Well I go for an extremist position of saying that everything is real i.e has some form of existence :) I prefer it to saying that nothing is real because then you can go down the road as to doubting your own existence.

If I'm counting squares then a squares a square. Squares combine to make other squares. How many squares are there on a chess board? its more than 64. I define a square as a shape with 4 linear sides of equal length. When I add them together I can create more from less. Somethings can be made from parts. A car is made of loads of parts but they still sum to make a car. A square can be made from parts but those parts can be squares :)

It depends on what you are adding together. 1+1 does not always equal 2. If your counting cuboids rather than squares, then you can add the cuboids together in such a way that they create 3.. It depends on what your unit is. If my unit is a square or a triangle for that matter then the result stands. Remember before you even get to the point of adding something together you have to have the identity that x=x but with some shapes this breaks down totally like triangles and squares i.e 4={4,5}. If i'm using circles for example then I can't arrange circles in any way that affects x=x no matter how I arrange them. With other shapes you can. All addition is is the act of adding 1 quantity to another and then summing the result. If I'm adding squares, triangles or cuboids then it all goes pear shaped so to speak..Your rats example is a different story. Your units are rats and initially the answer is 2 but this changes over time. With squares and triangles the result is instant and subject to other factors.

0.1 would be 1/2 to the right and 1/2 to the left to give us 1/4

1.1 would be 1/2 to the right and again 1/2 to the right to give us, again, 1/4

Both 0.1 and 1.1 are mapping on to the same fraction 1/4. — TheMadFool

01 goes first right which means it goes 1/2 the distance between 1/2 and 1 which are points on a line to first give 3/4. Then it left shifts halfway to the next point which is 1/2. This means it sits halfway between
1/2 and 3/4 which is 5/8. This value is halfway between 1/2 and 3/4. This is why I would normally use a square with a line going halfway down the middle. Each shift creates a new line and you only move halfway to your next line ok

(1) The sequence contains irrationals. The infinite sum remains rational.
(2) The sequence can consist only of rationals. The infinite sum can be irrational.
(3) The sequence can consist only of rationals, it can be strictly increasing or decreasing, but not converge in the rationals. (see 2)
(4) The sequence can consist only of irrationals, it can be strictly increasing or decreasing, but converge to a rational. — fdrake

Lets go through this proof again.

We have the sum 1=1/2 + sum of positive fractions - sum of negative fractions

So where does the 1 come from you might ask. Well if you take all shifts as positive regardless of direction then the distance the line travels = 1/2 +1/4 +1/8 etc which sums to 1.


Now this is equal to 1/2 + sum of the positive fractions - sum of negative fractions ok.

This gives 1/2 = sum of positive fractions - sum of negative fractions.

Now the sum 1/4 + 1/8 + 1/16 etc sums to a 1/2

I hope you would agree that no matter how I reorder this sum then it always = 1/2 ok
Also no fraction of the value 1/2^y will be irrational on its own
All numbers in the sum are rational
Now at all points in that sum the sum is always rational at any point we sum up to regardless of the order of the numbers. If this were not the case then the sum would not be rational.
If at all points in all sequences of this sum the product is rational then we can find a particular ordering with only the positive fractions being summed which will be rational from what has just been argued and with the remainder of the sequence being the negative fractions which must also sum to a rational value in order for the product (1/2) to remain the product.

Hence all irrational numbers are mapped to rational keys and my proof stands.

1.1 would give the number 11 in binary. This means we either shift twice to the left or twice to the right depending on what direction you give the 1,s and 0,s . Lets suppose its left. We start at 1/2 and left shift to 1/4 then we left shift again to 1/8

-1.1 would just be represented by the key -1/8

0.1 would shift right for zero going to 3/4 and then shift left to halfway between 3/4 and a 1/2

i.e A=0 B=1/2 C=1

This goes to point D halfway between B and C

A=0 B=1/2 D=3/4 C=1

The next shift is left so we have
A=0 B=1/2 E=5/8 D=3/4 C=1

So it terminates on 5/8

So at what point is a number of the form 1/2^y an irrational number.? I'm curious. The proof works if you understand it, what you've suggested is complete nonsence with regards to the sum you posted

Well here goes again

The sum is 1/2 + sum of positive fractions - negative fractions

The maximum sum of the positive fractions is a 1/2 from the sum 1/4 + 1/8 +1/16 etc

Now if the sum is a rational number then at no point in the sequence 1/4+1/8 etc can the sum be an irrational number otherwise the sum of the whole sequence will be irrational.

