Deeper than primes

[jsfisher]

The straightforward meaning of the word "no gaps" is "no interval".

I see you jump from one guessed meaning for the word gap to another. Perhaps you should consult a dictionary, because you are now zero for two trying to get this one right.

The mathematical meaning for no gaps between real numbers is precisely inline with the common English usage: the space between any two numbers is completely filled by other numbers (leaving no gaps).

In that case there is no room for z between x and y, if we deal with the non-finite collection of all members of set X.

Now we are back to you making baseless assertions. There is no immediate predecessor because there is always another real number between any two real numbers.

You have moved from drawing a false conclusion from a false premise to drawing the false conclusion from no premise at all. This is not helping to support your assertion that every real number has an immediate predecessor or an immediate successor.

 

[doronshadmi]

I see you jump from one guessed meaning for the word gap to another. Perhaps you should consult a dictionary, because you are now zero for two trying to get this one right.

The mathematical meaning for no gaps between real numbers is precisely inline with the common English usage: the space between any two numbers is completely filled by other numbers (leaving no gaps).

Completely filled by other numbers (leaving no gaps), means that by Standard Math, no single R member is missing in any collection of all non-finite R members of a given interval.

In other words, Y has an immediate predecessor by Standard Math, and it cannot be defined by a any collection of finite members of that interval, exactly because any finite collection is not the all R members of the given interval.

Once again jsfisher, Standard Math simply guesses that something that is based on a finite collection holds also in the case of a complete non-finite collection (where complete means that all non-finite elements of the given interval are included).

If you think that by using a categorical expression style in some formal mathematical definition makes the difference between right and wrong notions, then you are living in fantasy.

Now we are back to you making baseless assertions. There is no immediate predecessor because there is always another real number between any two real numbers.

jsfisher, you are using a baseless assertion taken from a finite collection, and force it on a complete non-finite collection.

This is a fact that you cannot deny, and because of this fact you avoid any detailed reply to http://www.internationalskeptics.com/forums/showpost.php?p=4749611&postcount=3158 .

 

[The Man]

They are synonyms as long as they are related to the same mathematical case, which deals with what exists (or not) between a collection of all non-finite ordered elements.


EDIT:

As for "no gap" in the case of x < z < y.

If there is really no gap, then x = z = y.

Since we deal with x < z < y, then "no gap" is meaningless in the real sense.

No Doron the word gap is not synonymous with ‘interval’, but it is defined as an ‘empty interval’. There is a distinction, one that you seem to be unable to make with your notions that you claim have distinction as a ‘fist order property’. A gap does require difference between the boundaries of that gap, but it is specifically where something could be between those boundaries and is not. When there are no gaps between those boundaries that extent is said to be continuous, or in other words can continue from one boundary to the other without gaps. When there are gaps the extent is said to be discontinuous or discrete.

The interval (1,2) in the real numbers has no gaps (or is continuous). A set representing that interval would contain an infinite number of elements, each with a finite value greater then 1 and less then 2. However in the integers that same interval is a gap (it is an empty interval or would result in the empty set) as both of those discrete and discontinuous boundaries (1 and 2) are excluded and there are no integers between those boundaries.

 

[The Man]

Completely filled by other numbers (leaving no gaps), means that by Standard Math, no single R member is missing in any collection of all non-finite R members of a given interval.

In other words, Y has an immediate predecessor by Standard Math, and it cannot be defined by a any collection of finite members of that interval, exactly because any finite collection is not the all R members of the given interval.

Once again jsfisher, Standard Math simply guesses that something that is based on a finite collection holds also in the case of a complete non-finite collection (where complete means that all non-finite elements of the given interval are included).

If you think that by using a categorical expression style in some formal mathematical definition makes the difference between right and wrong notions, then you are living in fantasy.

What the heck is a “non-finite R member” or “an immediate predecessor”, these are your expressions Doron you need to clearly define them first before you can claim anything about them other then they simply lack definition.

