Deeper than primes

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Doron, time of reckoning has arrived . . . LOL. Get your brain ready, for the nuts and bolts mathematics knocks on the door. This time a number is needed -- no more words dressed in a fancy evening gown. The number drives Infinity, and so all ten digits of a human hand are not enough to catch the number speeding. I shall paint the picture with no delay and no mercy. Either you pull the number like a rabbit from the hat of the deepest knowledge, or you are the Conjurer Grande.

First, we build the road that has no end . . .

1/1, 1/2, 1/3, 1/4, 1/5, . . .
2/1, 2/2, 2/3, 2/4, 2/5, . . .
3/1, 3/2, 3/3, 3/4, 3/5, . . .
4/1, 4/2, 4/3, 4/4, 4/5, . . .
5/1, 5/2, 5/3, 5/4, 5/5, . . .
. . . . .
. . . . .
. . . . .

The pattern of development is apparent -- anyone can continue the fractions in columns and rows all the way toward infinity.

So we have a set of distinct fractions -- there are no two identical fractions as far as identity between numerators and denominators is concerned, as the matrix grows larger and approaches infinity. And now we will create a subset. This subset is made of all fractions which can be reduced. 4/2 and 2/1 are two different fractions on the nominal level, but on the ordinal level 4/2 = 2/1.

And so, some of the fractions in the set can be reduced, some of them not. Now the question bursts in like a tornado through the closed curve -- I mean the window: What is the ratio between those fractions that cannot be reduced and those that can, as the value of the members of the set is approaching infinity?

I hope that you would be able to figure this out and come up with the number in the approximate form with the precision of ten decimal digits. Good luck.
 
epix said:
Doron, time of reckoning has arrived . . . LOL. Get your brain ready, for the nuts and bolts mathematics knocks on the door. This time a number is needed -- no more words dressed in a fancy evening gown. The number drives Infinity, and so all ten digits of a human hand are not enough to catch the number speeding. I shall paint the picture with no delay and no mercy. Either you pull the number like a rabbit from the hat of the deepest knowledge, or you are the Conjurer Grande.

First, we build the road that has no end . . .

1/1, 1/2, 1/3, 1/4, 1/5, . . .
2/1, 2/2, 2/3, 2/4, 2/5, . . .
3/1, 3/2, 3/3, 3/4, 3/5, . . .
4/1, 4/2, 4/3, 4/4, 4/5, . . .
5/1, 5/2, 5/3, 5/4, 5/5, . . .
. . . . .
. . . . .
. . . . .

The pattern of development is apparent -- anyone can continue the fractions in columns and rows all the way toward infinity.

So we have a set of distinct fractions -- there are no two identical fractions as far as identity between numerators and denominators is concerned, as the matrix grows larger and approaches infinity. And now we will create a subset. This subset is made of all fractions which can be reduced. 4/2 and 2/1 are two different fractions on the nominal level, but on the ordinal level 4/2 = 2/1.

And so, some of the fractions in the set can be reduced, some of them not. Now the question bursts in like a tornado through the closed curve -- I mean the window: What is the ratio between those fractions that cannot be reduced and those that can, as the value of the members of the set is approaching infinity?

I hope that you would be able to figure this out and come up with the number in the approximate form with the precision of ten decimal digits. Good luck.
Click to expand...

I predict more word salad forn Doron.
 
The Man said:
Well you just determined that it would take you infinitely many n values to label all your points on your closed curve. That would give it a cardinality of Alpha0 by your asserted determination of how many elements are required.
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No, you have just forced the term "all" on infinitely many distinct elements, that naturally have no accurate cardinality.

The Man said:
Conversly standard mathamatics would assert that between any two points (any finite length) are an infinite number of points.
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Between any pair of distinct elements there is ≠ that is free of the compared elements.

The Man said:
Since any finite length can be divided into a fintite number of finite lengths any one of those devisions also represents an infinte number of points.
Thus any finite length is itself a collection of a finite number of infinite collections of points (finite lengths).
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And each time when you deal with infinite number of points, you deal with an incomplete collection.

