Cont: Deeper than primes - Continuation 2

[doronshadmi]

Yes. We know.

Do you really believe that your "We" has any influence on mathematical results accept some common agreement about them that can easily become dogmatic?

 

[jsfisher]

Moreover your notations are undefined since you are not using any base

The irony of you accusing someone else of lack of necessary detail escapes no one except possibly you, doronshadmi.

Will you next be pointing out that it is equally undefined for not stipulating it to be a positional notation based on powers of the explicitly cited radix over some specific fixed symbol sequence?

Fortunately, in this case, your objection to perfectly acceptable notation fails. You neglected to formally declare the language, dialect, and font for your objection, and that renders it undefined.

 

[Little 10 Toes]

You are confused in step 3), and an equivalent confusion is seen in http://www.internationalskeptics.com/forums/showpost.php?p=10329336&postcount=88.

Moreover your notations are undefined since you are not using any base (you simply ignore http://www.internationalskeptics.com/forums/showpost.php?p=10655285&postcount=562).

You are the one that is confused. I asked which step confuses you. Not me, you. In addition, why are you using a message that does not deal with base₁₀ numbers.

Since you want to play a little game:

Please tell me which step confuses you:
Assume all numbers are base₁₀ unless otherwise noted.

1. X = 1.0
2. X / 3 = 1 / 3
3. 1 / 3 = 0.333...
4. X / 3 = 0.333...
5. 3 * (X / 3) = 0.333... * 3
6. 1.0 = X = .999...

 

[Little 10 Toes]

Please define "size". Please define "observed ... from the point of view".

Edit: Please tell me how you define "size". Please tell me how you define "observed ... from the point of view".

I use the definitions and reasoning of transfinite cardinality in both cases, which are not used by the definition of convergence.

Please note I did not ask how you use the terms. I asked you to define them. Please try answering the question that was asked.

 

[doronshadmi]

The irony of you accusing someone else of lack of necessary detail escapes no one except possibly you, doronshadmi.

Will you next be pointing out that it is equally undefined for not stipulating it to be a positional notation based on powers of the explicitly cited radix over some specific fixed symbol sequence?

Fortunately, in this case, your objection to perfectly acceptable notation fails. You neglected to formally declare the language, dialect, and font for your objection, and that renders it undefined.

You simply take the place_value_base method as a system of numerals.

I take the place_value_base method as a system of numbers, where transfinite cardinality can be used as a factor of it.

 

[doronshadmi]

In addition, why are you using a message that does not deal with base₁₀ numbers.

My posts are not restricted to any specific base > 1 (where base 10 is definitely one of them).

 

[doronshadmi]

Please note I did not ask how you use the terms. I asked you to define them. Please try answering the question that was asked.

Please look at this part, taken from http://en.wikipedia.org/wiki/Aleph_number :

The aleph numbers differ from the infinity (∞) commonly found in algebra and calculus. Alephs measure the sizes of sets; infinity, on the other hand, is commonly defined as an extreme limit of the real number line (applied to a function or sequence that "diverges to infinity" or "increases without bound"), or an extreme point of the extended real number line.

What I did is to use the different sizes of aleph numbers in order to construct numbers like 0.000...1₁₀ or 0.999...₁₀, as defined and explained in http://www.internationalskeptics.com/forums/showpost.php?p=10656499&postcount=567.

Exactly as ∞ in terms of aleph₁ is beyond the divergence of aleph₀, so is the case about convergence.

 

[Little 10 Toes]

My posts are not restricted to any specific base > 1 (where base 10 is definitely one of them).

Your post/picture mentions non base₁₀ numbers. I have asked a question about base₁₀ numbers.

Please tell me which step confuses you:
Assume all numbers are base10 unless otherwise noted.
X = 1.0
X / 3 = 1 / 3
1 / 3 = 0.333...
X / 3 = 0.333...
3 * (X / 3) = 0.333... * 3
1.0 = X = .999...

 

[Little 10 Toes]

Please look at this part, taken from http://en.wikipedia.org/wiki/Aleph_number :

What I did is to use the different sizes of aleph numbers in order to construct numbers like 0.000...1₁₀ or 0.999...₁₀, as defined and explained in http://www.internationalskeptics.com/forums/showpost.php?p=10656499&postcount=567.