In other words, Y has an immediate predecessor by Standard Math, and it cannot be defined by a any collection of finite members of that interval, exactly because any finite collection is not the all R members of the given interval.

Have you been missing the parts where we have been telling you that a set or collection representing an interval in the real numbers would not have a finite number of members? The members themselves would be finite (meaning each having a finite value) but there would be an infinite number of them in that collection.

 

[jsfisher]

Completely filled by other numbers (leaving no gaps), means that by Standard Math, no single R member is missing in any collection of all non-finite R members of a given interval.

No, it doesn't mean that at all. Why would you say such a thing? It is trivially false.

In other words, Y has an immediate predecessor by Standard Math

Not that that actually follows from your premise, but since your premise is false, your conclusion has no merit.

...
jsfisher, you are using a baseless assertion taken from a finite collection, and force it on a complete non-finite collection.

Yawn. You are the one with the baseless assertion that all real numbers have immediate predecessors. Care to try again to give your assertion some basis?

This is a fact that you cannot deny.

It is neither a fact nor impervious to denial.

 

[doronshadmi]

No Doron the word gap is not synonymous with ‘interval’, but it is defined as an ‘empty interval’. There is a distinction, one that you seem to be unable to make with your notions that you claim have distinction as a ‘fist order property’. A gap does require difference between the boundaries of that gap, but it is specifically where something could be between those boundaries and is not. When there are no gaps between those boundaries that extent is said to be continuous, or in other words can continue from one boundary to the other without gaps. When there are gaps the extent is said to be discontinuous or discrete.

The interval (1,2) in the real numbers has no gaps (or is continuous). A set representing that interval would contain an infinite number of elements, each with a finite value greater then 1 and less then 2. However in the integers that same interval is a gap (it is an empty interval or would result in the empty set) as both of those discrete and discontinuous boundaries (1 and 2) are excluded and there are no integers between those boundaries.

(let us use the term "ordered pair" for X<Y)

The Man, thank you for the accurate explanation (given by using Standard Math) about the difference between an empty interval (in the case of an ordered pair of integers) and a non-empty interval (in the case of an ordered pair of Q or R members).

Let us examine it very carefully.

The ordered pair of integers is finite, and because it is finite it is easy to show that there is no another integer between the ordered pair (what you call empty interval).

It is also easy to show that between any finite ordered pair of Q or R members, there are more Q or R members (what you call a non-empty interval), and it is easily shown exactly because we are using a finite amount of members.

But there is no way to use the finite Q or R case, in order to determine something about the non-finite case of all Q or R members of some complete and non-finite interval.

Standard Math is the framework that defines the complete non-finite collection of a given interval, and the same Standard Math tries to determine things about the complete non-finite collection of a given interval, by using terms that are based on a finite collection (which is something that cannot be done).

As a result, there must be an immediate predecessor to Y of [X,Y) or [X,Y] (because the interval is a complete non-finite collection), but it cannot be defined by any finite amount of Q or R members.

In other words, by Standard Math there is an immediate predecessor to Y, but is cannot explicitly be defined or disproved by the same Standard Math, because no finite amount of elements can be used in order to conclude anything about a complete collection of non-finite elements.

Once again:

We are in the same state of Godel's incompleteness theorems, where things must be true but cannot be proved or disproved within the deductive framework that deals with the non-finite.

 

[jsfisher]

The ordered pair of integers is finite

Finite? That's an odd way to use the term. An ordered pair is a single thing, so, in the sense of "how many", yeah, it is finite.

...and because it is finite it is easy to show that there is no another integer between the ordered pair (what you call empty interval).

Between the ordered pair? What's the difference between an orange? Did you mean to say, "between the members of the ordered pair"?

Double nonsense, at any rate. Your conclusion doesn't follow from the premise. And your conclusion is either gibberish or is patently false.

It is also easy to show that between any finite ordered pair of Q or R members, there are more Q or R members (what you call a non-empty interval), and it is easily shown exactly because we are using a finite amount of members.

