Anti-dialectics and Cantor's theorem

[red cat]

I have come across anti-dialecticians claiming that Cantor's theorem is "rubbish" .

http://www.revleft.com/vb/showpost.php?p=1529585&postcount=116

Given the tendency of the proponents of this pseudo-theory of creating strawmen and eventually proceeding to "negate" what they call Leninist or Maoist dialectics by mere word-play, I am interested in finding out exactly how they are attacking one of the most important theorems of mathematics.

Here is a relevant link:

http://www.ccas.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html

Discuss.

[Hit The North]

Cor blimey, Guv'nor, I couldn't make head nor tails of that article on Cantor! What does it have to do with the dialectic of the class struggle?

[red cat]

Cor blimey, Guv'nor, I couldn't make head nor tails of that article on Cantor! Neither could I. :lol:


What does it have to do with the dialectic of the class struggle?We should ask them to explain that. :)

http://anti-dialectics.co.uk/

http://anti-dialectics.co.uk/Godel_letter.htm

[mikelepore]

I couldn't make head nor tails of that article on Cantor!

In simpler English, Cantor introduced the idea that, in some cases, two cases of infinity are the same size, but, in some other cases, one infinity can be larger than another infinity. For example, the set "all integers greater than 5" and the set "all lintegers greater than 100" are the same size, because 6 can be mapped to 101, 7 mapped to 102, etc. The one-to-one mapping makes them the two infinitely-large sets the same size. However, the infinite number of number of points in a plane is a higher order of infinity than the infinite number of points in a line, because a one-to-one correspondence cannot be expressed -- every single point in a line can be associated with a whole infinity of points in the second dimension.


What does it have to do with the dialectic of the class struggle?

The only connection I can think of is: the two topics are alike in that both topics may be mentioned by people who like to impress other people with the fact that they went to college.

[red cat]

In simpler English, Cantor introduced the idea that, in some cases, two cases of infinity are the same size, but, in some other cases, one infinity can be larger than another infinity. For example, the set "all integers greater than 5" and the set "all lintegers greater than 100" are the same size, because 6 can be mapped to 101, 7 mapped to 102, etc. The one-to-one mapping makes them the two infinitely-large sets the same size. However, the infinite number of number of points in a plane is a higher order of infinity than the infinite number of points in a line, because a one-to-one correspondence cannot be expressed -- every single point in a line can be associated with a whole infinity of points in the second dimension.

What we are actually confused with is that how is Cantor's theorem a fallacy?

By the way, a one-to-one mapping from the line to the plane does exist and therefore they have equal cardinality.


The only connection I can think of is: the two topics are alike in that both topics may be mentioned by people who like to impress other people with the fact that they went to college. That was a nice one. :lol:

[Meridian]

I do not understand how there could exist a "higher order of infinity", compared to a "lower order of infinity". This seems like a paradox, given what is ordinarily meant by the word "infinity".

[mikelepore]

By the way, a one-to-one mapping from the line to the plane does exist and therefore they have equal cardinality.

Can you give me an example of unequal cardinality?

(The simplest example you can think of, otherwise I won't understand it.)

[red cat]

Can you give me an example of unequal cardinality?

(The simplest example you can think of, otherwise I won't understand it.)

Natural numbers and reals. Natural numbers form a sequence while reals don't. Moreover, the number of reals is equal to the number of subsets of the set of natural numbers. Hence, there are more reals than natural numbers.

[Rosa Lichtenstein]

Red Cat, we are quite right to question Cantor's mystical theory, based as it was on the defective diagonal 'proof'.

The 'proof' is based on a Platonist theory of classes/sets, and a novel use of the word "greater" -- so that if a set cannot be put in an equivalence relation with one of its own proper sub-sets, such as the set of counting numbers, it is uncountable and thus "greater" than that other infinite set.

I will, once my work on anti-dialectics is finished, publish a technical paper on this -- and one based partly on Hilbert's Hotel 'paradox'.

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

I won't say any more here since that will give my thesis away.

This is all part of my aim to show that traditional ways of looking at such 'problems' (and not just dialectics) are defective from start to finish -- as Wittgenstein pointed out.

Finally, what 'strawman' have I ever constructed? I have only ever relied on what dialecticians themselves say -- whereas you just ignore it.

And this can't be a 'strawman' if the 20th century's greatest mathematician, Poincare, had this to say about it:


"All Cantor's set theory is built on a sand [...]. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered. [...] Point set topology is a disease from which the human race will soon recover."

[Rosa Lichtenstein]

Mike:


The only connection I can think of is: the two topics are alike in that both topics may be mentioned by people who like to impress other people with the fact that they went to college.

And we could say the same about you scientists, couldn't we?

[red cat]

Red Cat, you are quite right to question Cantor's mystical theory, based as it was on the defective diagonal 'proof'.

The 'proof' is based on a Platonist theory of classes/sets, and a novel use of the word "greater" -- so that if a set cannot be put in an equivalence relation with one of its own proper sub-sets, such as the set of counting numbers, it is uncountable and thus "greater" than that other infinite set.

I will, once my work on anti-dialecitcs is finsihed, publish a technical paper on this -- and based on Hilbert's Hotel.

http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel (http://en.wikipedia.org/wiki/Hilbert%27s_paradox_of_the_Grand_Hotel)

I won't say any more here since that will give my thesis away.

