[[[MC]]] (title: What is your thinking on the following?)

His diagonal argument can show that |R| != |R|, since his method shows that for the set R he can construct a real number b such that b !E R.

[[[A♠]]]

No it doesn't. It shows that the set R is uncountable - you can't make a rule that lists all the real numbers, one after the other, and for any real number you can give its position in the list. How would you try applying it to R? Your "diagonal line" of digits would never hit most numbers, since not all numbers fit on the list.

[[[MC]]]

That is the point, right? I show that I can map every element in R, so how can he show a b E R that is not in my codomain if that is the same? :) Do you see what I'm getting at?

[[[A♠]]]

You haven't shown anything. By applying the diagonal method to your proposed bijection, you will find infinitely many numbers that you're missing.

[[[MC]]]

I am wondering thus the following. If I am able to show a bijective mapping g: R -> N. I mean this mapping g(x) gives a positive integer for any x E R. Thus R is the domain of that mapping. Then the inverse relation, g', is injunctive, and its codomain is R. Correct so far? Therefor, Cantor's diagonal argument cannot apply to my codomain, R?

[[[A♠]]]

Your "bijection" from N to R cannot exist. That's what Cantor's proof shows.

A bijection from R to R is perfectly fine. Cantor's diagonal only works on countable sets.

[[[MC]]]

My bijection is proven to have codomain of R. You propose you can apply the diagonal method to R? That is crazy talk!

[[[A♠]]]

Then show your proof. I can guarantee it'll be false, because Cantor has proven it's fakse. But right now, it's your word against hard mathematics.

[[[MC]]]

But don't you see? Consider this not from the subjective point of view that Cantor is infallible:

There is a proof that shows a bijection from R to N. Thus, Cantor's proof shows that his proof is a contradiction OR what you said. It only shows there is a contradiction in the model, that is the point of my conjecture, it is a PARADOX in the model! See what I mean? The model is relative as proved by Gödel.

[[[A♠]]]

You claim there is a proof, but you have not shown any proof. Please show the proof and explicitly describe the bijection. Your proof before was nonsense.

[[[MC]]]

I figured out a better way to word what I am talking about, paradox:

My proof shows that there is a function f:N->R. His proof shows that if the co-domain is not R you can show that there is no such function. My proof shows it is R, so his method doesn't work. I would say that doesn't mean his is wrong, it means ours together show a paradox with how we have defined infinite cardinality which is due to our il-defined number system (see: https://en.wikipedia.org/wiki/Natural_number#von_Neumann_construction).

[[[A♠]]]

Then show your proof. I can guarantee it'll be false, because Cantor has proven it's fakse. But right now, it's your word against hard mathematics.

[[[MC]]]

I agree that the proof isn't complete. I came to get the knowable unknowns known for me and fixed in the proof; thanks to many people's input, I have done that. Now I have to finish rewriting the function, and then that proof will be good enough for good mathematicians to understand the paradox it presents. Thanks for your help Ace!

However, it is not guaranteed false - such is guaranteed a naive statement to make. What I have discovered is called a paradox. It is a normal thing when the model is broken as Gödel proved to the nay sayers like DB6. There are many accepted paradoxes by science, and the concept is known and accepted (that our model is broken thus they have to exist). It is strange why so many mathematicians aren't aware of these basic facts that are well know now by smart people due to Gödel. (I'm talking about DB6 et al., not you.) It is obvious because this subreddit is mostly filled by pre-adolescent armchair mathematicians. It still surprises me the ignorance; obviously I will make sure to start a new math forum in my system I am building as /r/math is corrupted by immature children. I hope to see you in the new forum I am building to save the world by fixing moderation (no more brigades, downvoting because you are stupid and disagree, Etc.). Anyways, just wanted to say thanks and you seem pretty smart so keep up the pursuit of knowledge! Peace out!

[[[A♠]]]

The proof isn't just incomplete, it's absolute nonsense. Your statements do not follow from your premises, and you don't even prove what you wanted to.

Gödel said nothing about Cantor diagonalization. This is a profound misunderstanding of his work. You have not found a paradox since your proof is false. No matter what bijection you set up, I can guarantee that you are missing something. That is what Cantor showed. ANY function between N and R is not a bijection.

Science accepts no paradoxes in the sense of contradictions. None. Zero.