Finite ordered pair? You seem hung up on the word, finite, because you use it in so many inappropriate places. Wouldn't it have been far more direct to just say "between any two real numbers there is another real number" and "between any two rational numbers there is another rational number"? All the rest is just complication that serves no purpose.

But there is no way to use the finite Q or R case, in order to determine something about the non-finite case of all Q or R members of some complete and non-finite interval.

Nonsense. You continue to allege things without support. The mere fact an interval can be identified by it's two boundaries (a finite number of boundaries) doesn't prevent conclusions being drawn about it's infinite membership.

Standard Math is....

As demonstrated again and again, standard mathematics is something you know nothing about, so please spare us your misinformed lecture.

As a result, there must be an immediate predecessor to Y of [X,Y) or [X,Y] (because the interval is a complete non-finite collection)

I'll just add "complete" to the list of terms you don't understand. That aside, I see you again are using a irrelevant premise (infinite collection) to reach a false conclusion (existence of immediate predecessors).

...but it can never be defined by any finite amount of Q or R members.

Are you really saying an immediate predecessor is an infinite collection. How bizarre of you. If an immediate predecessor actually existed in R for some number, I'd need only one thing (a finite number of things) to name it.

 

[doronshadmi]

jsfisher, in my previous post I explicitly wrote that "ordered pair" means X<Y.

Completely filled by other numbers (leaving no gaps), means that by Standard Math, no single R member is missing in any collection of all non-finite R members of a given interval.

No, it doesn't mean that at all. Why would you say such a thing? It is trivially false.

I will use a simple proof by contradiction. As with all such proofs, it begins with an assumption then proceeds to construct a contradiction, thereby showing the assumption to be false.

Assume the set {X : X<Y} does have a largest element, Z.

For Z to be an element of the set, Z < Y.
Let h be any element of the interval (Z,Y).
By the construction of h, Z < h < Y.
Since h < Y, h is an element of the set {X : X<Y}.
Since Z < h, the assumption Z was the largest element of the set has been contradicted.

Therefore, the set {X : X<Y} does not have a largest element.

If your interval is [Z,Y) such that Y is not the least upper bound of [X,Z] then Z is the largest member of [X,Z] but it is not the immediate predecessor of Y.

If your interval is [Z,Y) such that Y is the least upper bound of [X,Z] then Z is the largest member of [X,Z] and the immediate predecessor of Y, and you cannot define Z by using a finite amount of R members.

 

[doronshadmi]

Finite? That's an odd way to use the term. An ordered pair is a single thing, so, in the sense of "how many", yeah, it is finite.



Between the ordered pair? What's the difference between an orange? Did you mean to say, "between the members of the ordered pair"?

Double nonsense, at any rate. Your conclusion doesn't follow from the premise. And your conclusion is either gibberish or is patently false.



Finite ordered pair? You seem hung up on the word, finite, because you use it in so many inappropriate places. Wouldn't it have been far more direct to just say "between any two real numbers there is another real number" and "between any two rational numbers there is another rational number"? All the rest is just complication that serves no purpose.



Nonsense. You continue to allege things without support. The mere fact an interval can be identified by it's two boundaries (a finite number of boundaries) doesn't prevent conclusions being drawn about it's infinite membership.



As demonstrated again and again, standard mathematics is something you know nothing about, so please spare us your misinformed lecture.




I'll just add "complete" to the list of terms you don't understand. That aside, I see you again are using a irrelevant premise (infinite collection) to reach a false conclusion (existence of immediate predecessors).



Are you really saying an immediate predecessor is an infinite collection. How bizarre of you. If an immediate predecessor actually existed in R for some number, I'd need only one thing (a finite number of things) to name it.

You did not understand this post.

Please refreash your screen and read all of it, before you air your view on any part of it.

If you do not read all of it before you you air your view on any part of it, I promise you that you again and again going to not get it.

 

[jsfisher]

jsfisher, in my previous post I explicitly wrote that "ordered pair" means X<Y.

(a) You added it after the fact in one of your legendary post edits.
(b) So what? What did I write that didn't take that into account?