This is all part of my aim to show that traditional ways of looking at such 'problems' (and not just dialectics) are defective from start to finish -- as Wittgenstein pointed out.

Actually I don't understand exactly why you claim Cantor's theorem to be false. I am looking forward to you explaining the counter-arguments.

[Rosa Lichtenstein]

Red Cat:


Actually I don't understand exactly why you claim Cantor's theorem to be false. I am looking forward to you explaining the counter-arguments.

I have never claimed it to be false. His 'argument' depends on a verbal sleight-of-hand, a bit like Anselm's Ontological proof. So, it's confused, not false.

[red cat]

Red Cat:



I have never claimed it to be false. His 'argument' depends on a verbal sleight-of-hand, a bit like Anselm's Ontological proof. So, it's confused, not false.

Why? In which step?

[Rosa Lichtenstein]

Red Cat:


In which step

In his enumerative definition of a set, in his use of "greater than", in his use of "infinite"..., and that's just for starters.

[red cat]

Red Cat:



In his enumerative definition of a set, in his use of "greater than", in his use of "infinite"..., and that's just for starters.

Just to be sure... are you talking about the proof of reals being uncountable, or the cardinality of a set being less than that of its power set in general, or both ?

[Rosa Lichtenstein]

Both (and more); the entire theory is based on what can only be called "mathematical mythology".

[red cat]

Both (and more); the entire theory is based on what can only be called "mathematical mythology".

Let's start with the uncountability proof. Where do you think it goes wrong?

[Rosa Lichtenstein]

As I have said already, if and when I want to publish something on this, it will either appear at my site first, or in a journal that will accept it. I only intervened here to answer a few general points, not give my thesis away.

You can find a clue as to what it will be in some of the references I give at my site, and here:

http://www.uea.ac.uk/~j339/Godelselfref.htm

http://plato.stanford.edu/entries/wittgenstein-mathematics/

[red cat]

As I have said already, if and when I want to publish something on this, it will either appear at my site first, or in a journal that will accept it. I only intervened here to answer a few general points, not give my thesis away.

You can find a clue as to what it will be in some of the references I give at my site, and here:

http://www.uea.ac.uk/~j339/Godelselfref.htm (http://www.uea.ac.uk/%7Ej339/Godelselfref.htm)

http://plato.stanford.edu/entries/wittgenstein-mathematics/

Too long and probably difficult to understand. If you don't explain things to us here then until you publish your thesis we will assume that both the proofs are correct.

[Rosa Lichtenstein]

Assume what you like...:)

[red cat]

Assume what you like...:)

We really will. :)

"Giving your thesis away" is not a reason big enough for not explaining why you think these proofs are false.

[Rosa Lichtenstein]

Yes; when I'm ready, I will publish it, and not before. The explanation will, anyway, be far too long for you; you seem to think that radically original work in such a difficult area can be summarised in a few lines.

And, I am far too busy zapping you Dialectical Mystics. When that work has been completed, I'll turn to other areas of fashionable mysticism, like Godel and Cantor.

[red cat]

Yes; when I'm ready, I will publish it, and not before. The explanation will, anyway, be far too long for you; you seem to think that radically original work in such a difficult area can be summarised in a few lines.

And, I am far too busy zapping you Dialectical Mystics. When that work has been completed, I'll turn to other areas of fashionable mysticism, like Godel and Cantor.

If you do not provideyour explanation, how do you know that I will find it too long to read? May be it will seem interesting to many of us and we will finally accept your claims ?

Until you do negate the proofs in a concrete manner, we won't accept your anti-dialectical claims or whatever you call them.

[Rosa Lichtenstein]

Red cat:


If you do not provide your explanation, how do you know that I will find it too long to read?

You already find my posts too long to read in the Mao thread -- you plainly skip past most of it. And they are quite easy.


May be it will seem interesting to many of us and we will finally accept your claims ?

Well, I'm not bothered either way; but even if I could talk you all round, that is still not enough reason for me to give my ideas away -- so stop asking. You will only ever get one answer.


Until you do negate the proofs in a concrete manner, we won't accept your anti-dialectical claims or whatever you call them.

Well, my criticisms of Godel and Cantor have very little to do with anti-dialecitcs; I only published the material in question at my site to challenge comrades to read more widely, and stop believing all the Platonist mysticism one finds in much of moderm mathematics.

My anti-dialectical arguments are, therefore, a completely separate matter.

And, as I have explained elsewhere (in fact, this is one of the first things I posted here, back in 2005), I do not expect to persuade a single Dialectical Mystic; here is why:

http://www.revleft.com/vb/showpost.php?p=1349779&postcount=2

[red cat]

Red cat:



You already find my posts too long to read in the Mao thread -- you plainly skip past most of it. And they are quite easy.



Well, I'm not bothered either way; but even if I could talk you all round, that is still not enough reason for me to give my ideas away -- so stop asking. You will only ever get one answer.



Well, my criticisms of Godel and Cantor have very little to do with anti-dialecitcs; I only published the material in question at my site to challenge comrades to read more widely, and stop believing all the Platonist mysticism one finds in much of moderm mathematics.

My anti-dialectical arguments are, therefore, a completely separate matter.

