The set of real numbers is uncountable, and by uncountable we mean that we cannot establish 1-1 correspondence with the set of natural numbers.

You can reject the axiom of infinity and then proceed to say that an infinite set does not exit but if you accept this axiom, cantor's proof of the fact that the cardinality of the uncountable set is greater than the cardinality of the countable set remains true.

If you'd read my proof you would see that you can create a 1-1 correspondence with the natural numbers and the reals. If there is something you do not understand about the proof let me know

If infinity exists then by definition an infinity of infinities must exist.

Err no. I only need one infinite quantity to map anything. If you can't map something when you have an infinite quantity then you are doing something wrong. Now you can have an infinity of infinities but I can map that to a single infinity

Just to be clear I will reiterate the proof in slightly more detail

We will map the number 5.7
First we turn the 5 into binary i.e 101. Then we map the .7 which equals 111
this creates the number 101.111

Now we draw a line of length 1 with the halfway point marked. We can represent this by three points

A B C

B is the halfway point.
Now we map the number. If the digit is a 1 then we left shift from our current point half way to the next line. If it is a 0 we right shift

101.111

We start at B

First digit is a 1 so we left shift halfway to the next point and create point D as follows

A D B C

D is halfway between A and B

Next digit is a 0 so we right shift halfway to the next point which is B giving

A D E B C

Next digit is a 1 so we left shift halfway to the next point which is D giving

A D F E B C

For arguments sake we will say the line terminates here at point E

Now I measure the distance from A to E. This distance will be some multiple of a 1/2 x some a/b

We know this has to be the case because we are always dividing our distances by 1/2 so the final distance will be some multiple of a 1/2 x a/b. This will be true even if the number has an infinite number of digits

Now each number will produce a unique a/b coordinate.

This produces a set of fractions that I can then list in a 1-1 correspondence with the natural numbers

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

We can now order our numbers and put all these numbers in a list because each has a unique a/b coordinate. This would order all the reals — Umonsarmon

And then Cantor can read down your list and diagonally generate a real number that isn't already on it.

Cantor's proof begins with a supposition that you've somehow produced a supposedly complete list of all the reals, and then shows how from such a list you can always generate yet another one that wasn't already on it.

First, to be clear, your argument is not, strictly speaking, against what you wrote in the title, that there isn't "more than one infinity," because you can't talk strictly about something as vague as that. Your argument is specifically directed against Cantor's proof that the reals cannot be put into a one-to-one correspondence with the integers. Which in Cantor's set theory means that the real set has a higher cardinality than the integer set. Which, informally, can be interpreted as the infinite real set being "bigger" than the infinite integer set, since cardinality is a generalization of the concept of set size.

Second, Cantor's proof is a theorem. The only way that you can invalidate it is if you find an error in his proof (good luck with that). If you cannot do that, then the right question to ask is not "Is Cantor wrong?" but "How is my proof wrong?" Is this what you wanted to ask?

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

Have you understood the proof? — Umonsarmon

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.

Just so you know, there's a bazillion cranks out there doing just what you are trying to do: attempting to prove Cantor wrong by proving something contrary to his result. They've been at it for decades: even before the Internet they've been inundating mathematicians and mathematical journals with their proofs. It is something like a perpetuum mobile for mathematical cranks. But none of them have managed to invalidate Cantor's proof yet.

Also a tip, since you are new on the forum: if you reply by clicking what looks like a crooked arrow underneath a post, or select some section of text and click on the "quote" prompt, then the person to whom you reply will get a notification about a reply in this thread.

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

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

How would you represent negative real numbers?

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

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. — Umonsarmon

It's not odd and it's not dogma. It's just straightforward logic: If Cantor's proof is correct, then his result is a theorem and therefore it is right. Cantor's proof is demonstrably correct, therefore his result is a theorem. You do understand that your result cannot be right if a theorem exists, according to which it is wrong, do you? If you are so confident about your result, then show us how Cantor's proof is wrong. It's as simple as that.

Just go through the proof. Then talk to me

How odd, you dismiss an argument you don't understand and don't even try to. — Umonsarmon

It's not odd and it's not dogma. It's just straightforward logic: — SophistiCat

Actually, if two proofs prove contradictory things, then there is a problem with one or both of the proofs. To say I understand one, but not the other, and I accept the one that I understand, therefore the other is wrong, as SophistiCat did, is illogical because the acceptance of the one may be based in a failure to see that its unsound, a mistaken understanding. Until you can exclude the possibility of mistake from your understanding, it is illogical to reject demonstrations which would show that your understanding is mistaken. .

3) Starting at the middle line do the following .If the digit in your string is a 1 move half the distance to the next line to the right. If the digit is a 0 move half the distance to the next line to the left. — Umonsarmon

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".



Sophisticat is right though. The diagonal argument does establish that no injection from the reals to the rationals exists. If your claim is correct, set theory is inconsistent (as it proves a contradiction), but it is provably consistent (within larger theories).

I'm not sure if the following is a proof that cantor is wrong about there being more than one type of infinity — Umonsarmon

Even if you grant your whole argument, it doesn't stop Cantor's theorem about powersets and cardinality from going through. And the broader claim about consistency rears its head again.

How is my proof wrong?

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.

What's the probability for each case. — Wittgenstein

50/50.

" Err no. I only need one infinite quantity to map anything. If you can't map something when you have an infinite quantity then you are doing something wrong. Now you can have an infinity of infinities but I can map that to a single infinity "


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

Sure, if you take Cantor and OP. If you take the countless mathematicians on one hand and OP on other hand, the outcome isn't that favourable. :wink:

Actually, there is a third possibility, as I said, and that is that both Cantor and the op are wrong. This raises the probability that the mathematicians are idiots to about 67 percent. The number of mathematicians involved is irrelevant, as herd mentality demonstrates. The number of people carrying out an action has no bearing on whether the action is correct.

Actually, if two proofs prove contradictory things, then there is a problem with one or both of the proofs. To say I understand one, but not the other, and I accept the one that I understand, therefore the other is wrong, as SophistiCat did, is illogical because the acceptance of the one may be based in a failure to see that its unsound, a mistaken understanding. Until you can exclude the possibility of mistake from your understanding, it is illogical to reject demonstrations which would show that your understanding is mistaken. — Metaphysician Undercover

By the same token, my hypothetical acceptance of the contrary proof could be "based in a failure to see that its unsound, a mistaken understanding." If that's the standard by which you propose to decide between the two proofs, then it cannot resolve anything. Examining the other proof wouldn't tell me anything that I didn't already know: that whatever opinion I form about the soundness of each proof, it might be mistaken.

Of course the possibility of mistake still exists, and if "resolution" requires removing that possibility, it would remain unresolved. But that's why I retained the option that both are wrong. However, broadening the mind to numerous options, presented from numerous different perspectives increases one's understanding of the subject, thereby lowering the probability of misunderstanding, even when there is no "resolution".. Therefore you are wrong to assert that doing this wouldn't tell you anything you didn't already know. The mistake is to assume that there is a "resolution" when there is not.

Is Cantor wrong about more than one infinity

Yes. Theoretical physics seems to demonstrate the infinitely iterated infinite universe.

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.

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. .

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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.

Err no. I only need one infinite quantity to map anything.[/quote]

If that were the case you would only need 1 infinite Universe to map our Universe.

Well it seems theoretical physics seems to disagree with you there. Apparently you need infinitely iterated universes to map our Universe.