Deeper than primes

[doronshadmi]

You did not answer the question. Please explain the process to determine "the existence of objects"

I did answer the question, and there is no process here.

 

[zooterkin]

Let us take this case, where the left side of the blue element is the right side of the red element:

________

Any one who claims ( fore example: The Man in http://www.internationalskeptics.com/forums/showpost.php?p=4714591&postcount=2841 ) that the point between the single blue element and the single red element, is both blue AND red, actually claims that a single point has sides, which is wrong.

He was only explaining that that's what you had defined.

For example, The Man does not understand that if y of [x,y][y,z] is considered as a point, then it is not blue in the case of [x,y] and not red in the case of [y,z].

So what colour do you say it is, if it is on both the blue and the red lines?

 

[integral]

Good god... this thread hurts.

 

[doronshadmi]

He was only explaining that that's what you had defined.

In order to do this you first have to distinguish between a point and a side.

So what colour do you say it is, if it is on both the blue and the red lines?

It is not blue AND not red, please to not reverse my definitions.

 

[jsfisher]

**bump**

Lest we drift too far from this, what did you mean by the following, Doron?

any collection of all distinct objects (the standard mathematical notion).

 

[doronshadmi]

A point has no sides w.r.t any element, including itself.

A line can have sides, such that no amout of sides that are on this line, is that line.

 

[doronshadmi]

**bump**

for any given immediate successor of some collection that its cardinal > 1 there must be an immediate predecessor, and for any given immediate predecessor of some collection that its cardinal > 1 there must be an immediate successor.

Note that the word all appears nowhere in the claim. Note also that Doron cannot support the claim.

Ok, let us write it like this:

Any element of the collection of all elements of the interval [x,y) including y element, must have an immediate successor OR an immediate predecessor.

This claim must be true, because we are using the universal quantifier "for all" on the interval [x,y).

 

[zooterkin]

Ok, let us write it like this:

Any element of the collection of all elements of the interval [x,y) including y element, must have an immediate successor OR an immediate predecessor.

This claim must be true, because we are using the universal quantifier "for all" on the interval [x,y).

Except that, by definition, [x,y) does not include y.

 

[jsfisher]

Ok, let us write it like this:

Any element of the collection of all elements of the interval [x,y) including y element, must have an immediate successor OR an immediate predecessor.

This claim must be true, because we are using the universal quantifier "for all" on the interval [x,y).

This is not your original statement. In it you simply required a collection with cardinality > 1. Are you retracting your original statement?

Moreoever, in the current version of your statement involving "all distinct objects", you have switched from collection to interval. Are you retracting the original version of your second statement.

And moreover, are you assuming the underlying domain for these intervals is the set of real numbers with the common order relationship? This is not implicit in your statement, but it is required to get to your statement about completeness.

 

[doronshadmi]

This is not your original statement. In it you simply required a collection with cardinality > 1. Are you retracting your original statement?

Moreoever, in the current version of your statement involving "all distinct objects", you have switched from collection to interval. Are you retracting the original version of your second statement.

And moreover, are you assuming the underlying domain for these intervals is the set of real numbers with the common order relationship? This is not implicit in your statement, but it is required to get to your statement about completeness.

In the last post I use the standard notion of using the universal quantifier "for all" on the elements of the interval [x,y).

An interval is an ordered collection of R members:

http://en.wikipedia.org/wiki/Interval_(mathematics)

In mathematics, a (real) interval is a set of real numbers with the property that any number that lies between two numbers in the set is also included in the set. For example, the set of all numbers x satisfying 0 ≤ x ≤ 1 is an interval which contains 0 and 1, as well as all numbers between them.

In that case, for example, 1 of [0,1) must have an immediate predecessor by the definitions of Standard Math, but Standard Math cannot explicitly define it.

----------------------------------------------------------------------------------------

On the contrary, Organic Mathematic explicitly defines the immediate predecessor of 1, as the non-local element that is both on 1 AND on any arbitrary member of [0,1) interval.

