Deeper than primes

[doronshadmi]

You may think that, but you are wrong.



This is what is known as confirmation bias. You are considering only examples that work. Here's one that doesn't: Let X be 1 and Y be 2.

I made this mistake for perpuse.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=4759564&postcount=3217 .

 

[doronshadmi]

Very good zooterkin, you start to get the point.

It is easy to show that my previous post is wrong, exactly because the integers' construction is based on a finite case of X < Y (because there is nothing between X and Y).

Only if Y = X+1, other then that particular circumstance in the integers there will be at least one integer between X and Y.

You have missed my point here.

The finite construction that shows a difference between integers, is based on X < X (such that Y=X+1), where there is no other integer between X and Y.

This finite construction has a meaning according to Standard Math (for example, in the case of a non-finite set of all integers ≤ X) exactly because this finite construction is the essential building-block of what integer is.

Well only each instance of Z is a “a single arbitrary R member” for example Z1 < Z2 < Z3… <Zn if we were to specify the indices of Z in that fashion. Remember a variable is, well, variable, we are not limited to just one particular instance of that variable hence the use of index notation. The set of all possible values for Z (Z1 ….Zn where n has the interval [1,[1,¥) ) in the integers) is in fact all real numbers between X and Y.

The Man, look what you are doing:

1) you are using an index that is based on all natural numbers, in order to define an uncountable set of all real numbers between X and Y. As a result, the claim "all real numbers between X and Y" is false.

2) In order to conclude that X < Y, there must d, such that d > 0 AND Y=X+d

In the case of integers d=1, and there is no problem to define a collection (finite or not) of all different elements (according to Standard Math).

In the case of the reals d can be any arbitrary value > 0, and as a result there is a non-finite regression of d values, such that X < X + (non-finite regression of d value).

Because of this non-finite regression of d value, the claim "all real numbers between X and Y" is false.

But if we claim that "all real numbers between X and Y exist in that interval, without a single exaction", then Y must have an immediate predecessor.

Since Standard Math claims that "all"="non-finite regression" its reasoning about the reals is based on a contradiction.

 

[jsfisher]

if we claim that "all real numbers between X and Y exist in that interval, without a single exaction", then Y must have an immediate successor.

No. And repeating this false claim isn't going to change its truth.

Since Standard Math claims that "all"="non-finite regression" its reasoning about the reals is based on a contradiction.

No. Standard Math makes no such claim, and so your conclusion is without merit.

 

[jsfisher]

I made this mistake for perpuse.

Please look at http://www.internationalskeptics.com/forums/showpost.php?p=4759564&postcount=3217 .

I did look at and have already responded to that post. It begins with a false premise, so nothing after that is of any merit.

 

[doronshadmi]

No. And repeating this false claim isn't going to change its truth.

Prove that this is a false claim:

"there is always another real number between any two real numbers" = "a non-finite regression of d value"

No. Standard Math makes no such claim, and so your conclusion is without merit.

Standard Math makes exactly this claim, which is:

"all" = "there is always another real number between any two real numbers" = "a non-finite regression of d value"

 

[doronshadmi]

I did look at and have already responded to that post. It begins with a false premise, so nothing after that is of any merit.

Respond but not understand.

 

[zooterkin]

"there is always another real number between any two real numbers" = "a non-finite regression of d value"

Are you saying that there isn't always another real number between any two real numbers?

Standard Math makes exactly this claim, which is:

"all" = "there is always another real number between any two real numbers" = "a non-finite regression of d value"

I don't think "Standard Math(s)" defines 'all' like that anywhere.

 

[jsfisher]

Prove that this is a false claim

Since the initial claim was yours, the burden of proof falls to you.

"there is always another real number between any two real numbers" = "a non-finite regression of d value"

Again, you attempt to move the goal posts. This isn't the claim currently at issue.

Standard Math makes exactly this claim, which is:

"all" = "there is always another real number between any two real numbers" = "a non-finite regression of d value"

Again, this is your claim, so the obligation to provide evidence is yours.

Permit me to provide a simple example: Please prove that in Standard Math, the "all" in

All Greeks are liars.​

is equivalent to "there is always another real number between any two real numbers" or "a non-finite regression of d value". If nothing else, the following doesn't quite seem right:

A non-finite regression of d value Greeks are liars.​

...but that's just me.

 

[The Man]

You have missed my point here.

The finite construction that shows a difference between integers, is based on X < X (such that Y=X+1), where there is no other integer between X and Y.

So I missed your point because you are confirming what I said that Y=X+1 is just a particular case of X < Y in the intervals where no integer is between X and Y just in that particular case?

