That is, there are real numbers not mapped to. — TonesInDeepFreeze

You don't need the diagonal proof to realize that. Every real number can be mapped from N^3. Every real number can be reached from N^3. N^3 can be mapped onto R.

You are exclusionary and dogmatic. — AgentTangarine

You have not shown any dogmatism by me. Nor any exclusion other than of ignorant confusion and misinformation.

That's not what the proof is about. It just shows that [0-1]is uncountable. — AgentTangarine

It shows that [0 1] is uncountable by showing that any map from N to [0 1] is not onto [0 1]..

Just now, and in the other thread that was deleted yesterday — TonesInDeepFreeze

On the contrary. I even told you I contradict myself in previous posts. I never told you to consider other posts. Anyhow... I'm truly tired and my beloved has awoken. Damned! 7 hours about infinities.. I'm off to bed. Gonna contemplate about mr. Gill's procedure. It was fun! :smile:

You don't need the diagonal proof to realize that. — AgentTangarine

True, there are other proofs of the uncountability of [0 1]. Cantor gave one of those other proofs.

Every real number can be mapped from N^3 — AgentTangarine

There is no map from N^3 onto R. And even if there were, it would prove the countability of R not the uncountability. You are again completely backwards and confused.

You have not shown any dogmatism by my. Nor any exclusion other than of ignorant confusion and misinformation. — TonesInDeepFreeze

Yeah, you are right about that! Sorry that I called you dogmatic and exclusionary! You certainly got me interested in this aleph topic! As a physicist I find it difficult to believe that the number of points on a line is the same as on a plane or in a volume. The number of directions are different though. Or not even that?

I never told you to consider other posts. — AgentTangarine

By "the rest of what you posted" I meant the rest of what you posted in that post, just as I was responding exactly to your complaint that I hadn't quoted more of your post.

Then I bet you really would not like Banach-Tarski.

Anyway, stepping back, do you understand the proof of the equinumerosity of N and NxN?

There is no map from N^3 onto R. — TonesInDeepFreeze

You can map N to all reals between 0.1 and 0.999999...
You can do this N times (for all smaller decimals). You can do this for all N size one intervals. Where am I wrong?

Then I bet you really would not like Banach-Tarski. — TonesInDeepFreeze

In fact, I like that theorem!

Example? — TonesInDeepFreeze

Off the top of my head, infinitesimals.

The definition you are seeing is the formal aspect. It's a kind of final touch to an idea that began as an interesting notion. — jgill

Yep, the formalism is necessary to ground what started off as an intuition, in logic; a necessity no doubt; after all, in math not having a crisp definition trying to find a missing person without either having a good description or photo of that person.

My point is that making a definition precise (usually) means losing some/all of the feelings that go with the intuition. The transition from just a vague notion to a clear-cut, well-defined idea is, for me, a heart-to-brain relocation of an idea and that's what bothers me. Math's link to the heart pops up occasionally though - I've heard of mathematicians being moved by the elegance and beauty of some formulae for example. I have a book titled The Heart of Mathematics which is on my reading list.

You can map N to all reals between 0.1 and 0.999999... — AgentTangarine

You keep saying that. It's dogmatism.

Where am I wrong? — AgentTangarine

In thinking that the fact that in your own mind you imagine that it must be so implies a mathematical proof. And in thinking that your disconnected and mathematically unsyntactical dribblings are too mathematical proof.

Yet you reject other theorems from the same axioms.

I think it's the continuum that confuses me, and its break-up into (onto?) its parts. I think the break-up of a square into lines is the same as a line into points. But maybe the break-up of a square into points is the same as a line into points.

In thinking that the fact that in your own mind you imagine that it must be so implies a mathematical proof. — TonesInDeepFreeze

I mean, where am I wrong if I say that N^3 can be mapped on R?

infinitesimals. — Agent Smith

That's almost a good example. But it's better described as the centuries-ago formulations being more than vague intuitions yet not adequately formalized. Then, centuries later, it was discovered with mathematical logic and model theory how to vindicate the notion rigorously.