The Man said:
However, a line or ray being an infinte number of finite lengths (each having an infinte number of points) is an infinte collection of infinte collections of points (finite lengths).
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Which is still an incomplete collection.

The Man said:
Standard math deals with your ‘potential infinity’ (finite length) and ‘actual infinity’ (infinite length) far better, easier, more consistently and concisely than you simply disagreeing with yourself and your “OM” notions
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No it does not, because a 1-dim element is a non-local atom, and it does not matter if it is closed or opened pointless 1-dim element.

Standard Math can't deal with the pointless 1-dim element.
 
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doronshadmi said:
No, you have just forced the term "all" on infinitely many distinct elements, that naturally have no accurate cardinality.
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Do you think that 'all' cannot apply to an infinite number of things? That I can't say "all the positive integers", and be quite clear in what I'm referring to?
 
The Man said:
That is just your "game" Doron of limiting a point to an edge and deliberately confusing a point with an edge, as well as inequality with a “gap”,
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The Man, you simply can't get the difference between local and non-local atomic aspects, and a complex (which is any given complex result of non-locality\locality interaction) where one of these complex results is a non-local atomic aspect (1-dim) with infinite 0-dim elements (local atomic aspect) along it.

No matter how many 0-dim elements there are (local atomic aspect) along an 1-dim, they are not a 1-dim element, and this difference is qualitative.

Because of this different quality, there is always a 1-dim domain between any arbitrary pair of distinct 0-dim elements along the 1-dim element, and as a result no infinite collection of distinct 0-dim elements along the 1-dim element, is complete.

I clearly showed that the term “all” ,if related to a collection of points along a closed curve, has a meaning only if each given distinct point of that collection is both a beginning AND an end point of that collection.

This property is possible only among a finite quantity of points (1,2,3,... etc.) along the closed curve, and it is impossible if the collection has infinitely many points.

Because of the impossibility of each given distinct point of an infinite collection along a closed curve, to be both a beginning AND an end point of an infinite collection, then an infinite collection is essentially incomplete and no transfinite cardinality is accurately determined.

In other words, an infinite collection is naturally incomplete and open (for example: it can’t completely cover a closed curve).
 
zooterkin said:
Do you think that 'all' cannot apply to an infinite number of things? That I can't say "all the positive integers", and be quite clear in what I'm referring to?
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You can also say that the sun rising in the middle of the night, so?
 
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The Man said:
Well since I made no “order definition” the failure remains yours again.
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The Man said:
http://www.internationalskeptics.com/forums/showpost.php?p=5485564&postcount=7914
You can ignore anything you want Doron, including that points along your “closed curve” are ordered (one between two others X1, X2, X3….). Changing that ordering around, or more specifically simply changing the ordering of your labeling, X3, X1, X2, does not make them any less ordered as one between two others. Without that ordering of one between two others you no longer have your “closed curve”.
Click to expand...

And also http://www.internationalskeptics.com/forums/showpost.php?p=5484258&postcount=7907
 
The Man said:
Two points is still a finite collection even in your so called 'case (1)'.
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A collection of two distinct points along a closed curve, has nothing to do with case 1, because any arbitrary point of that collection is both a beginning AND an end of that collection.
 
doronshadmi said:
The Man, you simply can't get the difference between local and non-local atomic aspects, and a complex (which is any given complex result of non-locality\locality interaction) where one of these complex results is a non-local atomic aspect (1-dim) with infinite 0-dim elements (local atomic aspect) along it.
Click to expand...

Please name 3 people, apart from yourself, who have ever 'got' the difference.
 
epix said:
Doron, time of reckoning has arrived . . . LOL. Get your brain ready, for the nuts and bolts mathematics knocks on the door. This time a number is needed -- no more words dressed in a fancy evening gown. The number drives Infinity, and so all ten digits of a human hand are not enough to catch the number speeding. I shall paint the picture with no delay and no mercy. Either you pull the number like a rabbit from the hat of the deepest knowledge, or you are the Conjurer Grande.

First, we build the road that has no end . . .