If your interval is [Z,Y)

It wasn't.

...such that Y is not the least upper bound of [X,Z]

It wasn't, and what's with this X?

...then Z is the largest member of [X,Z] but it is not the immediate predecessor of Y.

Since the premise is false, the conclusion is irrelevant.

If your interval is [Z,Y)

Still wasn't.

...such that Y is the least upper bound of [X,Z]

Still wasn't

...then Z is the largest member of [X,Z] and the immediate predecessor of Y

Still a false premise trailed by an irrelevant conclusion.

...and you cannot define Z by using a finite amount of R members.

One real number should be sufficient. That's a finite number of members, right?

 

[jsfisher]

You did not understand this post.

And you evade mine.

 

[The Man]

(let us use the term "ordered pair" for X<Y)

The Man, thank you for the accurate explanation (given by using Standard Math) about the difference between an empty interval (in the case of an ordered pair of integers) and a non-empty interval (in the case of an ordered pair of Q or R members).

No problem, but if you actually understood the ‘standard math’ you are claiming to supplant I would not have to explain it to you.

Let us examine it very carefully.

Well let’s be more careful in that examination then your usual ‘research’.

The ordered pair of integers is finite, and because it is finite it is easy to show that there is no another integer between the ordered pair (what you call empty interval).

The interval [1,3] in the integers would meet your qualification of an “ordered pair” as 1<3. Although the number of members in a set representing this interval would be finite (just 3 members that we can list as 1, 2 and 3) it is easy to show that there is another integer (2 in this case) between that “ordered” pair. Also this is a finite interval as it spans a finite difference between the boundaries (2 in this case)

It is also easy to show that between any finite ordered pair of Q or R members, there are more Q or R members (what you call a non-empty interval), and it is easily shown exactly because we are using a finite amount of members.

Again the interval [1,3] in the rationals or reals still meets your qualification for an ordered pair. However in this case a set representing that interval would not have a finite number of members. If you think otherwise it should be a simple matter to list those members as in the above example. So again the interval is finite that it spans a finite difference between the boundaries (again 2 in this case), but it certainly does not have a “finite amount of members”, again if you think so then list those members.

But there is no way to use the finite Q or R case, in order to determine something about the non-finite case of all Q or R members of some complete and non-finite interval.

Again a finite interval does not infer or require a finite number of members in that interval it simply refers to the finite expanse (or finite difference between the boundaries) of that interval, this seems to be the fact that you can not accept.

Standard Math is the framework that defines the complete non-finite collection of a given interval, and the same Standard Math tries to determine things about the complete non-finite collection of a given interval, by using terms that are based on a finite collection (which is something that cannot be done).

Well again if it is a “non-finite collection” as you claim above then it is not “a finite collection” as you claim above. The only misuse of terms is yours as usual.

As a result, there must be an immediate predecessor to Y of [X,Y) or [X,Y] (because the interval is a complete non-finite collection), but it cannot be defined by any finite amount of Q or R members.

Well again you have to define what constitutes your ‘immediate predecessor’ in order for anyone (including you) to have any basis to say what meets or does not meet your definition. The [1,3] interval in the integers before we can say that 2 is a predecessor of 3 in that interval and no other integer precedes 3 that is greater then 2. However that claim does not hold when that interval includes the rationals or reals.

In other words, by Standard Math there is an immediate predecessor to Y, but is cannot explicitly be defined or disproved by the same Standard Math, because no finite amount of elements can be used in order to conclude anything about a complete collection of non-finite elements.

Once again:

We are in the same state of Godel's incompleteness theorems, where things must be true but cannot be proved or disproved within the deductive framework that deals with the non-finite.

Once again the claim that there is an “immediate predecessor” in an interval of the reals or rationales is your claim so it is incumbent on you to explicitly define what would constitute an “‘immediate predecessor”. You are the only one in the ‘state’ of thinking what you can not prove or disprove “must be true” so much so that you do not seem much concerned with actually trying to do either.