This finite construction has a meaning according to Standard Math (for example, in the case of a non-finite set of all integers ≤ X) exactly because this finite construction is the essential building block of what integer is.

Um in the case of the set of all integers that would be an infinite construction as that would not be a finite interval, but of course that has been explained to your before.

The Man, look what you are doing:

1) you are using an index that is based on all natural numbers, in order to define an uncountable set of all real numbers between X and Y. As a result, the claim "all real numbers between X and Y" is false.

Sorry but that is how indices work, mostly for simplicity. An index of X_1.3 or Y_0.46756 would just get a very confusing. That is the flexibility of indices you have an infinite number of instances with each index and you can use an infinite number of indices, basically giving infinite infinities. However I should have been more specific about that in the previous post and that was my mistake. It still does not change the fact that your “finite case” for “a single arbitrary” value of Z is just a limitation you have chosen to impose.

2) In order to conclude that X < Y, there must d, such that d > 0. AND Y=X+d

See here you’ve got it all backasswords again X < Y is not the conclusion but the stipulation which of course requires some difference ‘d’ between X and Y that is simply trivial.

In the case of integers d=1, and there is no problem to define a collection (finite or not) of all different elements (according to Standard Math).

Just as there is no problem defining a set (finite or not) in the case of real numbers, you do realize that the set of all integers is just a subset of the set of all real numbers.

In the case of the reals d can be any arbitrary value > 0, and as a result there is a non-finite regression of d values, such that X < X + (non-finite regression of d value).

Because of this non-finite regression of d values, the claim "all real numbers between X and Y" is false.

What the heck is this “non-finite regression of d value” ?

But if we claim that "all real numbers between X and Y exist in that interval, without a single exaction", then Y must have an immediate successor.

That is your claim so again you must define how that real number ‘immediate successor’ is determined.

Since Standard Math claims that "all"="non-finite regression" its reasoning about the reals is based on a contradiction.

Please show use where any standard math text claims “"all"="non-finite regression"“ whatever that is suppose to mean

 

[doronshadmi]

Since the initial claim was yours, the burden of proof falls to you.



Again, you attempt to move the goal posts. This isn't the claim currently at issue.



Again, this is your claim, so the obligation to provide evidence is yours.

Permit me to provide a simple example: Please prove that in Standard Math, the "all" in

All Greeks are liars.​

is equivalent to "there is always another real number between any two real numbers" or "a non-finite regression of d value". If nothing else, the following doesn't quite seem right:

A non-finite regression of d value Greeks are liars.​

...but that's just me.

Jsfisher you still don't get it.

Let us do it this way:

Y=X+d

X < Y only if d is explicitly defined.

In other words, the existence of X<Y expression depends of the explicit value of d.

Since d's value is at non-finite regression (and therefore not explicitly defined), then X<Y expression does not exist.

As a result X < Z < Y expression does not exist.

Furthermore, no interval of R members exists.

 

[The Man]

Jsfisher you still don't get it.

Let us do it this way:

Y=X+d

X < Y only if d is explicitly defined.

No it does not need to be explicitly defined, it just needs to be a non-zero and non-negative value which are the limits placed on it by your “X < Y” requirement.

In other words, the existence of X<Y expression depends of the explicit value of d.

Again no, the limits of the variable ‘d’ depend on and are the result of your “existence of X<Y expression”, which sets those limits for that variable.

Since d's value is at non-finite regression (and therefore not explicitly defined), then X<Y expression does not exist.

As a result X < Z < Y expression does not exist.

Again what the heck does this “d's value is at non-finite regression” mean anyway. Again “d” is a variable specifically because its value is “not explicitly defined” although some of the limits on the possible values of d might be defined to varying degrees.

Furthermore, no interval of R members exists.

Great so now intervals of the real numbers do not exist.

 

[jsfisher]

Furthermore, no interval of R members exists.

Wow! That's quite the leap into the darkness, Doron.

 

[The Man]

Ok let’s use Doron’s ‘d’ variable and interpretations to check something.

Doron claims that without an explicit defined value for ‘d’ the expression X<Y does not hold where Y =X + d.

He also claims that any real number has an immediate predecessor and immediate successor in the real numbers.

That gives us four variables or just a ‘finite case’, by Doron’s assertions, as follows

X = A real number

X_P = Immediate real number predecessor to X

X_S = Immediate real number successor to X

d = X- X_P = X_S -X

Such that X _P < X and X < X_S with no other real numbers between those values (sounds like integers where d = 1 doesn‘t it?).

As noted before without an explicit defined value for d, X_P < X and X < X_S do not hold.

So Doron give us your explicit defined value for ‘d’ that is the result of a difference between real numbers such that X _P + d /2 is not in the interval [X_P, X] and X + d/2 is not in the interval [X, X_S]?