1/1, 1/2, 1/3, 1/4, 1/5, . . .
2/1, 2/2, 2/3, 2/4, 2/5, . . .
3/1, 3/2, 3/3, 3/4, 3/5, . . .
4/1, 4/2, 4/3, 4/4, 4/5, . . .
5/1, 5/2, 5/3, 5/4, 5/5, . . .
. . . . .
. . . . .
. . . . .

The pattern of development is apparent -- anyone can continue the fractions in columns and rows all the way toward infinity.

So we have a set of distinct fractions -- there are no two identical fractions as far as identity between numerators and denominators is concerned, as the matrix grows larger and approaches infinity. And now we will create a subset. This subset is made of all fractions which can be reduced. 4/2 and 2/1 are two different fractions on the nominal level, but on the ordinal level 4/2 = 2/1.

And so, some of the fractions in the set can be reduced, some of them not. Now the question bursts in like a tornado through the closed curve -- I mean the window: What is the ratio between those fractions that cannot be reduced and those that can, as the value of the members of the set is approaching infinity?

I hope that you would be able to figure this out and come up with the number in the approximate form with the precision of ten decimal digits. Good luck.
Click to expand...


There is a 1-1 correspondence between the reducible numbers and the non-reducible numbers.

Still both collections have no accurate cardinality, because both of them are incomplete collections along the closed curve.
 
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zooterkin said:
Because that doesn't happen, while it is perfectly normal usage to refer to 'all' of an infinite series.

Please explain why you believe it is not accurate to do so.
Click to expand...

Done without any need to believe to this fact. Cantor and you need to believe that there is a complete infinite collection.

But as a clearly show, there is no reasoning at the basis of that belief.
 
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zooterkin said:
Please name 3 people, apart from yourself, who have ever 'got' the difference.
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Any one that enables to get the notion of a pointless closed curve, for example:

closedcurve3.gif



This closed curve has exactly 0 cardinality, if cardinality is the number of points along the closed curve.

The term "all" for a given collection of points along a closed curve has a meaning only if any given arbitrary point of that collection is both a beginning AND an end of that collection. For example cardinality 8:

Image34.gif


This term is impossible in the case of an infinite collection.
 
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doronshadmi said:
Done without any need to believe to this fact. Cantor and you need to believe that there is a complete infinite collection.
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I don't need to believe anything, I'm merely stating a fact. It is perfectly acceptable to refer to 'all the positive integers', and it is well understood what is meant. There is no need to enumerate them all.
But as clearly show, there is no reasoning at the basis of that belief.
Click to expand...


Where do you clearly show this?
 
doronshadmi said:
Any one that enables to get the notion of a pointless closed curve, for example:

[qimg]http://dal.functionx.com/vcsharp/gdi+/forms/closedcurve3.gif[/qimg]


This closed curve has exactly 0 cardinality, if cardinality is the number of points along the closed curve.
Click to expand...

I hate to break it to you, but at every point on that curve there is, well, a point, whether you have put a large dot there or not.

The term "all" for a given collection of points along a closed curve has a meaning only if any given arbitrary point of that collection is both a beginning AND an end of that collection.
Click to expand...
Nonsense.
This term is impossible in the case of an infinite collection.
Click to expand...

Why?
 
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Little 10 Toes said:
I wonder if doronshadmi will admit he was lying?
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Avoidance noted. I have shown where you say you will apologize with conditions, and then later say that you won't apologize at all.

doronshadmi said:
zooterkin said:
Please name 3 people, apart from yourself, who have ever 'got' the difference.
Click to expand...

Any one that enables to get the notion of a pointless closed curve, for example:

[picture snipped]

This closed curve has exactly 0 cardinality, if cardinality is the number of points along the closed curve.

The term "all" for a given collection of points along a closed curve has a meaning only if any given arbitrary point of that collection is both a beginning AND an end of that collection. For example cardinality 8:

[picture snipped]

This term is impossible in the case of an infinite collection.
Click to expand...
Avoidance noted. Doronshadmi was asked a very basic question and has produced a red herring.
 
Little 10 Toes said:
Avoidance noted. I have shown where you say you will apologize with conditions, and then later say that you won't apologize at all.