And, as I have explained elsewhere (in fact, this is one of the first things I posted here, back in 2005), I do not expect to persuade a single Dialectical Mystic; here is why:

http://www.revleft.com/vb/showpost.php?p=1349779&postcount=2

Or may be that you don't have any satisfactory answer?

[Rosa Lichtenstein]

Red Cat:


Or may be that you don't have any satisfactory answer?

As I said, think what you like, I careth not....

[red cat]

Red Cat:



As I said, think what you like, I careth not....

Both the theorems are correct.

[Rosa Lichtenstein]

Red cat:


Both the theorems are correct.

Oh well that settles it. Short, sweet and logically water-tight...

No use asking you to think outside the Platonist box, is it?

[red cat]

Red cat:



Oh well that settles it. Short, sweet and logically water-tight...

No use asking you to think outside the Platonist box, is it?

Not until you give a complete proof of your claim. :)

[Rosa Lichtenstein]

Red Cat:


Not until you give a complete proof of your claim.

And, I'm happy for you to remain a Platonist mystic until then.:):)

[red cat]

Red Cat:



And, I'm happy for you to remain a Platonist mystic until then.:):)

Call me anything you like. Still nothing but a proof shall justify your claim. :)

[Rosa Lichtenstein]

Red Cat:


Call me anything you like.

I just did.


Still nothing but a proof shall justify your claim.

You will give up at line two, since it will be too long for you.

[red cat]

Red Cat:



I just did.



You will give up at line two, since it will be too long for you.

I won't. :)

[Rosa Lichtenstein]

Red Cat:


I won't.

Sadly, your track record suggests otherwise.:(

[Rosa Lichtenstein]

Duplicate post!

[Rosa Lichtenstein]

Duplicate post!

[Calmwinds]

If so Rosa, do you think Prinicipia Mathematica and its conclusions are fine as they are?

[Rosa Lichtenstein]

No, like Wittgenstein, I think it was a waste of paper and ink.

[black magick hustla]

No, like Wittgenstein, I think it was a waste of paper and ink.

now that is interesting. why? it convinced people that mathematics were not this odd intuitions, but the conclusions of rules we "agreed" upon.

[red cat]

Red Cat:



Sadly, your track record suggests otherwise.:(

I will read your proof. In mathematics, your usual tricks such as falsification of history or misinterpretation, won't work. So I think that there could really be something substantial in your proof.

[Rosa Lichtenstein]

Red Cat:


I will read your proof.

You have great difficulty reading my posts in Learning (see below), so I remain sceptical.


In mathematics, your usual tricks such as falsification of history or misinterpretation, won't work.

What have I falsified or misinterpreted?


So I think that there could really be something substantial in your proof.

You are never going to see it.

[red cat]

Red Cat:



You have great difficulty reading my posts in Learning (see below), so I remain sceptical.



What have I falsified or misinterpreted?



You are never going to see it.

Why are you avoiding giving your proof here ? :)

[Rosa Lichtenstein]

Red Cat:


Why are you avoiding giving your proof here?

Two reasons:

1) I'm not going to share original ideas, which challenge one of the most widely accepted and respected mathematical theses of the last 100 years with an unprincipled dissembler like you, who can't be bothered to get even simple posts of mine right, and who thinks that a complex problem like this can be solved in less than twenty pages.

2) I am more concerned to demolish that mystical theory of yours, so this will have to go on the back burner for several years.

[red cat]

Red Cat:



Two reasons:

1) I'm not going to share original ideas, which challenge one of the most widely accepted and respected mathematical theses of the last 100 years with an unprincipled dissembler like you, who can't be bothered to get even simple posts of mine right, and who thinks that a complex problem like this can be solved in less than twenty pages.

2) I am more concerned to demolish that mystical theory of yours, so this will have to go on the back burner for several years.


You can at least explain the first few pages of your proof. If you don't, it indicates some other reason.

[Rosa Lichtenstein]

Red Cat:


You can at least explain the first few pages of your proof.

May I refer the honourable dissembler to point one above.


If you don't, it indicates some other reason.

As I have said, several times: think what you like; I careth not.

[red cat]

Red Cat:



May I refer the honourable dissembler to point one above.



As I have said, several times: think what you like; I careth not.

How will we know whether your original idea really exist if we don't get a glimpse of even the first few pages?

[Rosa Lichtenstein]

Red Cat:


How will we know whether your original idea really exist if we don't get a glimpse of even the first few pages?

I do not care if you do not know this. What makes you think I value the opinions or views of a dissembler like you?

[red cat]

Red Cat:



I do not care if you do not know this. What makes you think I value the opinions or views of a dissembler like you?

It is amazing how you refuse to provide a proof at all when it comes to a subject as rigorous as mathematics. In other subjects, of course, you play with words and misinterpret stuff. But here that is not possible. Is that why you are trying to get away with such silly excuses? Try to admit when you lose debates. Admit that you don't have a proof at all.

[Rosa Lichtenstein]

Red Cat:


It is amazing how you refuse to provide a proof at all when it comes to a subject as rigorous as mathematics. In other subjects, of course, you play with words and misinterpret stuff. But here that is not possible.

I could say the same about your pathetic attempts to defend Mao; in fact I will.


Is that why you are trying to get away with such silly excuses? Try to admit when you lose debates. Admit that you don't have a proof at all

As I have said several times, and here it is again: think what you like; I careth not.

