f is onto Y if and only if (f is a function & range(f)= Y) — TonesInDeepFreeze

Indeed. So N can't be mapped 1-1 onto. You can at most map N to infinite points, infinite times. Say 1 into 2, 2 into 4, 3 into 6, etc. But into still. Then you still need to map into all inbetween intervals. On each interval an infinity of infinities of naturals is needed. So you need inf^3 times to map N into R. Pffffff.... I'm done! Thanks for the resistence! :smile:

I can map N onto R infinite times infinite times.

Infinite times onto

[0.1-0.99999...]
[0.01-0.09999...]
[0.001-0.009999...]
.
.
.
So infinite times onto [0-1]. The map even defines [0-1].

Times infinity for all length 1 intervals.

So in total there is an inf^3 involved. Hence aleph1.4. Merry Christmas!

No. You are right. Onto only for N^3! But into inf^3 times. I get sleepy.

Which means aleph1.4 is the one for R, and aleph2.6 is the one for RxR.

So? Not everything on the Internet is the sharpest formulation — TonesInDeepFreeze

There you go.

R is onto Y if and only if (R is a relation & Az(zeY -> Ej <j z> e R)) — TonesInDeepFreeze

Huh?

Anyway, now you can see why there is no function f with a singleton domain an d onto R. — TonesInDeepFreeze

Still, I can map N onto R. Inf^3 times even...

That's not quite right or at best awkward, — TonesInDeepFreeze

I got it from the net...

You guys need to get a hotel room. — TonesInDeepFreeze

Haha! In Hilbert's hotel! Wrong post...

Are you off your meds again? — jgill

I now even think R has cardinality aleph1.4!

You guys need to get a hotel room. — jgill

Haha! In Hilbert's hotel!

"The mapping of 'f' is said to be onto if every element of Y is the f-image of at least one element of X."

Y=R, N=X...

So which is it now? You claim that card(R) = aleph_1, thus asserting the continuum hypothesis? Or you deny that card(R) = aleph_1, thus denying the continuum hypothesis — TonesInDeepFreeze

No. I assert card(R)=aleph1.4. Call me crazy, like you wish. Card(RxR)=aleph2.6, more or less. 2^(1.4)=3, more or less, 2^(2.6)=6, more or less.

Then either you are insane or you don't know the meaning of 'map onto'. — TonesInDeepFreeze

I can send every cardinal on top of R. On top where I send it into.

I think we have discovered a real infinity in life, which is what this post is about. You and me commenting on each other. I may be in flat contradiction with what I thought previously, that's right. But not with my present self. But for sure with you....

So you are in flat out contradiction with yourself. — TonesInDeepFreeze

So?

Of course one can map a singleton set INTO R.

But there is no map of N ONTO R — TonesInDeepFreeze

I can map a singleton on R too. No problem.

Yet you say you disagree with the continuum hypothesis now, while you have been claiming it in dozens and dozens of posts — TonesInDeepFreeze

So?

The number 0 is not a riddle. — TonesInDeepFreeze

But if you say it's the number of times you can map it on R it is. I can even map a single element of N to R.

I looked it up when I wrote about the alephs, and it was even the reason for my post. Seems years ago.

So what? This is not a game with a scoreboard for how many questions I've answered. — TonesInDeepFreeze

Who says it is? I just said you have answered with a riddle.

So years ago you looked it up, but still don't understand it now. — TonesInDeepFreeze

I don't mean that litteraly.

I can map N infinite times infinite times on R.

I really would rather not know about the bedtimes of you and your wife. — TonesInDeepFreeze

Then ignore it. The only riddle you gave as an actual answer is that you can map N zero times on R.

I'm not surprised that your attention span wouldn't provide recalling my question: Why won't you look up 'continuum hypothesis' on the Internet? — TonesInDeepFreeze

I have done that years ago already. I don't agree though. I'm not a parrot like you.

Liza is getting tired and reduced to token replies. — TonesInDeepFreeze

Now you take refugee behind empty verbiage. Liza is tired indeed. My wife lies in bed for a couple of hours already... I told her to come too. Three hours ago...

That you don't understand the answer is not my problem. I did answer it. Now your turn to answer my question,. — TonesInDeepFreeze

That's no answer why it's false. You seriously believe you can map N zero times on R? What's your question?

The claim you made, as I quoted it, silly. — TonesInDeepFreeze

But why false?

Do you mean N onto R?

The answer is 0. — TonesInDeepFreeze

Ah, I didn't see that one! 0 times? Are you kidding?

Not the zillionth time, but it is the second time you have made that false claim. — TonesInDeepFreeze

Then what is false?

No one ever told you that mathematics is not throwing around words like 'conformal' while not knowing what they mean'. — TonesInDeepFreeze

Everyone knows what conformal means.

I answered one of your questions. — TonesInDeepFreeze

You haven't answered one!

You don't know what you're doing — TonesInDeepFreeze

Yeah, that's the zillionth time you mentioned this. I haven't seen one thing to conclude you do. You only make references.

That is purely arbitrary unfounded assertion. — TonesInDeepFreeze

If you consider the infinite line the discrete elements on the plane, like the natural numbers on R, than the relation between the lines and RxR, the infinite plane, is conformal to the relation between N and R.

You contradict yourself. You claim there is a bijection from N onto R, but above you admit that the interval [0 1] is uncountable. — TonesInDeepFreeze

You have to read what I write. There are infinite bijections between between N and [0-1]. Like there are an infinite times infinite between N and R. And there are an infinite times infinite between R and RxR.

Wrong. For any infinite set S. — TonesInDeepFreeze

And that's exactly where the failure lies. The relation between R and RxR is the same as the relation between R and N. R can be viewed as corresponding so a single cardinal. How many lines you need to construct RxR? The same number as the number of cardinals to construct R.

It's not so complicated as they make it. There are basically two types of infinities. Infinite countable infinities, like N and infinite times infinite types. Like the interval [0-1]. Uncountable. All alephs are multiplicities of them.

I have been referencing the more general theorem that for an infinite set S and natural number n>0, we have card(S) = card(S^n). — TonesInDeepFreeze

That's for countable sets. For R this doesn't hold.

I just asked you the question how many times you can map N on R.

How can his network be different if he believes the door is open? Is he lying? Have we wrongly identified his network? If he's not lying then his believe part should be the same.

If we examine person 1001, who claims to believe that the door is open, and do not find in them that specific neural network, do we conclude that we have not identified the correct specific network, or do we conclude that they do not really believe the door is open? — Banno

Are the other 1000 all the same, under the same conditions? If he believes the door is open, why should his network be different? Is he lying?

Sorry, I don't know what you're saying — frank

Ecactly what I wrote. It's plain English. You can exclude the worm and still see it with closed eyes and imagination.

Okay, N^2, as you wish. Same as inf^2.

Show the correct answer then. I already gave you a link to a supposed bijection between R and RxR. How many times can N be mapped on the real line? Just offer a tasty recipe.