Avoidance noted. Doronshadmi was asked a very basic question and has produced a red herring.
Click to expand...

Told you.
 
doronshadmi said:
There is a 1-1 correspondence between the reducible numbers and the non-reducible numbers.

Still both collections have no accurate cardinality, because both of them are incomplete collections along the closed curve.
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I forgot to mention that if you fail, your abode becomes infested with countable but infinite number of cockroaches.

Your answer is not the right one. But, there is a way to adjust the condition, so you would be correct. By doing so, we can prove that you were wrong about the 1 on 1 correspondence, or that the ratio between the fractions that can be reduced and those that cannot is 1:1.

Is there any set that can be divided on two subsets A and B so that the cardinality of A/B = 1?

I know that you are busy counting the cockroaches, but that's what it is: 1, 2, 3, 4, 5, 6, 7, 8, ...

The set of positive integers can be divided on even and odd integers and therefore there is 1 on 1 correspondence between them. Now when you regard the set of positive integers as nominators, then you need to come up with denominators for each integer in such a way that the 1 on 1 correspondence would remain intact. Since the difference between even and odd numbers is that even numbers are divisible by 2 , and odd numbers are not, you make sure that all distinct denominators whose value will be approaching infinity as well will be divisible only by 2 as well. That means the denominators will be in the form 2^k, where k = 1, 2, 3, 4, 5, ...

And so the matrix looks like this:

1/2, 2/2, 3/2, 4/2, 5/2, . . .
1/4, 2/4, 3/4, 4/4, 5/4, . . .
1/8, 2/8, 3/8, 4/8, 5/8, . . .
1/16, 2/16, 3/16, 4/16, 5/16, . . .
1/32, 2/32, 3/32, 4/32, 5/32, . . .
.
.
.

But since the original matrix is made of [k/(k+1)] and [(k+1)/k] fractions, it is unlike the above matrix, and therefore cannot have the same 1 on 1 correspondence. And that's the simple proof of you being wrong. But that doesn't really matter now, coz you need to call in an air strike. "Bravo, this is Aleph One. Unload within my perimeter. I repeat. Unload within my perimeter."
http://blogs.abcnews.com/photos/uncategorized/airplane_wirldfire_nr.jpg

Wow! That's what I call an air RAID.
 
epix said:
But since the original matrix is made of [k/(k+1)] and [(k+1)/k] fractions, it is unlike the above matrix, and therefore cannot have the same 1 on 1 correspondence. And that's the simple proof of you being wrong.
Click to expand...

epix, N is a proper subset of Q and still there is a 1-1 correspondence between N and Q.

It does not mean that N or Q collections are complete.

epix said:
even numbers are divisible by 2
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Operations like Subtraction or Division have no meaning in the Cantorean trasfinite system.
 
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zooterkin said:
I hate to break it to you, but at every point on that curve there is, well, a point, whether you have put a large dot there or not.
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Come on zooterkin, think abstract. Each large dot represents an 0-dim element along the closed curve.

zooterkin said:
Nonsense.
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Again your inability to get the notion of a pointless close curve is shown.
zooterkin said:
Why?
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Because you cannot get the notion of Non-locality.

The best that you get is up to a collection of 0-dim elements, but still you have no reasoning which explains what a collection is, in the first place.
 
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zooterkin said:
I don't need to believe anything, I'm merely stating a fact. It is perfectly acceptable to refer to 'all the positive integers', and it is well understood what is meant. There is no need to enumerate them all.
Click to expand...

It is wrongly understood, because any infinite collection is incomplete, where incompleteness is its essential property.

zooterkin said:
I don't need to believe anything, I'm merely stating a fact.
Click to expand...
All you have is a nonsense belief.
 
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Little 10 Toes said:
Avoidance noted. I have shown where you say you will apologize with conditions, and then later say that you won't apologize at all.
Click to expand...