[red cat]

Red Cat:



I could say the same about your pathetic attempts to defend Mao; in fact I will.



As I have said several times, and here it is again: think what you like; I careth not.

Too bad that you are depriving us of your intellectual marvel because you "careth not". :(

I will try to explore some other mathematicians' works which seek to disprove Cantor's theorem.

[Rosa Lichtenstein]

Red Cat:


Too bad that you are depriving us of your intellectual marvel because you "careth not".

I careth not about what you think, not about my proof.

[mikelepore]

Bob the Builder - "I couldn't make head nor tails of that article on Cantor! What does it have to do with the dialectic of the class struggle?

mikelepore - "The only connection I can think of is: the two topics are alike in that both topics may be mentioned by people who like to impress other people with the fact that they went to college."

Rosa Lichtenstein - "And we could say the same about you scientists, couldn't we?"

I don't see scientists citing any topic as abstract as proofs of theorems and claiming that it has any connection to the class struggle.

Eighth grade algebra is helpful in understanding how exploitation works. No mathematics more advanced than that is needed to communicate about the class struggle.

[red cat]

I don't see scientists citing any topic as abstract as proofs of theorems and claiming that it has any connection to the class struggle.


You won't find many mathematicians claiming so either. ;)

[Rosa Lichtenstein]

Mike:


I don't see scientists citing any topic as abstract as proofs of theorems and claiming that it has any connection to the class struggle.

Nor did I.

My point was aimed at your claim that this topic allows individuals to show off that they have been to college. The same can be said for scientists.

Much of modern physics is highly abstract -- e.g., quantum loop gravity, string (M) theory...

Recall, this thread was started by Red Cat, who has tried to link this with anti-dialectics, a link that exists only in his/her mind.

The material to which she/he linked at my site is only there to counter wild claims made by comrades in the UK-SWP concerning Godel and Cantor, claims which can only be accepted by Platonists -- i.e., those still in the closet and those who have come out.

[red cat]

I am beginning a discussion on Cantor's theorems and their proofs here. Those who try to bring in invalid points while not involving in any mathematical explanations are being requested to post elsewhere. Any query regarding the understanding of the proofs or definitions or anything mathematical is most welcome.

This discussion is meant for everyone, including people with knowledge of only day-to-day arithmetic. I will begin with elementary concepts. For explaining this, I will rely on the readers' intuition rather than on the formal methods. Rigorous proofs and definitions will pop up only later, if required at all. The posts will be kept as short as possible. I will also try to provide other applications of the techniques involved.

Feel free to ask about anything you don't understand, but please don't spam and bring in harder topics only to show that you know stuff..... :)

Let us start with some basic ideas:



Natural numbers : 1, 2, 3, 4, 5, 6, .......

The set of natural numbers is called N.



Integers: ....., -4, -3 , -2 , -1, 0, 1, 2, 3, 4, ......

In other words, natural numbers, zero and natural numbers with a minus in front of them.

The set of integers is called Z.



Rational numbers: An integer divided by another non-zero integer gives you a rational number. Every integer is automatically a rational since it is itself divided by 1.

Examples: 0, 98, 47/65, -349/762, 22/7

The set of rational numbers is called Q.



Real numbers : Any number that can be written in a decimal notation is a real. The decimal representation might be finite or infinite.

Examples: -3, 0, 56.7777....., -6.18181818, 1.01001000100001000001......, 1.414213562373095048801688......., 2.7182818284590452........

The set of reals is called R.


Continued at http://www.revleft.com/vb/showpost.php?p=1668410&postcount=56

[red cat]

Continued from http://www.revleft.com/vb/showpost.php?p=1667921&postcount=55

Classification of reals in terms of decimal representation:

There can be three kinds of real numbers in terms of decimal representation:



Terminating decimals: Those reals whose decimal representations terminate.

Examples: -1, 2.777, 34.8536



Repeating decimals: These do not terminate, but after a certain number of digits, a sequence of digits repeats itself to make the representation infinite.

Examples: 0.33333......, 76.5217634763476347364...., -6.98727272....


Non terminating - non repeating decimals: These neither terminate nor repeat.

Examples: 1.01001000100001000001......, -1.414213562373095048801688......., 2.7182818284590452........

Continued at http://www.revleft.com/vb/showpost.php?p=1668935&postcount=63

[Meridian]

Repeating decimals: These do not terminate, but after a certain number of digits, a sequence of digits repeats itself to make the representation infinite.

Examples: 0.33333......, 76.5217634763476347364...., -6.98727272....


Non terminating - non repeating decimals: These neither terminate nor repeat.

Examples: 1.01001000100001000001......, -1.414213562373095048801688......., 2.7182818284590452........ But, infinity is not a quantity. How could we operate with numbers with "endless" (as in, non-computable, non-estimable) decimals then? That is a contradiction. I notice that you write high amounts of (.....) to illustrate eternity, which is a very common fallacy -- indicating that infinity is something alike a "very large amount"!

[red cat]

But, infinity is not a quantity. How could we operate with numbers with "endless" (as in, non-computable, non-estimable) decimals then? That is a contradiction. I notice that you write high amounts of (.....) to illustrate eternity, which is a very common fallacy -- indicating that infinity is something alike a "very large amount"!