I predict that Doron will claim his d value as an atom (his indivisible and non-composite atom as opposed to his atoms composed as divisions) as one of his ‘non-local’ thingies. Claiming it is a superposition of the X _P and X or X and X_S ‘state‘ values. Not realizing of course that superposition is just a linier addition in this case resulting in his d value then being actually 2X-d or 2X+d. Making 2d = 2X thus d = X or d-d = 2X. Unfortunately this would also mean his d can never have an ‘explicitly defined value’ that meets the given requirements. Even in his usual misinterpretation of superposition meaning ‘having more then one value’. Thus Doron’s augment about the deficiencies of ‘standard math’ are just a projection of the very deficiencies within his own notions that he apparently can not bring himself to admit.

 

[doronshadmi]

Ok let’s use Doron’s ‘d’ variable and interpretations to check something.

Doron claims that without an explicit defined value for ‘d’ the expression X<Y does not hold where Y =X + d.

He also claims that any real number has an immediate predecessor and immediate successor in the real numbers.

You have missed it.

Y have no immediate predecessor in Standard Math framework, exactly because d value is a non-finite regression, such that this non-finite regression > 0.

If the term all is used, then there is no non-finite regression > 0, and d=0 (there is no other way, because if we use all on the non-finite, then d (which is an absolute value) must have the minimal absolute value, which is 0) .

In that case X<Y=X+0, is false.

Standard Math is based on a contradiction, in this case, because it claims that:

d > 0 (there is no immediate predecessor to Y) AND d=0 (the non-finite universe between X and Y, is completely filled).

EDIT:

"there is always another real number between any two real numbers" = "a non-finite regression of d value" = "the non-finite universe between X and Y is not completely filled"

 

[jsfisher]

Y have no immediate predecessor in Standard Math framework, exactly because d value is a non-finite regression, such that this non-finite regression > 0.

Ok, well, I think you've answered a previous question of mine. Regression is now the latest word de jour of Doronetics.

And, no surprise, you aren't using the term correctly.

...Standard Math is based on a contradiction, in this case, because it claims that:

d > 0 (there is no immediate predecessor to Y) AND d=0 (the universe between X and Y, is completely filled).

Standard Mathematics makes no such claim about values for some variable, d, being both positive and zero. That's just something you keeping trying to twist because you really don't have a very good understanding of infinite sets, or of the infinities, in general.

However, Mathematics does establish the fact that between any two real numbers there is a real number, and from this conclude that no real number has an immediate predecessor. The latter is not a contradiction; it is a simple consequence.

 

[jsfisher]

By the way, doron, note that the following expresses the notion that the real numbers are dense, and it does it without the appearance of any variable, d:

[latex]$$$ \forall x \forall y, (x < y) \Rightarrow \exists z, (x < z) \wedge (z < y) $$$[/latex]

(Ok, ok, I admit it. I just wanted to play with Latex.)

 

[doronshadmi]

Ok, well, I think you've answered a previous question of mine. Regression is now the latest word de jour of Doronetics.

And, no surprise, you aren't using the term correctly.



Standard Mathematics makes no such claim about values for some variable, d, being both positive and zero. That's just something you keeping trying to twist because you really don't have a very good understanding of infinite sets, or of the infinities, in general.

However, Mathematics does establish the fact that between any two real numbers there is a real number, and from this conclude that no real number has an immediate predecessor. The latter is not a contradiction; it is a simple consequence.

Since Standard Math claims that the universe between X and Y is (completely filled) AND (there is no immediate predecessor to Y), than Standard Math actually claims that d is both > AND = 0 (which is a contradiction under Standard Math framework).

 

[jsfisher]

Since Standard Math claims that the universe between X and Y is (completely filled) AND (there is no immediate predecessor to Y), that Standard Math actually claims that d is both > AND = 0 (which is a contradiction under Standard Math framework).

Repeating a false statement doesn't change its truth value. This "d dependency" is just your invention to cover your misunderstanding.

 

[doronshadmi]

By the way, doron, note that the following expresses the notion that the real numbers are dense, and it does it without the appearance of any variable, d:

[latex]$$$ \forall x \forall y, (x < y) \Rightarrow \exists z, (x < z) \wedge (z < y) $$$[/latex]

(Ok, ok, I admit it. I just wanted to play with Latex.)

It appears under the name z.

 

[doronshadmi]

Repeating a false statement doesn't change its truth value. This "d dependency" is just your invention to cover your misunderstanding.

It it ture jsfisher.

Actually I used here your phrases: (completely filled) AND (there is no immediate predecessor to Y).