The ball is in jsfisher's yard. I will apologize only if jsfisher will clearly claim that he does not agree with the paradigm that "pure" Math is Deductive context-dependent freamworks.
 
doronshadmi said:
epix, N is a proper subset of Q and still there is a 1-1 correspondence between N and Q.
Click to expand...
But I wasn't refering to set N and its proper subset Q -- sets which you use as an argument without introducing them in a necessary detail.
It does not mean that N or Q collections are complete.
Click to expand...
The ratio problem didn't involve any question of completeness.
Operations like Subtraction or Division have no meaning in the Cantorean trasfinite system.
Click to expand...
I used the division by 2 to define some members of the set (even numbers) -- it didn't enter the enumeration process.

Why didn't you google up the solution to the problem I presented?
 
doronshadmi said:
The ball is in jsfisher's yard. I will apologize only if jsfisher will clearly claim that he does not agree with the paradigm that "pure" Math is Deductive context-dependent freamworks.
Click to expand...


You lied by claiming I said something I never did. How does that in any way obligate me to make a counter claim? You lied, pure and simple. Be an adult. Admit it. Apologize. Move on.

Oh, right! You are still into that kindergarten mind-set, aren't you. Scratch the adult part. Be a well-behaved 5 year old. Admit the lie. Apologize. Move on.
 
doronshadmi said:
Cantor and you need to believe that there is a complete infinite collection.
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Since meanings and words are often disassociated by you, doron, exactly where do you think Cantor needed to "believe" that there is a complete infinite collection? Be specific.
 
doronshadmi said:
The ball is in jsfisher's yard. I will apologize only if jsfisher will clearly claim that he does not agree with the paradigm that "pure" Math is Deductive context-dependent freamworks.
Click to expand...

So were you lying when you said that you would never apoligize to jsfisher, or when you said you would apologize when "you [jsfisher] rigorously provide the needed evidence, which clearly shows that Contemporary “pure” Mathematics indeed has a concrete common basis (a “trunk)[sic]", or when jsfisher makes his claim? Inquireing minds want to know.
 
doronshadmi said:
Any one that enables to get the notion of a pointless closed curve, for example:

http://dal.functionx.com/vcsharp/gdi+/forms/closedcurve3.gif


This closed curve has exactly 0 cardinality, if cardinality is the number of points along the closed curve.

The term "all" for a given collection of points along a closed curve has a meaning only if any given arbitrary point of that collection is both a beginning AND an end of that collection. For example cardinality 8:

http://www.as.wvu.edu/phys/rotter/phys201/3_Motion_Kinematics/Image34.gif

This term is impossible in the case of an infinite collection.
Click to expand...


Heya! I was just popping out to have fun among the mountains and other wild area's of the world... but luckily it seems I have not missed much.

I just saw that post by Doron and I had to laugh...

Every curve is defined (closed or not) by at least a few points and a formula (mathematic or otherwise) that define that curve.

The coolest thing in that post is... if not for the definition of points... how did your computer render that graphic? It read your mind?

Great to see that Doron did not use my absence to quickly learn some real math.
 
doronshadmi said:
It is wrongly understood, because any infinite collection is incomplete, where incompleteness is its essential property.
Click to expand...

Cool, I can just continue where I left off...

Doron, which ones are missing from the collection then?

Edit: too lazy too look it up, but we already have done this and it went off at a tangent, let's see where we get this time.
 
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jsfisher said:
Since meanings and words are often disassociated by you, doron, exactly where do you think Cantor needed to "believe" that there is a complete infinite collection? Be specific.
Click to expand...

Not to have a stab at anyone here, but... I gather, from the postings above, that the definition used for 'complete' in this thread is 'when we can point at the last one without which we would not have the whole collection'.

But, that simply is not true. The definition of complete is not in what is in there, but what is missing.

Complete, to me (so as not to make people cry) means 'no elements have been left out'.

And with that definition, infinity is not at all a hard thing to see as complete.

Edit: I just thought of what is probably meant by 'complete'... Doron, I think you want to use the word 'exhaustive', which does more fit into your argumentation, meaning, enumerating all elements up until the very last one.
 
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realpaladin said:
Heya! I was just popping out to have fun among the mountains and other wild area's of the world... but luckily it seems I have not missed much.