What kind of computations are you talking about ? Can you give an example and point out where or what the contradiction is ?

And I don't really understand why you call a convention a fallacy. Consecutive dots are generally used in certain places to indicate infinity.

[Meridian]

What kind of computations are you talking about ? Can you give an example and point out where or what the contradiction is ?

And I don't really understand why you call a convention a fallacy. Consecutive dots are generally used in certain places to indicate infinity.
By something being computable I mean something that actually can be measured or constructed by humans (not "theoretically" or "abstractly"). Such as by counting, through geometrical measurement, etc. Something that can be determined, by a NASA-computer if need be.

The contradiction I was referring to is to operate with numbers supposedly having endlessly recursive decimals as if they were countable quantities (computable numbers). The whole point of them is that, supposedly, they are infinitely impossible to determine. Ending the decimal line with (....) describes very well the attitude many have towards these numbers, and the concept of infinity as something "very huge". How can we refer to something that can't be referred to? How can we count something that is not countable?

Things done by convention can just as easily be based on a fallacy as any other practice.

[red cat]

By something being computable I mean something that actually can be measured or constructed by humans (not "theoretically" or "abstractly"). Such as by counting, through geometrical measurement, etc. Something that can be determined, by a NASA-computer if need be.

The contradiction I was referring to is to operate with numbers supposedly having endlessly recursive decimals as if they were countable quantities (computable numbers). The whole point of them is that, supposedly, they are infinitely impossible to determine. Ending the decimal line with (....) describes very well the attitude many have towards these numbers, and the concept of infinity as something "very huge". How can we refer to something that can't be referred to? How can we count something that is not countable?

Things done by convention can just as easily be based on a fallacy as any other practice.

Every rational number can be geometrically constructed. In fact, the set of geometrically constructible numbers contains many other numbers too.

The formal notion of computability of a number is something like the number being computable up to any desired place of decimal. If this can be done, then other operations on the number are also computable.

Whether a rational number will have infinite representation or not depends on the system that we use. For example, 1/3= 0.333.... in the decimal system, but it is 0.1 in the trinary system. Also, 3*(0.333....) gives you 1.

So, whether these have a terminating representation or not is not related to computation on them.

[Meridian]

Every rational number can be geometrically constructed. In fact, the set of geometrically constructible numbers contains many other numbers too.
Whether intended or not, you are simply obfuscating the matter here.

Did I indicate that I was speaking of rational numbers or what you call "real numbers"? I did not mention rational numbers.


Whether a rational number will have infinite representation or not depends on the system that we use. For example, 1/3= 0.333.... in the decimal system, but it is 0.1 in the trinary system. Also, 3*(0.333....) gives you 1.

So, whether these have a terminating representation or not is not related to computation on them.
Again, these are rational numbers.

[red cat]

Whether intended or not, you are simply obfuscating the matter here.

Did I indicate that I was speaking of rational numbers or what you call "real numbers"? I did not mention rational numbers.


Again, these are rational numbers.

This is why I asked you to give an example. :)

I don't understand exactly what type of number you are talking about. If you tell me some numbers that fit the category that you have in your mind, I could answer your questions better.

[red cat]

Continued from http://www.revleft.com/vb/showpost.php?p=1668410&postcount=56

Claim: Rationals are exactly those reals which have either terminating or repeating decimal representations.

Let us look at the binary representation of 1/7 :


http://daugerresearch.com/pooch/LongDivision.gif

5x7=35..................40
.............................-35
7x7=49.....................50
...............................-49
1x7=7.........................10
................. .................-7

1/7=0.1428571.....


As observed, the whole process starts repeating as soon as a remainder (in this case, 1) repeats. Any number "n" can have at most "n" such remainders. Therefore after the decimal point has been encountered during division, a repeating pattern will either emerge after at most "n" steps, or the representation will terminate.


Now we will try to convert terminating and repeating decimals to rationals.
For terminating decimals, the process is relatively easy. Consider any terminating decimal, say 45.234568952. This equals 45234568952/1000000000 . In short, write the numerator without the decimal point, then divide by 1 followed by zeros equaling the number of digits after decimal in number.

Now consider a repeating decimal, say 624.72358635863586......

The repeating portion is 3586, and 4 digits long. The trick here is to to multiply the number by 10000.

Let x = 624.72358635863586......

=> 10000x= 6247235.86358635863586.....

=> 10000x - x = 6247235.86358635863586..... - 624.72358635863586......

=> 10000x - x = 6247235.86 - 624.72 ( since the repeating parts cancel each other)

=> 9999x = 6246611.14

=> x = 6246611.14/9999

Now, to finally eliminate the decimal point, we write

x = 624661114/999900

Every repeating decimal can be converted into a fraction form in this manner.




As above, we can convert every terminating or repeating decimal to a fractional form and vice-versa. Hence our claim is true.

Therefore the decimals which neither terminate, nor repeat, are not rationals. We call them irrational numbers.

Examples: 1.01001000100001000001......, -1.414213562373095048801688......., 2.7182818284590452........

Exercises for readers:



1) Prove that every decimal that terminates also has a repeating representation. Try to get the result by generalizing from the fact that 1=0.9999......