 

[doronshadmi]

Except that, by definition, [x,y) does not include y.

This is the reason of why I explicitly wrote "including y element".

 

[jsfisher]

In the last post I use the standard notion of using the universal quantifier "for all" on the elements of the interval [x,y).

Ok, but that does mean you made a sequence of statements that you have abandoned. There are now many, many constraints not present in your original statements. You have wasted a lot of bandwidth by stating something completely different from what you meant in the first place.

An interval is an ordered collection of R members

There's a little more to it than just that, but ok.

In that case, for example, 1 of [0,1) must have an immediate predecessor by the definitions of Standard Math, but Standard Math cannot explicitly define it.

This doesn't follow from anything in Mathematics. There is no such requirement. Why do you think otherwise?

 

[jsfisher]

This is the reason of why I explicitly wrote "including y element".

The point was that [x,y) doesn't include y, so tacking on "including y" is a contradiction, not an addition. I will assume this is an ESL problem.

If you meant to add y, you should have said something like "[x,y) and y". (That would be equivalent to saying [x,y], by the way.)

 

[doronshadmi]

Ok, but that does mean you made a sequence of statements that you have abandoned. There are now many, many constraints not present in your original statements.

Nonsense.

This doesn't follow from anything in Mathematics. There is no such requirement. Why do you think otherwise?

jsfisher you simply continue with your twisted style, in order to avoid the simple fact that Standard Math uses the universal quantifier "for all" on an interval [x,y) of R members, but cannot explicitly define the immediate predecessor of y.

The point was that [x,y) doesn't include y, so tacking on "including y" is a contradiction, not an addition. I will assume this is an ESL problem.

If you meant to add y, you should have said something like "[x,y) and y". (That would be equivalent to saying [x,y], by the way.)

It does not change the fact that Standard Math cannot explicitly define the immediate predecessor of y, whether we deal with [x,y) or [x,y].

Your twisted legend about 2^aleph0 members that cannot be expressed by aleph0 members is over, because Cantor's second diagonal is a direct proof of the incompleteness of any non-finite collection, ordered or not (the claim that there exists an interval of all non-finite members is false, because the Whole is greater than the sum of the Parts).

Pages 9,12,13 of http://www.geocities.com/complementarytheory/OMPT.pdf explicitly show the new notions about non-finite collections.

 

[jsfisher]

Ok, but that does mean you made a sequence of statements that you have abandoned. There are now many, many constraints not present in your original statements.

Nonsense.

Let's review: First you said it was true for any collection with cardinality > 1. Then you changed it to any collection of all distinct objects. Now, you have morphed it to requiring an interval over the reals along with the normal order property.

Are you honestly saying those different statements are saying the same thing?

This doesn't follow from anything in Mathematics. There is no such requirement. Why do you think otherwise?

jsfisher you simply continue with your twisted style, in order to avoid the simple fact that Standard Math uses the universal quantifier "for all" on an interval [x,y) of R members, but cannot explicitly define the immediate predecessor of y.

I see your "according to standard mathematics" claim has also morphed. It is now an "according to doron" claim. I also see you have introduced a subtle change. Before, we were discussing whether [X,Y) had an immediate successor. It does, and Y is one such example. But now, you've shifted to the immediate successor of just Y -- the interval has become irrelevant.

Why did you even introduce the interval if you were going to ignore it?

No, Y has no immediate successor. This is true for any real number, and it is also true for any rational number. Why do you think it must be otherwise?

The point was that [x,y) doesn't include y, so tacking on "including y" is a contradiction, not an addition. I will assume this is an ESL problem.

If you meant to add y, you should have said something like "[x,y) and y". (That would be equivalent to saying [x,y], by the way.)

It does not change the fact that Standard Math cannot explicitly define the immediate predecessor of y, whether we deal with [x,y) or [x,y].

As already pointed out, by asking about the immediate predecessor of Y, you have made irrelevant the intervals.

By the way, you are misusing the word, define. You don't "define the immediate predecessor of Y". You define what "immediate predecessor" means, then the rest follows.

