend to change, though, especially when they are as useful as this one is.

...which is wrong exactly because infinity is not involved.

That doesn't make it wrong. It is one of those things that makes it useful, though.

As a result 0 < |a_n - L| is perfectly acceptable

Not just acceptable. It is entirely consistent with the definition of 'limit'. (That is not to say it cannot be zero, but that 'limit' allows it to be positive.)

...and so is 0.999...₁₀ < 1, simply because they are inseparable of each other.

Repeating a bogus statement makes it no less bogus. You'd need to provide, you know, one of those proof things to show how your conclusion follows.

On the other hand, if you'd prefer something besides 'limit' to explore, all you need do is define that something first.

 

[doronshadmi]

Current? Well, I suppose. These things don't tend to change, though, especially when they are as useful as this one is.

Usefulness in not fixed, it can be developed beyond its current state, for example the usefulness of the difference between 0.999...₁₀ and 1, if infinity is involved in its relative or absolute forms, exactly as explained in http://www.internationalskeptics.com/forums/showpost.php?p=10622312&postcount=358.

That doesn't make it wrong. It is one of those things that makes it useful, though.

The current definition is one of those things that is restricted only to finitism.

As a result it can't distinguish, for example, between the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, 0.001₁₀) = 1 and the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) < 1

Not just acceptable. It is entirely consistent with the definition of 'limit'. (That is not to say it cannot be zero, but that 'limit' allows it to be positive.)

The current definition of limit is inconsistent exactly because 0 < |a_n - L| and, for example, 0.999...₁₀ < 1 are wrongly defined as separable of each other (and as a result 0 < |a_n - L| AND 0.999...₁₀ = 1).

Repeating a bogus statement makes it no less bogus.

It is a perfectly clear and valid statement that gets rid of the separability (as described above) that is at the heart of your currently accepted definition, which is wrong because of this separability.

 

[jsfisher]

Usefulness in not fixed, it can be developed beyond its current state

We all anxiously await your definition of 'doron-limit' to facilitate this great development.

...The current definition is one of those things that is restricted only to finitism.

I have no idea what you mean. It handles, for example, infinite series quite well.

As a result it can't distinguish, for example, between the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, 0.001₁₀) = 1 and the sum over (0.9₁₀, 0.09₁₀, 0.009₁₀, ...) < 1

I find it comforting that Mathematics provides the result, 0.999... = 1.

The current definition of limit is inconsistent exactly because 0 < |a_n - L|

In what way is it inconsistent? You have show no contradiction. Moreover, the greater than 0 is not a requirement of the definition of 'limit'.

...and, for example, 0.999...₁₀ < 1 are wrongly defined as separable of each other

No where in Mathematics is 0.999... defined to be less than 1.

(and as a result 0 < |a_n - L| AND 0.999...₁₀ = 1).

It is a perfectly clear and valid statement that gets rid of the separability (as described above) that is at the heart of your currently accepted definition, which is wrong because of this separability.

Your disdain for some definitions in Mathematics is irrelevant.

 

[Little 10 Toes]

There is no doron-level, but only the different levels of place value method, for example:

1 in 0.001₁₀ is in higher level than 9 in 0.0009₁₀

Please show someone else's work that mentions "levels" in the place value method.

Edit:

here's more proof for you that 0.999999... = 1.0

Let x=1.0
1/3 x = .33333333...
1/3x + 1/3x = 2/3x = .6666666...
2/3x + 1/3x = x = .999999999999...

 

[doronshadmi]

Please show someone else's work that mentions "levels" in the place value method.

Edit:

here's more proof for you that 0.999999... = 1.0

Let x=1.0
1/3 x = .33333333...
1/3x + 1/3x = 2/3x = .6666666...
2/3x + 1/3x = x = .999999999999...

0.333... < 1/3 x, so your example does not work.

 

[doronshadmi]

Your disdain for some definitions in Mathematics is irrelevant.

Your failure to understand the fallacy of 0 < |a_n - L| AND 0.999...₁₀ = 1, is indeed disdain for Mathematics.

We all anxiously await your definition of 'doron-limit' to facilitate this great development.

It is not 'doron-limit' but simply 'limit'.

Let X be a placeholder for that is summed.

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

 

[doronshadmi]

Please show someone else's work that mentions "levels" in the place value method.

The validity of notions is independent of the amount of those who understand them.