DB6 and the majority of /r/math are perfectly fine. We downvote not because we dislike you, but because you are incoherent, unwilling to listen to facts, and demonstrably wrong.

Gödel's theorem did not create any paradoxes. It simply states that no mathematical system can be both complete and consistent. There is nothing paradoxical about that.

/r/math is no more corrupted by immature children than the rest of Reddit. You will not save the world. Other people are not being ignorant when they disagree with you. And to be honest, your "new system" doesn't seem promising if it's as coherent as your "proof". An incomplete proof proves nothing, and Cantor already showed that your "proof" is incompletable since it tries to prove a false statement.

You are not creating a paradox. You are not revolutionizing anything. You are wrong. I do not say this out of blind faith or hatred, but out of knowledge. Mathematics, unlike science, is built upon absolute proof. Once a statement has been shown to be true, and there are no gaps, it is true and it is impossible to disprove, since that is what proof is. You are trying to argue with what we know is absolute fact: it's just not gonna work.

I'm sorry if I seemed harsh, but you don't seem to be understanding anything I've said. I don't want to be rude or mean - in fact, I admire your willingness to experiment - but you are completely deluded in what you think you have accomplished.

[[[MC]]]

It is okay, I didn't read what you said past the first bolded phrase because it started sounding like you can't get past the concept of paradox. I understand. You will just have to discover it for yourself. Thanks for your input though, I will remember you as helpful by ignoring you from that bolded point on until you do. :) Later.

[[[A♠]]]

Please read what I actually said. I'm trying to explain the flaw in your "proof".

[[[MC]]]

Sorry, to clarify, I then skipped forward to just after the horizontal line and read the first phrase before the comma (so good you said that) - hence my response. But peace out until then still.

[[[MC]]]

When you said something was 'absolute nonsense' you proved that you don't understand what Gödel proved already a long time ago. Thus you proved you don't know as deeply into the specific subset of knowledge that is necessary to understand the nature of paradox, what I'm talking about, Etc. Therefor, until you can either prove Gödel wrong, or show that you have learned why the statement 'absolute nonsense' is always invalid, then you aren't in a position of understanding to argue from about this specific topic. I do appreciate your answering my questions and trying to help. I have already found the only remaining question thanks to everyone here, so no more need to try to convince me of things I already know to be false. I am aware of those positions. I have read them, disproved them tomyself, Etc. I was looking for new stuff I had not, and that is where I am now, the definition of a number. Do you think it is absolute infallible certainty that 0 = {} ? Because that is what math says today. So I would say the contradiction is there. Gödel proved that is just as possible as in my proof. 1=1, Gödel proved that is consistent and complete. Addition is not, you cannot do it in a complete and consistent way. Any proof that requires addition, or multiplication, (arithmetic), is either incomplete or invalid. Incomplete means you cannot rationally say it is valid because it is proven incomplete (meaning unprovable)! You must go research this area of understanding first before you come to me with your current expertise. You are ahead of me in many areas, but your application of that knowledge is not valid in the domain I am investigating unless you are constraining it to the facts revealed by Gödel et al.

I wish you luck, I like objective people who disagree with me. We are more in common than many people who agree with me. For then you and I are the same: we both value the truth above all else.

Take care Ace!

[[[A♠]]]

I know what Gödel proved. **Gödel proved that no sufficiently

complicated system of axioms can be both complete and consistent.** That says NOTHING about any specific proofs. It does NOT say that any proof involving addition is incomplete or inconsistent. Proofs by themselves cannot be "incomplete or inconsistent" in the sense of Gödel: Gödel was talking about the properties of systems of axioms.

**Incompleteness means that there is a true statement that those axioms cannot prove.**

**Inconsistency means that there is a false statement that those axioms can prove.**

Neither of those says *anything* about making proofs with addition and multiplication. You can make absolutely true proofs that use addition or multiplication.

For instance, here is an absolute proof of the statement

"In a system using the Peano axioms, S(0) + S(S(0)) = S(S(S(0)))."

Here is a proof:

---

S(0) + S(S(0)) = S(S(0)) + S(0) by the definition that states a+S(b) = S(a+b).

... = S(S(S(0))) + 0, again by the definition that states a+S(b) = S(a+b).

... = S(S(S(0))), by the definition stating that a+0=a.

By transitivity of equality, we conclude that S(0) + S(S(0)) = S(S(S(0))).