 

[jsfisher]

Is it worth asking what's the point of this latest tangent? Other than Doron wanting to claim some moral superiority for something he's imaged in his fictitious OM, where is this all going?

On the other hand, will Doron understand that his invented immediate predecessor for some particular real number isn't a real number and therefore isn't an immediate predecessor? And once he wiggles into the shelter of denial over that, will we respond favorably to a question about what's the immediate predecessor to the immediate predecessor?

Train wrecks are such fascinating events.

 

[The Man]

Is it worth asking what's the point of this latest tangent? Other than Doron wanting to claim some moral superiority for something he's imaged in his fictitious OM, where is this all going?

On the other hand, will Doron understand that his invented immediate predecessor for some particular real number isn't a real number and therefore isn't an immediate predecessor? And once he wiggles into the shelter of denial over that, will we respond favorably to a question about what's the immediate predecessor to the immediate predecessor?

Train wrecks are such fascinating events.

Indeed, as observed before with Dorn’s claims that he is ‘bridging’ logic with ethics; moral superiority seems to be one of his goals or fantasies. However, logically and ethically one would need to first understand and communicate effectively ‘standard math’ before one claims to be able to supplant or even argue against that standard. Thus the train wreck continues.

 

[doronshadmi]

Let us go back to the beginning of the last discussion of immediate successors or predecessors to some R member.

At post http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 jsfisher writes this:

Assuming X < Y < Z, and any reasonable definition for "immediate successor", then, like it or not, doron, Y is in fact an immediate successor to [X,Y). So is [Y,Z]. So is ....

And it does matter if we deal with [X,Y] or [X,Y).

So jsfisher, by your own words you determine that "Y is in fact an immediate successor to [X,Y)".

Moreover, by your own words you determine that " … it does matter if we deal with [X,Y] or [X,Y)".

Now, at post http://www.internationalskeptics.com/forums/showpost.php?p=4736076&postcount=2974 you determine that "No real number has an immediate predecessor or immediate successor.

These two determinations of yours clearly contradict each other.

Let are focused only on Standard Math (OM is not used at this part of the discussion).

Jsfisher, (by ignoring the contradiction that is derived from your two determinations above) since you explicitly say (in your first determination above) that Y is an immediate successor of [X,Y] or [X,Y) , then Y is an immediate R member to some another R member, which is not Y.

In this case, and by following your own determination, this another R member (which is not Y) must be the immediate predecessor of Y.

By using only Standard Math, we follow your own determinations along the discussion, as follows:

1) By standard Math, any interval of non-finite R members is complete (all R members of that interval are included, without any exceptions (otherwise the interval is incomplete)) (here http://www.internationalskeptics.com/forums/showpost.php?p=4749909&postcount=3161 we can define your own determination to completeness, which is related to this case).

2) By Standard Math completeness of a non-finite collection and no gap (where "no gap" by Standard Math means "there is a room for more R members") is a contradiction, because something can't be (R complete) AND (enables a room for more R members).


Now, let us examine again why Standard Math is derived to this contradiction.

Standard Math is derived to this contradiction, because it tries to understand the non-finite by notions and examples that are based on the finite.

Let us see how Standard Math doing it:

First, we shell examine the case of two integers a and b, such that a is the immediate predecessor of b.

It is easy to get that a is the immediate predecessor of b, because we deal here with the finite case of distinct a and distinct b (this is exactly the construction of the integers).

Now let us examine the case of Q or R members.

By using a finite amount of Q or R members, it is easy to show that there is a room for more Q or R members between distinct a and distinct b.

This room exists exactly because we are using a finite amount of Q or R members, and we cannot conclude that this room still holds when we deal with the non-finite amount of all Q or R members.

This is exactly where Standard Math fails in its own framework, because what can be concluded about integers (by using a finite case) cannot be concluded about Q of R (by using a finite case).

 

[zooterkin]

So jsfisher, by your own words you determine that "Y is in fact an immediate successor to [X,Y)".
.

He doesn't determine it, that's what it means. Do you understand the phrase, 'by definition'?

