Deeper than primes

[doronshadmi]

This post is a perfect example of Doron’s inability to distinguish his misinterpretations from ‘Standard Mathematics’. It does not matter that we are talking about ‘Standard Mathematics’ (or in this case intervals) Doron will simply assert his misunderstanding as some failing in ‘Standard Mathematics’. Again Doron uses a notation that indicates his point ‘y’ is included in both of his red and blue intervals and presents his misinterpretation that this indicates point ‘y’ is “in more than a one location (blue AND red)” . Doron’s misunderstanding is simply a result of him not reading (or understanding) the post he is responding to, indicating that his ascriptions of red and blue, as shown above, both include the same location ‘y’ while the intervals [x,y) (y,z] would not include that location. The illusion Doron is that you understand ‘Standard Mathematics’ or common notation and you are the only one (at least on this thread) succumbing to that illusion. Again a boundary is one of those ‘"well defined" agreed definitions, terms, concepts, etc …” while your application of the words ‘edge’ and ‘direction’ are still undefined.

If y is a point, ( notated as [x,y) (y,z] ) then it is not blue and not red, because a point does not have a direction (it is not left and not right).

The standard Math wrongly understands [x,y) (y,z] because it does not get that a line segment is not a collection of points.

Since a line segment is not a collection of points, then [x,y) (y,z] means that y point is not the left edge of [x,y] line segment, and y point is not the right edge of [y,z] line segment.

Look how limited is standard math by get anything only in terms of local elements.

Furthermore, by Standard Math [x,y) (y,z] is not accurate because there is no way to define the exact element which is the predecessor of y in the case of [x,y) or the successor of y in the case of (y,z].

In order to ignore this inability Standard Math will tell you about non-countability or ugly creatures like supremum or infimum instead of simply understand that no collection of local elements (points) can be a non-local element (a line segment).

 

[zooterkin]

If y is a point, ( notated as [x,y) (y,z] ) then it is not blue and not red, because a point does not have a direction (it is not left and not right).

I'm not familiar with this notation - [x,y) (y,z]. Can you explain it? It seems rather circular; why do you have to bring in x and z to define y?


ETA: How's that definition of 'crisp' coming on, plus some examples?

 

[Little 10 Toes]

[x,y] means all points between x and y inclusive. (x,y) means all points between x and y, but not x and y.

Final example: [-5,0),(0,5], all numbers between -5 and 5, but not zero.

Don't worry doronshadmi, i'll respond back to you later.

 

[doronshadmi]

[x,y] means all points between x and y inclusive. (x,y) means all points between x and y, but not x and y.

Final example: [-5,0),(0,5], all numbers between -5 and 5, but not zero.

Don't worry doronshadmi, i'll respond back to you later.

This is the standard notion, where anything is understood only in terms of local elements like points.

By OM's non-standard notion of [x,y] (where x<y) [x,y] is a non-local element because it is not less than both x AND y element, where x element OR y element cannot be x AND y element.

This notion of the difference between the non-local and the local, saves the problem of the inability to define the immediate predecessor of y in [x,y) case, or the immediate successor of y in (y,z] case (where y<z) (this problem exists iff [x,y) or (y,z] understood in terms of local elements like points).

The solution is based on the notion that [x,y] or [y,z] is not made of points, but it is a difference type of building-block that
it is at x AND y ( = ____) OR y AND z (= ____ ) location, which is a qualitative property that no point has (a point must be in a one and only one location).

 

[The Man]

If y is a point, ( notated as [x,y) (y,z] ) then it is not blue and not red, because a point does not have a direction (it is not left and not right).

No it is simply not blue and not red because it is not included in those intervals defining what constitutes blue and red in this consideration.

The standard Math wrongly understands [x,y) (y,z] because it does not get that a line segment is not a collection of points.

As sets containing, well, points those intervals are by definition a collection of points. However a line segment is defined by (or bounded by) two points such as the interval [x,y] thus the interval [x,y) simply lacks that end point ‘y’ as part of the included interval but is still bounded by that point.

Since a line segment is not a collection of points, then [x,y) (y,z] means that y point is not the left edge of [x,y] line segment, and y point is not the right edge of [y,z] line segment.

Again simply a matter of the notation indicating which points are included in the given intervals.

Look how limited is standard math by get anything only in terms of local elements.

Limited? I gave you four different examples of your red-blue line each with specific considerations and utility.

Furthermore, by Standard Math [x,y) (y,z] is not accurate because there is no way to define the exact element which is the predecessor of y in the case of [x,y) or the successor of y in the case of (y,z].