---

I just proved, absolutely, that under the Peano axioms 1 + 2 was equal to 3. There is nothing incomplete about my proof; there are no holes in it. Cantor's proof is on that level. It has been checked time and time again, by mathematicians amateur and professional. Gödel said nothing about any individual proofs with his Incompleteness Theorem: he only made a statement about *systems of axioms*.

As for the "0={}" thing, that is only one of many models of Peano arithmetic. We have axiom systems (which tell us the "rules" we can play by, and pinpoint what we're talking about) and models of those axiom systems, which let us more concretely work with them. In most formalizations of mathematics, we adapt the model that defines 0 as the empty set, 1 as {{}}, 2 as { {{}}, {} }, and so on. That does not mean that 0 is the empty set. 0 is a symbol that can mean whatever we want. In Peano arithmetic, 0 is just a number with no predecessor - there's nothing in there about empty sets.

Representing 0 with the empty set is not a contradiction in any way. It is simply a definition of what is meant by the symbol "0" in a certain model of a certain set of axioms.

You seem to have several misunderstandings of what Gödel actually said. Gödel didn't say that there were *any* contradictions in mathematics. He just proved a claim about what certain systems of axioms are able to prove. If the axioms are the rules, and mathematics is the game, Gödel showed that no set of rules could get to every space without being able to go off the board.

I definitely appreciate the search for truth. However, what you're saying about Gödel's Incompleteness Theorem is plainly false, and it betrays a lack of understanding. I suggest rereading my bold explanation above, and if necessary, looking up more explanations, or even Gödel's original paper.

[[[MC]]]

First off, my apologies, I meant to say multiplication. Addition is possible, science does know that already. It is multiplication that is impossible. Gödel proved that very simply and no one who understands questions it.

I notice you mention Peano axioms? Consider the following quote from wikipedia (sorry no time right now):

"In 1931 and while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g., the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:

- If the system is consistent, it cannot be complete.

- The consistency of the axioms cannot be proven within the system."

Do you understand why I am not listening to your conjectures that my paradox are invalid, when your proofs are based on incomplete unprovable models? My proof shows a paradox in your model. Your axioms are pure hand waving. Show the whole system, complete. You can't. Are you saying Russell's paradox doesn't exist? Your system is broken my friend. You have to pick and choose which broken model you are using based upon what the problem is you are trying to apply it to - there's even no most correct model, never mind an actually correct one, LOL. Please understand this before arguing from your non-axioms.

So I am very sorry about my mistake there (saying addition) wasting your time with your proof there. If you are truly interested in this topic, then do it again but for multiplication - and you will see soon realize what I am talking about.

[[[MC]]]

You are clearly the one who didn't know enough about Gödel to realize my error when I said addition instead of multiplication. You seriously need to actually understand what he showed, as it invalidates your paradigm. I am sorry that means you have to rethink a lot, but it will be worth it to be correct, trust me.

[[[MC]]]

Observe that I noticed and fixed the error when copy pasting my message to you into another post (removing your name to speak to the general audience); whereas you wrote a whole proof and post and still didn't realize the error I said. Therefor, you likely had no clue of a basic fact of Gödel's incompleteness proofs. Therefor, you have little grounds to be asserting anything really.

You should really read up on it, it is very interesting. Seriously look at the first and second proof. They are incredibly basic actually, I guarantee you will understand the proof itself. It will likely take you a while to understand the many multitude of results however. It is fun going though, good luck!

[[[A♠]]]

Okay, here's a proof that 2×2 = 4 by the Peano axioms and definition of multiplication:

S(S(0)) × S(S(0)) = S(S(0)) + ( S(S(0)) × S(0) )

= S(S(0)) + S(S(0)) + ( S(S(0) × 0 )

= S(S(0)) + S(S(0)) + 0

= S(0) + S(S(S(0)))

= S(S(S(S(0))))

There. An absolute proof that 2×2=4. You are simply wrong about the multiplication thing too.

---

Proofs in incomplete models are *still proofs*. Just because there is some other statement that we can't prove doesn't mean that the ones we can are wrong.

Russell's Paradox is entirely unrelated to this.

>My proof shows a paradox in your model. Your axioms are pure hand waving. Show the whole system, complete. You can't.

That first statement is false, since you haven't actually shown any proof. The second is false: you don't seem to know what axioms are. The third is true, but I'm perfectly aware of it. **Just because there are some statements that we can't prove doesn't mean that we can't prove anything.**