By definition, the successor of [X, Y) is Y.

ETA:

Y is not the immediate successor of [X, Y].

 

[jsfisher]

Let us go back to the beginning of the last discussion of immediate successors or predecessors to some R member.

At post http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 jsfosher writes this:


So jsfisher, by your own words you determine that "Y is in fact an immediate successor to [X,Y)".

Yes. If the successor concept is extended to include intervals, then Y is an immediate successor. So is [Y,Z] for any Z>Y.

Moreover, by your own words you determine that " … it does matter if we deal with [X,Y] or [X,Y)".

Yes, it does matter. [X,Y) has immediate successors while [X,Y] does not.

Now, at post http://www.internationalskeptics.com/forums/showpost.php?p=4736076&postcount=2974 you determine that "No real number has an immediate predecessor or immediate successor.

Yes, exactly right.

These two determinations of yours clearly contradict each other.

Nope. Not in the slightest. Within the context of the real numbers (or the rational numbers), no number has an immediate predecessor or and immediate successor. If you extend that to include intervals as well, then things change.

Let are focused only on Standard Math (OM is not used at this part of the discussion).

Jsfisher, (by ignoring the contradiction that is derived from your two determinations above)

There is no contradiction, just a lack of comprehension by you.

...since you explicitly say (in your first determination above) that Y is an immediate successor of [X,Y] or [X,Y)

More lack of comprehension by you. I did not ever say Y was a successor, immediate or otherwise, to [X,Y].

...then Y is an immediate R member to some another R member, which is not Y.

No, not true. [X,Y), for example, is not a real number (i.e. not a member of R).

In this case, and by following your own determination, this another R member (which is not Y) must be the immediate predecessor of Y.

Your faulty premises have led you to a faulty conclusion.

By using only Standard Math, we follow your own determinations along the discussion, as follows:

1) By standard Math, any interval of non-finite R members is complete (all R members of that interval are included, without any exceptions (otherwise the interval is incomplete)) (here http://www.internationalskeptics.com/forums/showpost.php?p=4749909&postcount=3161 we can define your own determination to completeness, which is related to this case).

That's a circular definition you have for complete, or didn't you notice? Also, in the link you provide, I provide no definition for completeness.

2) By Standard Math completeness of a non-finite collection and no gap (where "no gap" by Standard Math means "there is a room for more R members") is a contradiction, because something can't be (R complete) AND (enables a room for more R members).

Ok, so not only do you prove, again, you don't understand what complete means, you have also reverted back to misrepresenting the phrase, no gap. You also assert a conclusion that doesn't follow from your premises (even if they weren't faulty).

Now, let us examine again why Standard Math is derived to this contradiction....

Since you are completely wrong to this point, there is no need to continue.

 

[doronshadmi]

He doesn't determine it, that's what it means. Do you understand the phrase, 'by definition'?

By definition, the successor of [X, Y) is Y.

ETA:

Y is not the immediate successor of [X, Y].

And on what this definiton is based?

 

[zooterkin]

And on what this definiton is based?

What do you mean, what is it based on? That's simply what the notation [X, Y) means. It refers to the finite interval starting with X, and including everything up to, but not including, Y.


(I despair, really. This is a notation I'd not encountered a week ago, yet here I am explaining it to someone who claims to know conventional maths well enough to state that it doesn't work.)

 

[doronshadmi]

More lack of comprehension by you. I did not ever say Y was a successor, immediate or otherwise, to [X,Y].

Moreover, by your own words you determine that " … it does matter if we deal with [X,Y] or [X,Y)".

At post http://www.internationalskeptics.com/forums/showpost.php?p=4721582&postcount=2864 jsfisher writes this:

Assuming X < Y < Z, and any reasonable definition for "immediate successor", then, like it or not, doron, Y is in fact an immediate successor to [X,Y). So is [Y,Z]. So is ....

And it does matter if we deal with [X,Y] or [X,Y).

Yes, it does matter. [X,Y) has immediate successors while [X,Y] does not.

In other words, you do not follow your own words.