Exactly as ∞ in terms of aleph₁ is beyond the divergence of aleph₀, so is the case about convergence.

The Wikipedia link does not use the phrase "point of view". It uses the word "size" twice when referring to cardinality of sets.

Your referenced post does not define or explain anything.

Again:
Please note I did not ask how you use the terms. I asked you to define them. Please try answering the question that was asked.

Please tell me how you define "size". Which number has the biggest "size"? 2 or 3?

Please tell me how you define "observed ... from the point of view".

 

[Slings and Arrows]

Mathematical Association of America

Does 0.99999.. equal one or does it not?

https://www.youtube.com/watch?v=x-fUDqXlmHM

 

[doronshadmi]

Mathematical Association of America

Does 0.99999.. equal one or does it not?

https://www.youtube.com/watch?v=x-fUDqXlmHM

The observer is a factor of the results.

 

[jsfisher]

The observer is a factor of the results.

Not the way you mean it. The metaphorical "if you believe" had nothing to do with belief and everything to do with underlying mathematical model.

 

[doronshadmi]

Please tell me how you define "size".

As the number of elements of a given collection.

Please tell me how you define "observed ... from the point of view".

Y is deduced from X mapping.

Let X the bijection between .000... and .999... symbols.

This bijection is done at |N| size, so if Y is not in the range of this mapping (for example |R| size is not in the range of |N| size), then Y is observed as a single element from the point of view of X.

The difference between X and Y enables the construction of numbers like 0.999...₁₀ (where the number of its symbols is |N|, and it is some X example), and its complement number to number 1 is number 0.000...1₁₀, which is a combination of X and Y, such that 0.000... is at the size of countably infinite cardinality (X), and ...1₁₀ (Y, which is at the size of uncountably infinite cardinality) is observed as a single element from the point of view of X.

More details are already given in http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587.

 

[doronshadmi]

Not the way you mean it. The metaphorical "if you believe" had nothing to do with belief and everything to do with underlying mathematical model.

Being a factor of the results is not necessarily involved with beliefs (please do not determine what I mean, please speck for yourself).

For example, observing |R| from |N| does not need any belief in order to be done, and yet it is used to construct numbers like 0.000...1₁₀ as a complement of number 0.999...₁₀ to number 1.

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=10660072&postcount=587.

 

[Little 10 Toes]

Backround post: Please tell me how you define "size".

As the number of elements of a given collection.

So you mean cardinality. Why don't you just say cardinality?

Y is deduced from X mapping.

No. Y cannot be deduced from X mapping and X and Y are undefined. X and Y already have to exist. How can you map two things when you have no things?

Let X the bijection between .000... and .999... symbols.

No, let's not. For one, .999... is not a symbol. More importantly, it's undefined (see, two can play this stupid game). In addition, it's not in a set.

Let's look and see what Wikipedia says about bijection^WP

In mathematics, a bijection (or bijective function or one-to-one correspondence) is a function between the elements of two sets, where every element of one set is paired with exactly one element of the other set, and every element of the other set is paired with exactly one element of the first set. There are no unpaired elements. In mathematical terms, a bijective function f: X → Y is a one-to-one (injective) and onto (surjective) mapping of a set X to a set Y.

X cannot be the bijection.

This bijection is done at |N| size cardinality, so if Y is not in the range of this mapping (for example |R| size cardinality is not in the range of |N| size cardinality), then Y is observed as a single element from the point of view of X.

Fixed that for you.

Y is undefined. What are you trying to compare?

The rest of your post is gibberish because you are trying to compare undefined stuff. We'll get back to the "observed ... from the point of view" once you answer those questions.

I also see that you keep avoiding answering simple questions about your mathematical ability. I'll post it here since you'll answer this post and only quote what you want to.

Please tell me which step confuses you:
Assume all numbers are base10 unless otherwise noted.
X = 1.0
X / 3 = 1 / 3
1 / 3 = 0.333...
X / 3 = 0.333...
3 * (X / 3) = 0.333... * 3
1.0 = X = .999...

 

[doronshadmi]

No, let's not. For one, .999... is not a symbol. More importantly, it's undefined (see, two can play this stupid game). In addition, it's not in a set.

Only one person does his\her best in order to play some stupid game, and it is not me.