Now I can rearrange this sequence in all possible combinations and at all points in all these rearrangements the sum is rational. If it were not then the product would be irrational ok.

Now I can arrange the sequence 1/4+1/8+1/16 in such a way that the positive fractions in our sum are in a sequence. But this sequence has to sum to to a rational number from what was argued earlier. This means that the sum of the positive fractions is always rational...Hence the negative fractions are also rational because the produce is a 1/2.

Let me just clarify my proof to fdrake. The sum breaks down to 1 = 1/2 + sum of positive fractions - sum of negative fractions. The sum of the positive fractions is equal to a rational number and hence the negative fractions are also rational. Why is this the case, well here goes.

If we have the sequence 1/4 + 1/8 + 1/16 etc etc no matter how many numbers I remove from that list the product will be rational. If we sum them all up then the product is 1/2. If the sum of the total is rational than any total in the sequence at any point also has to be rational. If I knock out numbers from that sequence than that still leaves me with a rational total otherwise I could add up those numbers in a different order and somehow produce an irrational number. This is impossible from that perspective. If the sum of the total is rational than how can we subtract an amount from this to create an irrational number if the only numbers we are subtracting are rational numbers. So no I stand by my proof ;)

To be fair i will have to give that some thought as to whether I can produce a proof that they all terminate on rational numbers. Not an easy task I know. However the total length travelled by an irrational across the line is equal to a 1 if its an irrational number because each time we move between two points we are shifting the point by some 1/2^y. All those sums are rearrangments of the sum 1/2 + or - 1/4 + or -1/8 and so on. Now if the sum of the distance is always 1 which we get by simply measuring the length travelled regardless of whether we are left or right shifting then then the sum breaks down to 1 = (sum of positive fractions - sum of negative fractions). Both these values will be rational numbers because all they consist of are fractions of the form 1/2^y. .

Ok here we go remember we start at 1/2 and then shift either left or right by half. Lets say that 1 is left and 0 is right.

1 goes to 1/4 (i.e halfway between 0 and 1/2
-1 just gets represented as the key -1/4
0 right shifts from 1/2 to 3/4

No you have a valid point, however if I'm just halving distance between 2 rational numbers I'm not summing them like those sequences all do. Any number 1/2 way between 2 rational numbers will always be a rational number. I can just sum those 2 numbers and then divide by 2 and that will give me a new rational number

Err no. 0 would be a shift from 1/2 to either left or right depending on what direction you wanted to use. 1 maps to a left or right shift that would be the opposite of 0. This means that -1 would map to either -1/4 or -3/4 depending on what you did

Now that is an interesting equation. I would have to see the break down of all the calculations for that. If it is correct then I will accept your point

Ok the termination point of the number is a rational number, thats what I'm arguing. This rational number is a key for the binary number so to speak. I put the rational number in a set of other rational key numbers which as we all know can then be listed by the naturals. The convergence is to do with whether an infinite binary number would still converge on a rational key. You can think of the key as the address of the binary number.. The point is that each binary number will have a unique key that it terminates on regardless of its size , this is critical, this is what allows them to be listed.

Yes that is true from our perspective I agree with you on that however all I'm asking you to do is to not keep on projecting how our reality works onto a hypothetical universe which intuiitvely should be much more simple than this one. The point is that if conditions before the big bang were homogeneous and there was some sort of constraint that made this a condition then the only degree of freedom that I can think of which can occur in that state is a change in color of the background. What the homogeneous state is is not something I can say at the moment, questions as to whether its a vacuum or space are not important really in the context of what I'm saying

I think your splitting hairs personally. People can tend tend to get bogged down in details which actually mean nothing. I prefer geometric proofs which I can visualise. I will not apologise for doing that as I feel that is a much more powerful form of maths than anything I can scribble down on a piece of paper. If. If we cut through all of the banter it all boils down to whether you believe that a rational number + or - a rational number equals a rational number. No matter how many times I perform that operation it will always result in a rational number. The equation for fractions will prove that. So to throw the ball back in your court what changes when I perform that operation ad infinitum, well the honest answer is that nothing does.

