Deeper than primes

[doronshadmi]

I never thought that, nor would I. You really, really must do something about your reading comprehension. Did you really think I said [X,Y] was a finite interval and (X,Y) was not? They are both finite intervals.

Here we go again.

jsfisher, [X,Y] or (X,Y) have infinitely many R members, whether your community likes it or not.

Furthermore, all of these members are in these intervals, and as a result there is an immediate predecessor to Y, whether Y is excluded or included, it does not matter.

 

[zooterkin]

Here we go again.

jsfisher, [X,Y] or (X,Y) have infinitely many R members, whether your community likes it or not.

Who is saying there is not an infinite number of real numbers in the interval? I think you misunderstand the meaning of the term 'finite interval'.

Furthermore, all of these members are in these intervals, and as a result there is an immediate predecessor to Y, whether Y is excluded or included, it does not matter.

Really? If Y is 7.3, what is its immediate predecessor?

 

[doronshadmi]

No Doron again and as usual the contradiction is simply in your own assertions



“infinitely many finite cases” has “the quality of a non-finite collection” because you specifically gave it that ‘quality’ by declaring that you are concerned with “using infinitely many finite cases”. Then you say "no finite case alone has the quality of a non-finite collection", but your assertion was about "infinitely many finite cases” not any one "finite case alone".

Organic Mathematics does not have this problem, because by it only Fullness is actual infinity.

Your framework cannot get the notion of the incomplete non-finite collection, because its abstraction can't get things beyond the existence of a collection, and it also force the term all on a non-finite collection, which clearly shows that your community simply does not understand the non-finite.

Stop forcing Weak Emergence on OM's non-standard Strong Emergence, because each time that you do that you get the "spring of contradiction" right back in your face.

 

[zooterkin]

In case you overlooked it, I'm still waiting for the definition of 'crisp' and some example crisp ids.

 

[doronshadmi]

Who is saying there is not an infinite number of real numbers in the interval? I think you misunderstand the meaning of the term 'finite interval'.

So why to call to an ordered collation of infinitely many elements 'finite interval'?

By using this style of reasoning, the words have no meaning, so instead 'finite' or 'non-finite' we can use any meaningless sting of symbols, for example 'yoklo' and 'non-yoklo'.

In that case the meaning of 'yoklo' is derived form the axioms, definitions, etc. of the given framework, where the axioms, definitions must have a meaning, otherwise we are under a meaningless non-finite regression.

Really? If Y is 7.3, what is its immediate predecessor?

This is exactly my claim.

Standard Math cannot explicitly define or disproves the existence of the immediate predecessor of 7.3, if 7.3 is a value used at [X, 7.3] or [X, 7.3) non-finite intervals.

 

[doronshadmi]

Let's try this again, but only one question at a time:

If B is in the open interval (A,C), then is B < C ?​

Simple enough question, isn't it?

By B < C we mean that any member of (A,C) is < C.

 

[zooterkin]

So why to call to an ordered collation of infinitely many elements 'finite interval'?

Because the interval is finite?

In that case the meaning of 'yoklo' is derived form the axioms, definitions, etc. of the given framework, where the axioms, definitions must have a meaning, otherwise we are under a meaningless non-finite regression.

Quite. How's that definition of 'crisp' coming on?

This is exactly my claim.

Standard Math cannot explicitly define or disproves the existence of the immediate predecessor of 7.3, if 7.3 is a value used at [X, 7.3] or [X, 7.3) non-finite intervals.

So, by your OM, what is the immediate predecessor of 7.3?

 

[doronshadmi]

Because the interval is finite?

Why, because we explicitly know X and Y of the interval [X,Y]?

 

[zooterkin]

Why, because we explicitly know X and Y of the interval [X,Y]?

Um, exactly because X and Y are defined, the interval is finite. It's from X, to Y. Finite.

 

[doronshadmi]

Um, exactly because X and Y are defined, the interval is finite. It's from X, to Y. Finite.

Well, on the the contrary to you I do not ignore the non-finite elements of [X,Y] interval.

An interval can be called 'finite' iff each element of the interval [X,Y] is clearly defined exactly as X and Y are clearly defined.

 

[doronshadmi]

So, by your OM, what is the immediate predecessor of 7.3?

http://www.internationalskeptics.com/forums/showpost.php?p=4736517&postcount=2995

 

[zooterkin]

http://www.internationalskeptics.com/forums/showpost.php?p=4736517&postcount=2995

Using OM, tell us what the predecessor of y is in the expression [x,y] when using a number line.