So Doron show us how your notions “define the exact element which is the predecessor of y” “ or the successor of y” on the real number line. By allowing for infinite decimal representation it is as accurate as any number can possibly be and quite accurately excludes ‘y‘ from those intervals as given. The ‘successor’ of the interval [x,y) is ‘y’ just as it is the ‘predecessor’ of the interval (y,z]. How much more exact can you get then one point?

In order to ignore this inability Standard Math will tell you about non-countability or ugly creatures like supremum or infimum instead of simply understand that no collection of local elements (points) can be a non-local element (a line segment).

As always Doron the ‘instability’ and ignorance remain yours.

 

[doronshadmi]

However a line segment is defined by (or bounded by) two points such as the interval [x,y] thus the interval [x,y) simply lacks that end point ‘y’ as part of the included interval but is still bounded by that point.

So we take point y ( which is not in [x,y) ) and somehow we look beyond it and claim that y is the boundary of [x,y).

1) The thing that enables us to define y as the boundary of [x,y) is exactly a non-local element (a line segment, which is an element that is not made, defined, etc … by points) that exists both in AND out of [x,y) set of points.

2) No point has this property, and also infinitely many points together do not have this property exactly because it is a qualitative difference and not a quantitative difference.

3) Standard Math cheating because it claims that a line segment is the result of 'dragging' a point. This is a fundamental failure exactly as shown in http://www.internationalskeptics.com/forums/showpost.php?p=4709081&postcount=2811 , and this cheating has to be stopped as soon as possible.

Limited? I gave you four different examples of your red-blue line each with specific considerations and utility.

Because of these four different examples, the universe of infinitely many examples, which are the result of the bridging between Non-locality and Locality, cannot be defined.

The ‘successor’ of the interval [x,y) is ‘y’ just as it is the ‘predecessor’ of the interval (y,z].

y is not in [x,y) or (y,z] so it cannot be the successor of [x,y) or the predecessor of (y,z] (see (1),(2),(3) above).

So Doron show us how your notions “define the exact element which is the predecessor of y” “ or the successor of y” on the real number line.

Very simple, by using the non-local element, which is both any arbitrary local element in [x,y) AND out [x,y) (on local y element) there cannot be an immediate predecessor of y ( in the case of in [x,y) AND out [x,y) ) and there cannot be an immediate successor of y ( in the case of out (y,z] AND in (y,z] ) exactly because a non-local element is not made of local elements.

This notion is possible exactly because there is a qualitative difference between non-local and local elements.

On the contrary, Standard Math has no rigorous answer for this case, and it claims that there is a function that somehow enables to "visit" any real number along the real line, including the immediate predecessor or the immediate successor of y.

 

[jsfisher]

So we take point y ( which is not in [x,y) ) and somehow we look beyond it and claim that y is the boundary of [x,y).

You have such trouble with basic concepts. You invent your convoluted set of inconsistent, contradictory notions just because you "can't get" something as simple as half-open intervals.

 

[zooterkin]

y is not in [x,y) or (y,z] so it cannot be the successor of [x,y) or the predecessor of (y,z] (see (1),(2),(3) above).

Sorry? What is the successor of [x,y), then?

 

[doronshadmi]

You have such trouble with basic concepts. You invent your convoluted set of inconsistent, contradictory notions just because you "can't get" something as simple as half-open intervals.

Not at all, because I understand the fundamental failure of the reasoning of clopen or open set, I can provide the solution to this fundamental failure by distinguish between the qualitative difference between local and non-local elements, and how exactly they are bridged without derived from each other.

By your fundamental failure a line is the result of a dragged point, which is an utter nonsense that does not hold water, as I clearly show in http://www.internationalskeptics.com/forums/showpost.php?p=4709081&postcount=2811 .

The whole idea of clopen or open set is based on this utter nonsense.

 

[doronshadmi]

Sorry? What is the successor of [x,y), then?

I show that there is not such a thing like the immediate local successor or the local immediate predecessor of local y.

Also I show that y has an immediate non-local successor or an immediate non-local predecessor, where local y cannot belong both to the immediate non-local successor AND the immediate non-local predecessor (the local y belongs to the immediate non-local successor OR to the immediate non-local predecessor, and in both cases it is not a building-block of any one of them).

 

[jsfisher]

Not at all, because I understand the fundamental failure of the reasoning of clopen or open set...

Oh, dear! You've circled back to topology. That ended so tragically for you before, Doron. Do you really want to venture there again? There really is no need. We were discussing intervals, remember? Let's stay focused on one topic at a time, please.