ETA:
On the other hand, if you do want to continue intervals as a vital part of this discussion, then the term, immediate predecessor must include intervals as well. In this case, Y does have immediate predecessors, all of the form [X,Y) or (X,Y).​

Your twisted legend about 2^aleph0 members that cannot be expressed by aleph0 members is over, because Cantor's second diagonal is a direct proof of the incompleteness of any non-finite collection, ordered or not.

Cantor's second diagonal? Boy you twisted that one, didn't you. Did you mean Cantor's second uncountability proof?

You have alleged this before, and you failed before to prove the allegation. Care to try again? Perhaps you'd like to disprove Cantor's first proof, too.

 

[Little 10 Toes]

"magnitude of existence" definition

I did answer the question, and there is no process here.

Lets recap what you say about the phrase "magintude of existence":

1) magnitude is a measurement unit
2) it does not measure the number of distinct objects in a collection (aka Set_(mathematics)^WP) but it does however measure "the existence of objects"
3) there is no way to determine "the existence of objects"

 

[doronshadmi]

Lets recap what you say about the phrase "magintude of existence":

1) magnitude is a measurement unit
2) it does not measure the number of distinct objects in a collection (aka Set_(mathematics)^WP) but it does however measure "the existence of objects"
3) there is no way to determine "the existence of objects"

http://www.internationalskeptics.com/forums/showpost.php?p=4727965&postcount=2918 is my exact ansewr to your question.

If you don't like it you have to explicitly show why you disagree with it.

So please do it in details.

 

[doronshadmi]

Are you honestly saying those different statements are saying the same thing?

Yes, and your inability to get it is the reason of why you don't get my answers about this case.

No, Y has no immediate successor. This is true for any real number, and it is also true for any rational number. Why do you think it must be otherwise?

This is a good example of how you totally ignore what you read.

I am talking about the immediate predecessor of y, whether we deal with [x,y] or [x,y), and you insist to talk about y's immediate successor.

Actually (and this is the main point of my argument) it does not matter if we are talking about an immediate predecessor or an immediate successor of any arbitrary given R member along the real-line.

My claim is this, if y is an immediate successor of some R member, then this member is the immediate predecessor of y. This claim must be true both in [x,y) or [x,y] cases, since the universal quantifier "for all" is used in both cases.

Standard Math uses the twisted legend, that any representation method is limited to aleph0, and as a result there is no way to represent the immediate successor or the immediate predecessor of any arbitrary given R member along the real-line.

But this twisted legend is collapsed because Standard Math (as you wrote) can't show the immediate successor or the immediate predecessor of any arbitrary given Q member along the real-line, even if there are aleph0 Q members along the real-line.

We do not need more than that in order to show that the use of the universal quantifier "for all" on a collection of non-finite elements, does not hold.

Cantor's second diagonal? Boy you twisted that one, didn't you. Did you mean Cantor's second uncountability proof?

You have alleged this before, and you failed before to prove the allegation.

No, you have failed to get the fact that Cantor explicitly used a way to define the exact member that is not mapped with any member of N, and by using this method, the conclusion and the premise are under a circular reasoning.

jsfisher, as long as your community uses the pinky garbage can called "proper classes" as an ad hoc problems' solver, you are nothing but a religious community that has nothing to do with modern science.

 

[Little 10 Toes]

http://www.internationalskeptics.com/forums/showpost.php?p=4727965&postcount=2918 is my exact ansewr to your question.

If you don't like it you have to explicitly show why you disagree with it.

So please do it in details.

http://www.internationalskeptics.com/forums/showpost.php?p=4733964&postcount=2957

 

[The Man]

You are right, let us correct it.

A single element that there is more than a one location on it, each one of those locations is called a side w.r.t this single element.

Furthermore, no amount of the sides is exactly this single element (the Whole is greater than the sum of the Parts).

For example:

A line segment is the minimal example of a single element that has sides.

A point is en example of a single element that has no sides.

OK, so as locations, points still meet your requirement for being your ‘sides’.