2) When in a fraction, we cancel out the common factors of the numerator and denominator, it is said to be in its simplest form. While referring to a rational number, we often refer to its simplest form. For example, 56/35 = 8/5 , 123/216 = 41/72. Prove that the rationals with terminating decimal representations are exactly the ones that have denominators (in the simplest form) whose prime-factorization consists only of 2s and 5s.

3) Prove that the square-root of 2 is irrational.

Continued at http://www.revleft.com/vb/showpost.php?p=1671202&postcount=71

[Rosa Lichtenstein]

Now that Red Cat has gone into lecturing mode, can a mod split this and move it to Research?

[red cat]

This will involve understanding Cantor's theorems and possibly examining proofs that claim to negate it.

Please do not move or split this thread. I have taken the topic back to the original purpose of the thread despite others' empty wordplay, and some posters have shown interest in mathematical discussion.

EDIT: Sorry, for a moment I thought research meant original work here. Still, given that the topic concerns purely mathematics, I think this thread should remain here.

[Rosa Lichtenstein]

In fact, such material belongs in Research, whatever you think.

[Meridian]

This is why I asked you to give an example. :)

I don't understand exactly what type of number you are talking about. If you tell me some numbers that fit the category that you have in your mind, I could answer your questions better.
Examples are whenever anyone tries to directly operate with infinite amounts, including infinite amounts of decimals.

These are non-ostensible, and can only be represented figuratively, i.e. metaphorically.

[red cat]

Examples are whenever anyone tries to directly operate with infinite amounts, including infinite amounts of decimals.

These are non-ostensible, and can only be represented figuratively, i.e. metaphorically.

Let us consider ( 3 + 2^0.5 ) and ( 3 - 2^0.5 ). Is your point that representing these two numbers in decimal form and then multiplying them might give inaccurate result while if we do the multiplication before we represent them in decimal form we will get the correct result?

[Meridian]

Let us consider ( 3 + 2^0.5 ) and ( 3 - 2^0.5 ). Is your point that representing these two numbers in decimal form and then multiplying them might give inaccurate result while if we do the multiplication before we represent them in decimal form we will get the correct result?
Go ahead, "represent them" in decimal form for me.

[red cat]

What I want to say is that decimal representations for these do exist, as in, we can find a finite algorithm that will compute the nth digit after decimal place for both of these two numbers, and any operations concerning them can be correct till the nth digit where n is supplied by the user.

For handwritten computations, of course, we will use elementary algebra to make things simpler, and this sort of algorithms are applicable to machines too, but whenever the formal notion of computability comes in, we are only interested in correctness up to a range supplied by the user.

So, the decimal representation of such irrationals should not be taboo to us.

[red cat]

Continued from http://www.revleft.com/vb/showpost.php?p=1668935&postcount=63

Mappings and bijections

Consider two sets (sets of any objects; we will go deep into set theory later to explore which type of set constructions make sense). Call one A and the other B. Now associate with each element of A exactly one element of B. This association is a map or a function. A and B are called the domain and range of the function respectively.

Example:

A={ rabbit, tiger, leopard, lion, human, baboon }
B={ carnivora, lagomorpha, rodenta, primata }
we can define a function f by the following association represented by tuples:
{(rabbit, lagomorpha) , (tiger, carnivora) , (leopard, carnivora) , (lion, carnivora) , (human, primata) , (baboon, primata)}



Now consider the following function:

A= { France, Spain, Canada, Afghanistan, Myanmar, Egypt, Brazil, Sudan, Vietnam }
B= { Asia, Europe, Africa, South America, North America}
f= { (France, Europe), (Spain, Europe), (Canada, North America), (Afghanistan, Asia), (Myanmar, Asia), (Egypt, Africa), (Brazil, South America), (Sudan, Africa), (Vietnam, Asia) }

In this function, for every element in B, there is at least one element in A that maps to it. In other words, every element in B has a pre-image in A. We call such functions surjective or onto functions.

Next example:

A= { dog, cat, duck, lion }
B= { kitten, puppy, calf, cub, duckling }
f= { (dog, puppy), (cat, kitten), (duck, duckling), (lion, cub) }

Here, every element in B has a unique pre-image in A. Such functions are called one-to-one or injective functions.

If a function is both surjective and injective, then it is called a bijective function.

Example:

A={0, 1, 2, 3, 4, 5, 6, 7}
B={8, 32, 128, 1, 4, 64, 2}
f= {(0, 1), (1, 2), (2, 4), (3, 8), (4, 16), (5, 32), (6, 64), (7, 128)}

Claim: If there exists a bijection between two finite sets, then they have equal cardinalities.

Exercise: Prove the above claim.

Continued at http://www.revleft.com/vb/showpost.php?p=1674160&postcount=75

[Rosa Lichtenstein]

Mods, why has this not been moved to Research?

[red cat]

Mods, why has this not been moved to Research?

Stop spamming this thread and PM some mod if you're so hell-bent on moving this thread to research.

[Rosa Lichtenstein]

Good idea!:)

[red cat]

Continued from http://www.revleft.com/vb/showpost.php?p=1671202&postcount=71

Bijections between some infinite sets


We will now see bijections within some infinite sets



Example 1:

A= N
B= Z

Bijection: Take an element x from A to (x-1)/2 if x is odd, otherwise to -x/2, in B.



Example 2:

A= N
B= Any infinite subset of N.