I can provide the solution to this fundamental failure by distinguish between the qualitative difference between local and non-local elements, and how exactly they are bridged without derived from each other.

And, yet, you cannot demonstrate any failure. You are grossly ignorant of so many simple mathematical concepts - probably willfully in many cases. You then use whatever misunderstandings you concoct from your misunderstandings to "disprove" unrelated mathematical facts as faulty.

You now seem fixated on the simple fact for any given number, y, there is no largest number, x < y. Big deal. You may not like the fact, but it doesn't discount it in any way. Nor can you disprove it. Nor can you use it to reject something simple like half-open intervals.

The set { x : 0 <= x < 1 } is perfectly acceptable with no demonstrable failure, as you put it. (And since you are not particularly fond of the real numbers, we can consider the set over just the rationals, if you like.)

...<more Doronetics nonsense>...

 

[jsfisher]

y is not in [x,y) or (y,z] so it cannot be the successor of [x,y) or the predecessor of (y,z]

Rubbish. Absolute illogical rubbish.

Were we to apply this example of doronetics to the integers, say, we'd find that 7 cannot be a successor to 6 because 7 is not in 6.

By any reasonable meaning of the Doron-tortured terms successor and predecessor, y is a successor (but not the successor) to [x,y) and y is a predecessor to (y,z]. The fact y isn't included in either interval is a requirement for this - exactly the reverse of what Doron alleges. (No surprise there.)

 

[doronshadmi]

You now seem fixated on the simple fact for any given number, y, there is no largest number, x < y. Big deal. You may not like the fact, but it doesn't discount it in any way. Nor can you disprove it. Nor can you use it to reject something simple like half-open intervals.

No.

I provide the exact reason of why local y (an element that has a single value) cannot be also x (an element that has a single value) , because no one of them is the non-local element that is both on x AND y, but it is not made of (not derived from) x or y.

The set { x : 0 <= x < 1 } is perfectly acceptable with no demonstrable failure, as you put it. (And since you are not particularly fond of the real numbers, we can consider the set over just the rationales, if you like.)

I do not care if set { x : 0 <= x < 1 } is perfectly acceptable or not.

I clearly show that x is a local element that cannot be = AND > 0 , which is a property that only a non-local element has.

 

[doronshadmi]

Rubbish. Absolute illogical rubbish.

Were we to apply this example of doronetics to the integers, say, we'd find that 7 cannot be a successor to 6 because 7 is not in 6.

Rubbish. Absolute illogical rubbish.

The real-line is not limited to whole numbers, so, for example, the set of [0,7) can't take 7 as the immediate successor of 6, because there are infinitely many local numbers between 6 and 7.

Furthermore, 7 is not a member of the collection of local elements of [0,7).

The immediate successor of 6 is the non-local element that is both on 6 AND 7.

Let us notate this notion like this:

[0,6_]7

 

[zooterkin]

I show that there is not such a thing like the immediate local successor or the local immediate predecessor of local y.

Also I show that y has an immediate non-local successor or an immediate non-local predecessor, where local y cannot belong both to the immediate non-local successor AND the immediate non-local predecessor (the local y belongs to the immediate non-local successor OR to the immediate non-local predecessor, and in both cases it is not a building-block of any one of them).

If I understand what was described before, by definition, y is the successor of [x, y). If it isn't, then what is [x,y)?

 

[doronshadmi]

If I understand what was described before, by definition, y is the successor of [x, y). If it isn't, then what is [x,y)?

[x,y) = [x ]y

The immediate successor of any local element in [ ] is __ ( and so is the case of [x,y] that is = [x y] )

 

[The Man]

So we take point y ( which is not in [x,y) ) and somehow we look beyond it and claim that y is the boundary of [x,y).

Look beyond? No, it is quite specifically defined as the boundary in that interval notation.

1) The thing that enables us to define y as the boundary of [x,y) is exactly a non-local element (a line segment, which is an element that is not made, defined, etc … by points) that exists both in AND out of [x,y) set of points.

No just the interval allows us and requires us to define the boundaries of that interval and that interval can represent a line segment.

2) No point has this property, and also infinitely many points together do not have this property exactly because it is a qualitative difference and not a quantitative difference.

By your own assertions Not red is blue. Thus in the example I gave with the closed intervals [x,y] is red and [y,z] is blue (or not red by your ascriptions) the point ‘y’ is both in red and not in red (or blue by your own assertions). The only difference Doron is self-consistency, standard math has it while your notions do not.