I just saw that post by Doron and I had to laugh...

Every curve is defined (closed or not) by at least a few points and a formula (mathematic or otherwise) that define that curve.

The coolest thing in that post is... if not for the definition of points... how did your computer render that graphic? It read your mind?

Great to see that Doron did not use my absence to quickly learn some real math.
Click to expand...

The chance of Doron learning some real maths is infinitely small.
 
jsfisher said:
Since meanings and words are often disassociated by you, doron, exactly where do you think Cantor needed to "believe" that there is a complete infinite collection? Be specific.
Click to expand...

When your premise is: "There is the collection of all elements" , you have to show that the first arbitrary element of that collection is both a beginning AND an end of that collection. Only then you can conclude that some collectionis complete (the term "all" is valid).

Since Cantor did not show it he can't conclude that this collection is complete (the term "all" is valid).

In other words, Cantor's reasoning about the completeness of infinite collections is nothing but a belief, because he forces his premise on the conclusion and therefore uses a circular reasoning about the completeness of infinite collections.

Since he has no valid reasoning about this subject, all he has is nothing but a belief about this subject.

realpaladin said:
Edit: I just thought of what is probably meant by 'complete'... Doron, I think you want to use the word 'exhaustive', which does more fit into your argumentation, meaning, enumerating all elements up until the very last one.
Click to expand...

The term "all" valid only if any arbitrary element of that collection is both a beginning AND an end of that collection.

realpaladin said:
But, that simply is not true. The definition of complete is not in what is in there, but what is missing.
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The property of any arbitrary element of that collection to be both a beginning AND an end of that collection, is indeed missing.

realpaladin, exactly as we do not need to enumerate elements in order to conclude that some collection is complete, we also do not need to show what is missing in that collection simply because if we refer to some element it is immediately not missing (we know exactly what is this element) and it does not matter if this recognized element is a member of our collection, or not.

For example a collection of 4 distinct elements is complete (the term "all" is valid) exactly because any given arbitrary element of that collection is both a beginning AND an end of that collection. We do not care if there are other elements of the same property that are not in (what you call missing) the collection of the 4 distinct elements.

By checking the property of being both a beginning AND an end of a given collection, we realize that this general property holds only in the case of finite collections, and this property is more general than any other property that is used in order to determine the entities of some collection, simply because this property (of being both a beginning AND an end of a given collection) does not care what are the properties of the elements of some given collection.

The name of the game is Generalization, and the Cantorean transfinite system fails in this game, simply because any arbitrary element of some infinite collection is nor both beginning AND an end of that collection (as a result, the cardinality of any infinite collection is undetermined).
 
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epix said:
I used the division by 2 to define some members of the set (even numbers) -- it didn't enter the enumeration process.
Click to expand...

Again, if you wish to talk Transfinite, then you have to know that Subtraction or Division have no meaning in the Cantorean trasfinite system.
 
jsfisher said:
You lied by claiming I said something I never did.
Click to expand...
Jsfisher, I argue that your paradigm about "pure" Math (deductive context-dependent frameworks) is in agreement with the majority of "pure" mathematicians.

If you do not agree with this paradigm, then and only then I will apologize in front of you.
 
doronshadmi said:
When your premise is: "There is the collection of all elements" , you have to show that the first arbitrary element of that collection is both a beginning AND an end of that collection. Only then you can conclude that some collectionis complete (the term "all" is valid).
Click to expand...

An element is both beginning AND end only if it has circular (self-referal is included) relations.

The term "all" simply means that every element pertaining to a certain formulation (element is 'green') is not outside of that collection.
I.e. if we inspect "any" element and find that it's properties are matching that formulation, then it is included.

That is what "all" means. In any translation and in any way.

doronshadmi said:
Since Cantor did not show it he can't conclude that this collection is complete (the term "all" is valid).

In other words, Cantor's reasoning about the completeness of infinite collections is nothing but a belief, because he forces his premise on the conclusion and therefore uses a circular reasoning about the completeness of infinite collections.

Since he has no valid reasoning about this subject, all he has is nothing but a belief about this subject.
Click to expand...