Bijection: Arrange the numbers in B in increasing order. Take x from A to the "X"th number in B arranged thus.

Exercise 1: Prove that if for any three sets A, B and C, A is bijective to B and B is bijective to C, then A is bijective to C.

Exercise 2: Prove that any infinite subset of Z is bijective to Z.



Example 3:

A= Z
B= Q

Bijection: Recall that every member of Q is an integer divided by a non-zero integer. Henceforth by a rational, we will refer to its simplest form. Now consider any rational of th form p/q. We also assume that if p/q < 0, then p<q, if p/q= 0, then p=0 and q=1, if p/q >0, then p>0 and q>0. Now map p/q in B to (2^p)*(3^q) if p is not negative. If p is negative, then map p/q to -(2^(-p))*(3^q). Note that the minus in front of p is to make its value positive. But clearly, the set of numbers that are thus bijective to Q is infinite, and a subset of Z. Therefore by exercises 1) and 2), Z is bijective to Q.

Exercise 3: From the above, show that N is bijective to Q.



Example 4: This is a more straight forward and elegant bijection between N and Q.

http://upload.wikimedia.org/wikipedia/commons/thumb/8/85/Diagonal_argument.svg/429px-Diagonal_argument.svg.png
Arrange all positive rationals in the given manner, cross out all but the simplest form of each rational, then draw a diagonal as shown in the picture. The "x"th black(not crossed out) rational along the diagonal is mapped to x in N. By now it shouldn't be difficult for readers to seee how this can be extended to the whole of Q. This beautiful construction is due to Cantor.


Thus we have proved that N, Z and Q have equal cardinalities.

Example 5:

A= (0, 1), as in all reals between 0 and 1, excluding both of them.
B= R

Exercise 4: Construct the bijection for this example.

Hint: Consider (0, 1). Bend it into a semicircle (where the two endpoints are missing. Now make this semicircle touch the real line from above, say at 0. Make sure that equal portions of the semi circle are now at both sides of zero, i.e. now it is rather in a balanced position on the real line. Now draw straight lines through the center of this semicircle to intersect the real line. Each such line will cut the semi-circle at a point too. That point in (0, 1) is mapped to the pointing R where the line cuts it. Try to write down a mathematical equation for this.


Continued at : http://www.revleft.com/vb/showpost.php?p=1674958&postcount=76

[red cat]

Continued from: http://www.revleft.com/vb/showpost.php?p=1674160&postcount=75

Meaning of countability and a theorem of Cantor.

A set S is said to be countable, if all of its members can be arranged in a sequence. This means that we have a sequence in which every term of S occurs at some point.

In other words, S is bijective to N.

Those infinite sets that are not countable are called uncountable.

Now we state a theorem of Cantor:



Theorem: R is uncountable.



Proof: We will assume that R is countable, and then find a contradiction.

Let us assume that R is countable. Then, as we(or rather you :))earlier showed that (0, 1) is bijective to R, (0, 1) is countable. Then there is a sequence in which all members of (0, 1) occur. Let that sequence be something like this:

0.333333......
0.453657647
0.978954723232323....
0.78
0.7654321111....
...
.
.
.

Consider the terminating decimals here to have a sequence of zeros in the end. So 0.78 should look like 0.7800....

We will construct a real K in (0, 1) that lies nowhere in this sequence. Here it is:

It starts with 0. (obviously), then its (K's) "x"th digit after the decimal point will be:

3 if the "x"th digit of the "x"th number in the sequence is not 3.
5 if the "x"th digit of the "x"th number in the sequence is 3.

For example, consider the sequence here. We reveal K upto the "x"th digit after decimal simultaneously:

1) 0.333333.................. K= 0.5.....
2) 0.45365764700.......... K= 0.53...
3) 0.978954723232323.... K= 0.533...
4) 0.7800..................... K= 0.5333...
5) 0.7654321111............. K= 0.53335....
.
.
.

Do you see why this number isn't in the sequence?

Because we have constructed it in such a way that its "x"th digit after decimal differs from the "x"th digit after decimal of the "x"th number in the sequence. In other words, this number K differs from every number in the sequence ! So K is not in the sequence.

Now, the way we constructed K, given any sequence of reals in (0, 1), the above property holds true. That is, for any such sequence, we can construct a K which is not in it. So, we have a contradiction to our initial assumption that (0, 1) is countable. Therefore (0, 1) must be uncountable. As it is bijective to R, R must be uncountable too.

This proves the theorem.

I hope everyone who followed this thread had no difficulty in understanding things so far. The theorem above is one of the theorems that some self proclaimed mathematicians call "rubbish". After we get familiar with more mathematics in this thread, we will try to negate some of their spurious proofs.

Next we will see a more general theorem of Cantor.

Continued at: http://www.revleft.com/vb/showpost.php?p=1675789&postcount=79

[Dimentio]

I was forced to read up on who Cantor was.

http://en.wikipedia.org/wiki/Georg_Cantor

Now I feel somewhat wiser. *yawn*

[Rosa Lichtenstein]

Dimentio -- exactly; wtf has this got to do with Marxism/Leftism?

Red Cat is just showing off how good he/she is at cutting and pasting stuff off the web.

[red cat]

Cantor's Theorem

Now we will discuss the more general Cantor's Theorem. The special case proved in the earlier post will also be derived.