That is a tricky proof, I will have to think about that however all the distances travelled are rational numbers. A rational number + or - a rational number is a rational number. So even if I do this infinetley it will end up on a rational number. I will again say that you are cutting Cantor more slack than me :) . Would you have asked him to have written out an infinite decimal number to prove his theory even though that's impossible..

only time will tell. I doubt you are correct. Why wouldn't that light escape the area that had the homogenous state. When light is released into a vaccuum it will probably head out of the original space quite possibly perpetually. — christian2017

Again whether the change in color is by light or the flying dutchman is not important. The conditions I'm talking about are not the same as the conditions we live in. One might ask where light comes from in the first place. What causes the colour might have nothing to do with light. We are talking about speculative conditions before the big bang

Remember each time the point moves between the halfway point of 2 rational numbers. This in itself would be a rational number. That's just simple maths. Now I understand what your saying but even if I do that infinetley it should end up on a rational number. Now I will admit I can't be 100 certain of that so that is a bone of contention.

It would converge on a rational number just as the sequence 1/2+1/4+1/8 converges on 1 after an infinite number of steps. Don't get me wrong, it would be quite fiendish to try and calculate where those numbers terminate but in theory they would

Those numbers were originally spaced out alot more in my post. The terminating point is F which is 5/16

I will explain the proof again. I am aware that my presentation of it is not brilliant so here goes.

Firstly Cantor himself uses infinite decimal numbers in his diagonal argument so I don't understand why you cut him some slack but not me :)

Now first convert the real number into a binary number. We can convert all numbers to base 2 people if we need to.or we could just turn each digit into a base 2 representation

We draw a line of length 1. Point A = 0 Point B = 1/2 and point C = 1

To keep this simple I will use the binary string 101.

A B C
0 1/2 1

Now if the digit in your number is a 1 we left shift 1/2 the distance to the next point.
If the digit is 0 right shift 1/2 the distance to the next right point.

We start at B
So the first digit is 1. We left shift to the new point D

A D B C
0 1/4 1/2 1

Next digit is 0 so we right shift from D halfway to the next point which in this case is B


A D E B C
0 1/4 3/8 1/2 1


Next digit is 1 so we left shift from E halfway to the next point which in this case is D

A D F E B C
0 1/4 5/16 3/8 1/2 1

Number terminates here at 5/16 which is a rational number a/b.
Each binary number will terminate on its own unique rational a/b as will each irrational number. This can then be mapped into the natural numbers, in other words each number fed into the system comes out with a rational key regardless of how many numbers you feed into it.

If you don't understand how my proof works then just say so. I am well aware of that sort of stuff, I did study maths for my degree but unlike you it seems I do question all the assumptions that I was taught and if I think something does not make sense then I will think about it. I mean you are aware that set theory has been troubled by paradoxes in the past right. This is not really anything new from that perspective

Great! Now try π. — tim wood

Just to explain myself more clearly with pi you can simply covert each digit of pi into binary and do this with all the other numbers as well to generate your string.
i.e 3.141 etc = 3 in binary+1 in binary +4 in binary + 1 in binary etc etc

Great! Now try π. — tim wood

Pi would just have a never ending trail of digits but the procedure is exactly the same. That's just an infinite binary string.Cantors argument relies on numbers with infinite numbers of decimals and thats what he uses in his argument as well so if you want to cobble me for that then you have to cobble Cantor as well.Cantors argument specifically relies on having infinite strings with his slash argument. It wont work with finite strings because these can all be converted into rational fractions which we can list

When something changes color there are moving parts. Colors are produced by different wavelengths of light. Visible light is above Infrared and below Ultra Violet. — christian2017

Your getting bogged down in projecting what we currently understand to a situation that would predate the laws that we currently experience. The situation that I proposed is a universe where the only degree of freedom is an instant and total color change. What other parameters. that would be neccessary to produce this I do not know but there would be no moving parts in a homogeneous state. There would be no things to move. For example if I handed you a blank piece of paper and said whats on that paper you would say nothing. If I draw a line or a boundary then there is something because that breaks the homogeny. All I'm pointing out is a theoretical way that you could produce a clock from a homogeneous state.under a certain set of circumstances

How? — tim wood

Well the number 51.32 would be broken up into the binary for 51 and the binary for 32 and then combined into a single string.

That's what this should be about. Either all the mathematicians since Cantor have been idiots and retarded to not notice the fatal flaw in the proof or OP is wrong. What's the probability for each case. — Wittgenstein

Well rather than trying to deal with phantom probabilities why don't you just read the proof. The point you are forgetting is that how many people have tried to debunk Cantors proof after it was published, most people just accept it as fact. The logic is very water tight but set theory has produced paradoxes in the past and this just might be one of those cases.I only stumbled on it when i was working on a different problem.I am not closed to it being wrong it just I don't see many people trying to understand the proof. I can explain it in an even simpler way if need be.