You are simply believing that because it fits with your belief.

All you have done here is shown that you have found a loophole in a semantic definition. Nothing real emerges from your reasoning.

The nice thing about maths though, is that anyone is allowed to augment or fix holes.

So, try your same story about the Cantorean system again with my definition of "all". How far do you get?


doronshadmi said:
The term "all" valid only if any arbitrary element of that collection is both a beginning AND an end of that collection.
Click to expand...

No, it is not. Don't be silly. When you read this, you actually know that you are wrong. Just don't try to cover it up by defending it.


doronshadmi said:
The property of any arbitrary element of that collection to be both a beginning AND an end of that collection, is indeed missing.
Click to expand...

And for good reason, as you already know.


doronshadmi said:
realpaladin, exactly as we do not need to enumerate elements in order to conclude that some collection is complete, we also do not need to show what is missing in that collection simply because if we refer to some element it is immediately not missing (we know exactly what is this element)
Click to expand...

/me offers Doron some coffee.

It is not about what the element is, it is whether it has the properties of that collection and whether it is included or not.

Let me try to explain to you how this works:

* P is the collection of all points on an infinite line in a 2-pimensional plane.
* Q is the collection of all points NOT on that line in a 2-dimensional plane.
* each element q in the collection Q does not belong to P
* each element p in the collection P does not belong to Q

Both collections contain an infinite number of points, and both are complete, since no arbitary p could be outside of P and belong to Q and vice versa.

And I can say this without inspecting ANY element.

doronshadmi said:
and it does not matter if this recognized element is a member of our collection, or not.
Click to expand...

Can I be your grocery salesman?

doronshadmi said:
For example a collection of 4 distinct elements is complete (the term "all" is valid) exactly because any given arbitrary element of that collection is both a beginning AND an end of that collection.
Click to expand...

Why?

doronshadmi said:
We do not care if there are other elements of the same property that are not in (what you call missing) the collection of the 4 distinct elements.
Click to expand...

The formula for this collection is then 'the collection of these 4 distinct elements', so nothing could be missing. So?

doronshadmi said:
By checking the property of being both a beginning AND an end of a given collection, we realize that this general property holds only in the case of finite collections
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By checking? How? Show? And realize does not equal prove.


doronshadmi said:
, and this property is more general than any other property that is used in order to determine the entities of some collection, simply because this property (of being both a beginning AND an end of a given collection) does not care what are the properties of the elements of some given collection.
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The color green does not care if it is on an apple or not. This whole stuff says absolutely nothing about properties of a collection.
It only states one common property (that I think is wrong anyway) that all elements have. So, you know how to say 'element' in more words. What of it?

doronshadmi said:
The name of the game is Generalization, and the Cantorean transfinite system fails in this game.
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You have not shown this. You have stated that because you miss some lines of tekst, the system is invalid. That is weird, just... plain... out there man....

The best generalisation is 'all things are things'. That is basically and to the core of it what you are saying.

It has been said over and over again, through the ages by wannabe sages and wannabe wise men, but it actually says nothing.

Doron, the Cantorean transfinite system does not fail because it left out some words that you need as a proof.

Maybe this helps you a bit to find your way again: http://www.apronus.com/provenmath/ordinals.htm
 
doronshadmi said:
Again, if you wish to talk Transfinite, then you have to know that Subtraction or Division have no meaning in the Cantorean trasfinite system.
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But that does not prohibit creating a solution using Subtraction and Division.

Example:

- finding a reasoning line:

a+a = 4b, therefore b = 1/2a

- Rewriting it

a+a = 4b, therefore a = 2b


The system of proofs says nothing of which path you take to get to the solution. Only that you can prove the solution to be correct.

This is all old and accepted stuff, you know... the Theorem thingy?
 
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realpaladin said:
Heya! I was just popping out to have fun among the mountains and other wild area's of the world... but luckily it seems I have not missed much.
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Yes, your right, you did not missed much :D
I missed your contribution to the entertainment though :(
I was worried when you suddenly disappeared...seeing I was the last person to reply to your last post.
How were the mountains then?
 
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