Definitions: The power-set P(S) of a set S is the set of all subsets of S.

For infinite sets, we measure their cardinalities using cardinal numbers. By |A| for an infinite set A, we refer to its corresponding cardinal number. Two infinite sets are said to have the same cardinality( or cardinal number) if there is a bijection between the two. The cardinal number of an infinite set A is said to be strictly greater than that of another infinite set B if there is no surjection from B to A. Notice how this coincides with our notion of cardinality in case of finite sets.

Example 1:

Let S be the empty set, 0 (or phi, more commonly).

P(S) = { 0 }


Example 2:

S= {1, 2, 5 }

P(S)= { 0, {1}, {2}, {5}, {1, 2}, {1, 5}, {2, 5}, {1, 2, 5}}




Exercise 1: For any finite set S with cardinality n (will be denoted by |S|=n henceforth), prove that |P(S)|=2^n.

Claim: R = P(N)


Proof: We start with a small observation; if there is an injection from A to B, then |B| is at least |A|.

First we will construct an injection "f" from R to P(Q). For any x in R, f(x)={ set of all rationals less than x }.

Next we construct an injection from P(Q) to (0, 1). For any x in P(Q),
start constructing a real by the following method:

Our real f(x) starts with 0 followed by a decimal point. Now consider the sequence of all rationals. If the "n"th rational is in x, then the "n"th digit after decimal will be 3, and 5 otherwise.

Exercise 2: Complete the details of the proof and conclude that |P(N)|=|R|.


Cantor's theorem: For any set S, |P(S)|>|S|.


Proof: For finite sets, the solution to Exercise 1) will be the proof.

For infinite sets, let us assume the contrary. Then a surjection must exist from S to P(S). Let the surjection be f.

Notice that an element "x" of S might or might not be in f(x). Construct a set A = { all x such that x is not in f(x) }. Clearly, A is a subset of S. So, some "x" in S must map to A under f (since f is a surjection). But then, if x cannot be in A because if it is, then by the definition of A, x cannot be in f(x), that is, A. On the other hand, if x is not in A, then x is not in f(x) (which is A itself). But then by the definition of A, A must contain x. So we have a contradiction which proves such an f is not possible. Therefore the cardinal number of P(S) must be strictly greater than that of S.

From the two parts of the proof above, we conclude that |P(S)|>|S|.

We have now seen how simple and elegant the proof of this beautiful theorem is. Now we have to deal with the people who claim it to be "rubbish" and either present fallacious negations or nothing at all! :lol:

Note that the theorem and claim combined with the fact that any two countably infinite sets are bijective, proves the special case of Cantor's theorem given in my previous post.

Within the next few days I will try to disprove some "negations" of this theorem.

[Dimentio]

What does this have to do with disproving or proving dialectics?

[red cat]

What does this have to do with disproving or proving dialectics? Nothing. Just pointed out that certain anti-dialecticians don't understand mathematics either. :)

[Dimentio]

Nothing. Just pointed out that certain anti-dialecticians don't understand mathematics either. :)

Alright...

Should this thread be locked?

[black magick hustla]

Man, Cantor. I don't think Cantor is of any relevance beyond the most abstract of the abstract mathematicians who specialize in things like foundational theory

[red cat]

Alright...

Should this thread be locked?

Not now. I wish to go into more detailed mathematics and negate some counter proofs too.


Man, Cantor. I don't think Cantor is of any relevance beyond the most abstract of the abstract mathematicians who specialize in things like foundational theory

Actually Cantor's theorems have had their impact on branches of mathematics with widespread applications. The conclusions and techniques involved in the proofs are equally important.

[Dimentio]

But no one wants to debate mathematics with you. People want to debate dialectics.

[red cat]

But no one wants to debate mathematics with you.

Meridian already did.


People want to debate dialectics.

True. Still I want to have a thread to debate this theorem so that people claiming nonsense about the same have no excuse later.

Plus we should have some spurious proofs negated here.

[Rosa Lichtenstein]

Scaredy Cat:


Just pointed out that certain anti-dialecticians don't understand mathematics either.

Well, you certainly claim this, but we have yet to see your proof.:lol:

And, considering you did not understand the proofs I linked to, the presumption is that your comprehension of mathematics, beyond the basics of number and set theory, is not too good.:(

[red cat]

To all serious posters:

This thread has already seen a well described proof of Cantor's theorem. We will wait till the middle of August for someone to defend any mathematical negation of the same. If no one responds accordingly, then we shall proceed with some more mathematical and logical results related to the theorem.

In the meantime, if anyone has any queries regarding this proof, then feel free to ask here. PLEASE DO NOT MAKE POSTS DEVOID OF ANY SPECIFIC MATHEMATICAL QUERY OR RESULT RELEVANT TO CANTOR'S THEOREM .

Also, no matter how taunting, insulting, idiotic or worthless any troll-post may be, DO NOT FEED TROLLS.

[Rosa Lichtenstein]

On the other hand, genuinely materialist comrades can follow the links (posted at my site) to several refutations (which, oddly enough, Red Cat does not seem to be able to understand) of this mystical, Platonist 'theorem':

http://anti-dialectics.co.uk/Godel_letter.htm

And, as far as Red Cat here is concerned, 'troll' appears to be synonymous with 'anyone who disagrees with Red Cat'.