Well the theoretical phycists are welcome to have their opinion. But how can that be the case if the universe is finite in size as it would be if it were expanding from the big bang. That just sounds like alot of mumbo jumbo to me.

If someone came up to me and presented a proof that 1=2, l would immediately discard the proof. The OP obviously didn't present something that ridiculous but it does amount to saying that the prove Cantor gave was wrong as it proves the opposite.There isn't a third possibility here. It isn't about herd mentality here since it is mathematics.In mathematics, we stand on the shoulders of giants and it does not tolerate any weakness that we find in philosophy, religion or social sciences. I understand where you are coming from but you have to see for yourself that in this sub section, we need to be more objective and avoid beating around the bush as we normally do — Wittgenstein

I would be a bit cautious about statements about dismissing proofs like that. I can easily produce an argument using triangles and how they combine that proves 1+1 = 3. I would even go as far to say that using a very simple technique that involves very simple geometry and counting that 4=5 in some circumstances.. People get fixated with what they treat as absolutes and they will come a cropper later.

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Well it seems theoretical physics seems to disagree with you there. The infinitely iterated infinite universe is exactly what they are proposing. — ovdtogt

eply="ovdtogt;360080"]

Sorry but mapping an infinite number of infinities to 1 infinity is easy.

Here is how you do it.

Create from the natural numbers lists composed only of a prime number and its powers. for each prime number. This creates an infinite number of lists each with an infinite number of numbers with no number in any list overlapping. I can repeat this process again with each of those lists by mapping them back to the total number of natural numbers and creating a 2nd tier of infinite infinities. I can do that infinetly All of this though is contained in the infinity of the natural numbers.

I think your procedure does produce an injection between the sets, but the initial set you're feeding into the injection is actually uncountable. You're mapping the real numbers to the real numbers rather than the real numbers to the rationals.

If you wanna see this, there's an uncountable infinity of real numbers whose first digit right of the decimal point is 1 in the binary expansion. The same goes for any binary digit. I think you're not registering the distinction between "the set of sets of real numbers with x in a given digit in their decimal expansion" and "this real number has x with a given digit in their decimal expansion". — fdrake

Well as far as I can tell any number fed into this procedure should result in a terminating rational length which will produce a set of rationals which map to the natural numbers regardless of whether it is an irrational number or not. Now I understand the point that you could argue that the set your feeding in is uncountable but this leaves us in a strange position because the two proofs directly contradict. I think its a problem with infinity personally. Now a better way to do this would be to feed in just the values between 0 and 1. You then have terminating values a/b for +1/x -a/b for -1/x, ai/b for x and -ai/b for -x which can all be grouped and listed hence covering the reals. I must admit to being a little hasty in thinking it could map the complex numbers. I can interleave the two parts of a complex to create a single unique value which covers a +bi and -a-bi but as of yet I cannot map in -a+bi and a-bi cannot be covered.

If you think about it though Cantors proof is really paradoxical because if you have an infinite quantity then the only thing that determines your ability to list those values is the algorithm you use to sort the numbers. .

Just go through the proof. Then talk to me

I would just use a -a/b value and then list that next to its a/b twin

No, I haven't read your proof. I don't need to, because I have read and understood Cantor's diagonal proof. That's all I need to know that Cantor is right. Unless you can show how the diagonal proof is wrong, Cantor's result stands. — SophistiCat

How odd, you dismiss an argument you don't understand and don't even try to. That sounds like some sort of dogma to me. The argument once you understand it is very straight forward and you can put the reals into a 1-1 correspondance with the natural numbers using it

Err no because none of the a/b numbers are irrational numbers. hence his diagonal argument wont work. The a/b numbers are all rational numbers and therefore can be listed as rationals can

Sure its not an exact description of "nothing" in the absolute sense but if there was nothing in the beginning then I don't think we would be having this conversation as there would be nothing to cause the big bang in the truest sense. This is about as simple a structure as I think you can get to besides a reality that would be both homogeneous in both space and time in which case absolutely nothing would happen at all.

Have you understood the proof?. I can put the reals into a 1-1 correspondence with the natural numbers. That proves Cantor is wrong. I can also put the complex numbers into a 1-1 correspondence with the natural numbers. If someone finds a way to list the reals then cantors work collapses and that i have done. IF there is an error in the proof then show me, I am completely open to it being wrong if someone can demonstrate this

That should have been point F that we measured the line from not E