Of course, one can look back centuries, even to the ancients, to see that their mathematics has seen been formalized.

What I thought you had in mind though are cases where mathematicians had vague notions and then they or their contemporaries formalized those notions themselves.

It seems, as always, mine is a case of one-sided love. Math, for some reason, refuses to let me unravel its secrets. Cosmic censorship or I'm just plain stupid! Likely the latter. Good day. See you around!

making a definition precise (usually) means losing some/all of the feelings that go with the intuition. — Agent Smith

(1) How do you know that unless you've interviewed mathematicians about it?

(2) I highly doubt that mathematicians very much regret whatever such loss of feelings there might be, as I would think mathematicians are primarily eager to communicate their notions clearly and objectively to other mathematicians and to prove their results.

Math, for some reason, refuses to let me unravel its secrets. — Agent Smith

I cherish the mysteries of mathematics. That is not impaired by formalization. On the contrary, formalization leads to even deeper mysteries for me.

where am I wrong if I say that N^3 can be mapped on R? — AgentTangarine

I told you. You don't have proof of it. You only think you do.

Also, you have a serious problem when it's pointed out to you that you are in contradiction with yourself and your response is "So?".

I have nothing more to say (for now). I'll get back to you later.

think it's the continuum that confuses me, — AgentTangarine

Start by adopting a specific definition of 'the continuum'. The term is often used flexibly, but I would settle on the continuum understood to be the pair of the set of reals with the standard ordering on the reals:

c = <R less_than_on_R>

Or, if you prefer to think of it as merely the x-axis:

{<x 0> | x e R}

Or when we refer to "the cardinality of the continuum" we are thinking of the continuum as merely the set of real numbers.

I told you. You don't have proof of it. You only think you do. — TonesInDeepFreeze

But where am I wrong in my proof? Cannot N be mapped onto 0.1-1? You ñeed N numbers for that: 1-99999999.... What number do I leave out here? Or do I leave numbers out between 0.1-0.9999999....?

Cannot N be mapped onto 0.1-1? — AgentTangarine

Do you mean to suggest that there is a 1-1 function from N onto 0?

By the way, you claimed that I have no sense of humor. Well, you haven't said anything funny. Neither have I very much, since I don't find this context with you to motivate me to make jokes. That is not a lack of sense of humor. You don't know me.

By the way, you claimed that I have no sense of humor — TonesInDeepFreeze

I was a bit angry when I wrote that! Forget it! I don't even know you!

There is a 1-1 function from N onto [0.1-1] (the interval from 0.1 to 1). Isn't there?

Given any function f from N to [0 1], the diagonal proof constructs a member of [0 1] that is not in the range of f.

I feel pretty safe in thinking that you don't know the diagonal proof.

Moreover, even if the diagonal proof were found to be incorrect (it won't be) then that would not constitute a proof of its negation.

I've told you a couple of times already: To prove your claim, you must prove that there is a function whose domain is N and whose range is R and that it is 1-1. And I won't even ask that your proof be constructive by showing a particular such function, only that one exists, even though Cantor's proof is constructive: showing, for any given function from N to R, a particular real number not in the range of that function.

The diagonol proof doesn’t apply here. I was thinkiñg that too. The interval I talk about is 0.1 to 1. Not zero to 1.

You say it doesn't apply, yet you mentioned it directly in connection.

I might think you are trolling me, but even worst trolls don't usually have your endurance.

Okay, last time. Between 0.1 and 0.99999.... you use all numbers of N; 1,2,3,....9999999..... Are there more numbers?

It's a 1-1 map!

Strangely enough, you need more numbers between 0 and 0.1.

I asked you first:

Cannot N be mapped onto 0.1-1?
— AgentTangarine

Do you mean to suggest that there is a 1-1 function from N onto 0? — TonesInDeepFreeze

Do you mean to suggest that there is a 1-1 function from N onto 0? — TonesInDeepFreeze

No. I mean: why can't N be mapped onto 0.1-1 (or 0.1-0.9999999999.....). I use every member of N one time.