3) Standard Math cheating because it claims that a line segment is the result of 'dragging' a point. This is a fundamental failure exactly as shown in http://www.internationalskeptics.com/forums/showpost.php?p=4709081&postcount=2811 , and this cheating has to be stopped as soon as possible.

Again Doron simply claiming is not ‘showing’

Because of these four different examples, the universe of infinitely many examples, which are the result of the bridging between Non-locality and Locality, cannot be defined.

What, so you can’t define any of the ‘infinitely many examples’ in your notions? That sounds like your notions are extremely limited.

y is not in [x,y) or (y,z] so it cannot be the successor of [x,y) or the predecessor of (y,z] (see (1),(2),(3) above).

The fact that it is not included makes it the successor and predecessor to those intervals, as jsfisher also noted.

Very simple, by using the non-local element, which is both any arbitrary local element in [x,y) AND out [x,y) (on local y element) there cannot be an immediate predecessor of y ( in the case of in [x,y) AND out [x,y) ) and there cannot be an immediate successor of y ( in the case of out (y,z] AND in (y,z] ) exactly because a non-local element is not made of local elements.

So you can not define ‘y’ as a successor and a boundary to the interval [x,y) in your notions, again it makes your notions extremely limited.

This notion is possible exactly because there is a qualitative difference between non-local and local elements.

On the contrary, Standard Math has no rigorous answer for this case, and it claims that there is a function that somehow enables to "visit" any real number along the real line, including the immediate predecessor or the immediate successor of y.

No function that I am aware of just the result of defining your interval properly so that there is an isolated point as I have already said. Which was basically your claim about your blue-red line segment.

 

[jsfisher]

I provide the exact reason of why local y (an element that has a single value) cannot be also x (an element that has a single value) , because no one of them is the non-local element that is both on x AND y, but it is not made of (not derived from) x or y.

No, local and non-local are these bogus terms you introduce without definition. Even if they were defined, however, they cannot be used to disprove something else when that something else is outside their provence.

I do not care if set { x : 0 <= x < 1 } is perfectly acceptable or not.

I clearly show that x is a local element that cannot be = AND > 0 , which is a property that only a non-local element has.

No, you didn't. Local and non-local are your inventions, supposedly to "fix" a deficency in Mathematics. If Mathematics is truly deficient, then you need to show that there is an inconsistency or contradiction within the existing Mathematics. All you have done is invent something different (which is itself inconsistent and contradictory) then claimed since regular Mathematics isn't doronetics, that regular math must be in error. There is another, more likely possibility.

There is nothing wrong with the interval, [0,1), and 1 is one of its boundaries. If you have a problem with either of those statements, show how they generate a problem within the "pre-doronetics" world.

 

[jsfisher]

Rubbish. Absolute illogical rubbish.

The real-line is not limited to whole numbers, so, for example, the set of [0,7) can't take 7 as the immediate successor of 6, because there are infinitely many local numbers between 6 and 7.

The only person who has tried to equate whole numbers to the interval [0,7) is you, doron. The "absolute illogical rubbish" is entirely yours. The rest of us agree with you.

Furthermore, 7 is not a member of the collection of local elements of [0,7).

This is more of the same "absolute illogical rubbish". Perhaps if you actually read and understood what I had written you wouldn't have gone off on this ridiculous aside.

The immediate successor of 6 is the non-local element that is both on 6 AND 7.

Since the original post restricted the domain to the integers (and was about successors, not immediate successors), your statement above is just that much more ludicrous. Try to pay attention. You will never keep up if not.

 

[The Man]

Rubbish. Absolute illogical rubbish.

The real-line is not limited to whole numbers, so, for example, the set of [0,7) can't take 7 as the immediate successor of 6, because there are infinitely many local numbers between 6 and 7.

Do you not know what an integer is, since jsfisher specifically referred to integers?

Furthermore, 7 is not a member of the collection of local elements of [0,7).

The immediate successor of 6 is the non-local element that is both on 6 AND 7.

Let us notate this notion like this:

[0,6_]7

[x,y) = [x ]y

The immediate successor of any local element in [ ] is __ ( and so is the case of [x,y] that is = [x y] )

Well typical Doron problem solving just claim that it “is both on 6 AND 7” and make up some equally undefined representation.

Furthermore, by Standard Math [x,y) (y,z] is not accurate because there is no way to define the exact element which is the predecessor of y in the case of [x,y) or the successor of y in the case of (y,z].

.

So unlike standard math, even in the case of integers, Doron can not “define the exact element which is the” “ the successor of y in the case of (y,z]” since his ‘immediate successor’ is both y and y+1.