If you stop playing your stupid game, then you can easily define the bijection between two sets of |N| distinct place values, which are used in order to construct numbers like .999...₁₀ or 0.000...1₁₀ (where the ...1₁₀ part of 0.000...1₁₀ is at |R| that is observed as a one element from |N| size).

More details are given in http://www.internationalskeptics.com/forums/showpost.php?p=10662309&postcount=593 , where Y and X are defined there as mappings between sets and not as sets (as defined in http://en.wikipedia.org/wiki/Bijection).

X cannot be the bijection.

Well, this is some concrete example of how to play stupid games instead of use your energy for fruitful discussion.

When you stop play stupid games please let me know.

Till then, you are on your own in such games.

 

[doronshadmi]

In my opinion, Mathematics is one of the concrete examples of the art of abstraction (for example, http://link.springer.com/chapter/10.1007/978-94-007-6534-4_14#page-1 , http://link.springer.com/chapter/10.1007/978-94-007-6534-4_18#page-1 , http://scholar.uwindsor.ca/cgi/viewcontent.cgi?article=2004&context=ossaarchive).

What you did to X and Y in http://www.internationalskeptics.com/forums/showpost.php?p=10663462&postcount=595 does not follow that art.

Any way, you can always can enjoy Vi Hart ( https://www.youtube.com/watch?v=X1E7I7_r3Cw ).

 

[doronshadmi]

Please tell me which step confuses you:

1. X = 1.0
2. X / 3 = 1 / 3
3. 1 / 3 = 0.333...
4. X / 3 = 0.333...
5. 3 * (X / 3) = 0.333... * 3
6. 1.0 = X = .999...

Let's demonstrate your reasoning.

X = 1.0

(or maybe we can't do that, because according to your reasoning X must be a set, as you claim in http://www.internationalskeptics.com/forums/showpost.php?p=10663462&postcount=595).

 

[ZeeGerman]

The proof that the number of primes is infinite is very simple. If you are interested:

This is a proof by contradiction. We will begin by assuming the opposite of what we wish to prove and show how that leads to a contradiction: Assume the number of primes is finite. There must then be a largest prime number. Call it P.

Let Q be the product of all the integers between 1 and P (inclusive). (So, Q = 1 x 2 x 3 x ... x P).

Now, consider the number Q+1. Because of how Q was constructed, dividing Q+1 by any prime number will have a remainder of 1. So, Q+1 must be prime number.

It is clear that Q+1 is greater than P, so if Q+1 is prime and greater than P, then our original assumption is contradicted. There can be no largest prime number, and so the number of primes is infinite.

That's a bit too coarse an explanation and therefore not quite correct.
(renaming Q from Q = P! to Q = P!+1)

You state that for P prime, the number Q =P!+1 would be prime as well , since dividing it by any number <= P (all other primes <= P included) would leave a remainder of one.
This is not correct:
Smallest counter example: P = 5 , Q = 5!+1 = 120+1 = 121 = 11*11
Just multiplying all the primes <= P and adding one doesn't work either
Smallest counter example: P= 13 Q = 2*3*5*7*11*13+1 = 30031=59*509

But both examples lead to the correct statement of the proof:
Either Q is prime itself or it must have a prime factor P' that divides Q. Since P' cannot be one of the primes <= P, it must be larger than P, contradicting the assumption that P is the largest prime.

For this to be 100% water proof, you need to assume the uniqueness of the prime factorization for any given number (http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic).

Just a nit pick, I have no idea what the rest of the discussion is about, so carry on
Zee

 

[Slings and Arrows]

In my opinion, Mathematics is one of the concrete examples of the art of abstraction (for example, http://link.springer.com/chapter/10.1007/978-94-007-6534-4_14#page-1 , http://link.springer.com/chapter/10.1007/978-94-007-6534-4_18#page-1 , http://scholar.uwindsor.ca/cgi/viewcontent.cgi?article=2004&context=ossaarchive).

What you did to X and Y in http://www.internationalskeptics.com/forums/showpost.php?p=10663462&postcount=595 does not follow that art.

Any way, you can always can enjoy Vi Hart ( https://www.youtube.com/watch?v=X1E7I7_r3Cw ).

Since you mentioned it, here she is; fast-talker Vi Hart:

9.999... reasons that .999... = 1
https://www.youtube.com/watch?v=TINfzxSnnIE