Organic Mathematic explicitly defines the immediate predecessor of y, as the non-local element that is both on y AND on any arbitrary member of [x,y] interval.

So, what is that value in the case of y=7.3?

 

[zooterkin]

Well, on the the contrary to you I do not ignore the non-finite elements of [X,Y] interval.

Who's ignoring the infinite number of elements? The term 'finite interval' refers to the interval as being finite, because it is fixed in size. Why are you having trouble understanding that?

An interval can be called 'finite' iff each element of the interval [X,Y] is clearly defined exactly as X and Y are clearly defined.

You really don't understand, do you?

 

[doronshadmi]

Since you can't get the red photon analogy, then the is no choice but to show it to you step-by-step according to your "proof".

Here is your "proof":

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.

Now let us analyze it row by row:

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

By this assumption we get Z as the immediate predecessor of Y, but this is the point
where the assumption does not hold (even before the Z < h < Y constriction is used as a part of the "proof").

The reason that it is immediately does not hold is very simple:

If the element is clearly defined, then we deal with the interval [Z,Y), and Z is obviously not the immediate predecessor of Y (because to any "most left" distinct member of some interval, there must be members that are greater than it) and therefore not the largest element of the interval that is < Y.

Z<Y is nothing but finite case that can't be used in order to conclude anything about the immediate predecessor of Y.

In other words, your "proof" dies on its own assumption, and as a result there is no conclusion.

 

[The Man]

Organic Mathematics does not have this problem, because by it only Fullness is actual infinity.

Your framework cannot get the notion of the incomplete non-finite collection, because its abstraction can't get things beyond the existence of a collection, and it also force the term all on a non-finite collection, which clearly shows that your community simply does not understand the non-finite.

Stop forcing Weak Emergence on OM's non-standard Strong Emergence, because each time that you do that you get the "spring of contradiction" right back in your face.

Doron you are the one who referenced “This inability to conclude something about the non-finite, by using infinitely many finite cases” and then claimed it was “because no finite case alone has the quality of a non-finite collection”. You simply can not make up you mind whether you want to talk about using “infinitely many finite cases” or just some “finite case alone”. Stop trying to pin your crap on others, because each time you do that you will get your crap sprayed right back in your face.

 

[zooterkin]

In other words, your "proof" dies on its own assumption, and as a result there is no conclusion.

That's the point.

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.

(My bolding.) Try reading all the words, in order.

 

[doronshadmi]

Doron you are the one who referenced “This inability to conclude something about the non-finite, by using infinitely many finite cases”

No, at OM I use the actual infinity of fullness, in order to conclude that there is no such a thing like, the interval of all non-finite elements.

Standard Math is the framework that claims that there is such a thing like the interval of all non-finite elements.

What I show is that because of this claim, standard Math can't explicitly define or disprove the existence of the immediate predecessor of Y.

In other words, you simply do not understand what you read.

 

[doronshadmi]

That's the point.



(My bolding.) Try reading all the words, in order.

You have totally missed my argument once again.

What I show is that if Z is distinct, then it is not the immediate predecessor of Y.

It does not mean that the immediate predecessor of Y does not exist.

It simply shows that Standard Math can't use any formal method, based on symbols, in order to explicitly define of disprove the existence of the immediate predecessor of Y.

 

[The Man]

No,

No? You mean you did not make those claims I quoted?

at OM I use the actual infinity of fullness, in order to conclude that there is no such a thing like, the interval of all non-finite elements.

So now you’re saying the claims you made before were not the claims you made before because you are now making some other claim about disproving “the interval of all non-finite elements” which no one has claimed anything about except you.

Standard Math is the framework that claims that there is such a thing like the interval of all non-finite elements.

Please show where that claim is made.

What I show is that because of this claim, standard Math can't explicitly define or disprove the existence of the immediate predecessor of Y.

In other words, you simply do not understand what you read.

Again, Doron you are only showing that you make claims about standard math without understanding the claims you make or that standard math.

 

[jsfisher]

Since you can't get the red photon analogy....

How would you know? Your analogy has not been discussed at all except for a side reference you completely failed to understand. So, the only evidence available so far is that you are the only one lacking some understanding of the red photon analogy.

...
By this assumption we get Z as the immediate predecessor of Y...

By this assumption we get Z as the largest element of the set.

...
In other words, your "proof" dies on its own assumption, and as a result there is no conclusion.

It doesn't "die" at all. That is exactly how proofs-by-contradiction work. The assumption leads to a contradiction, and therefore the assumption must be false. And thus, the set {X : X<Y} has no largest element. QED