I got it from the net... — AgentTangarine

So? Not everything on the Internet is the sharpest formulation.

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

There you go.

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.

I redid the post.

There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject.

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:

There you go what? I am the first to say that one has to use great caution trying to pick up math on the Internet. There are some excellent Internet sources, but usually the best approach is in books. I recommended the Internet to you only because I know you wouldn't bother to read a proper book on this subject. — TonesInDeepFreeze

There are a lot of good books indeed. Thanks for the references. I prefer the math use in physics though. And the alephity of the continuum has implications for particles moving in it. That's why I think the aleph of the volume is different from the line and plane. I have booked a hotel for us... Just kidding! Gnight!

You still demonstrate that you don't understand these basic concepts.

There are a lot of good books indeed. — AgentTangarine

And you desperately need one if you are not to remain mired in your terrible confusions.

And you desperately need one if you are not to remain mired in your terrible confusions. — TonesInDeepFreeze

Well... I don't take it too seriously... You are probably right. Still, I can't see how R and RxR can have the same cardinality. There are just inf^3 times as many points in RxR as there are in R.

Thanks for the resistence! :smile: — AgentTangarine

I resist misinformation.

I don't take it too seriously — AgentTangarine

The problem is not so much that you don't take it seriously, but that you take it seriously enough to stubbornly persist in claims that are false or just ersatz gibberish from your own mind uninformed about anything other than itself.

I can't see how R and RxR can have the same cardinality. — AgentTangarine

You could ask for more details about the proof mentioned by jgill and about the proof in the Quora thread.

For a proof in greater generality for any infinite S, as I said, it requires learning set theory.

L

You could ask for more details about the proof mentioned by jgill and about the proof in the Quora thread. — TonesInDeepFreeze

The point is that the proof in quora is incorrect. It's making use of decimal expansions also but overlooks the majority of them.

It's making use of decimal expansions also but overlooks the majority of them. — AgentTangarine

Name one.

I resist misinformation. — TonesInDeepFreeze

You should edit Wikipedia. We need Wikipedia and Wikipedia needs you i.e. we need you!

No offense intended AgentTangarine. I don't think your're guilty of misinformation. TonesInDeepFreeze is conflating facts with opinions.

Name one. — TonesInDeepFreeze

Well, the point made is that a pair of numbers (x,y) say (0.678567..., 0,98678...) is contained in a single number 0.65456456.... The infinite number behind the 0 should contain both the infinites behind the 0 of x and y. This is not so.

Take the square {(x,y):0<x<1,0<y<1} and map it one-to-one to the line {r:0<r<1) by using the procedure implied by the simple example (.329576914..., .925318623...) <-> .39229537168961243...

You can figure it out if you stay off the Xmas grog long enough. Although you are a smart physicist and may be pulling our legs. You and Agent Smith can work this out. It cropped up in the course I used to teach in Intro to Real Analysis.

Hence, there are exactly the same "number" of points in the (section of) the plane and on the unit interval. Same cardinality.

conflating facts with opinions — Agent Smith

No way. One can offer alternative systems; I enjoy reading about them if they are rigorous. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion.

Saying in a case like this "Oh, it's all opinion anyway" is intellectual dereliction.

.

a number (x,y) — AgentTangarine

That is not a real number, you understand, right?

No way. One can offer alternative system; I enjoy reading about them. And one can even stipulate one's own terminology, and if it is rigorous, then we can accept it for purpose of discussion. But whether a proof is correct from given axioms is not a matter of opinion. Indeed, in principle, it is machine checkable. And the matter of what, in fact, mathematicians mean by the terminology is empirical fact, not opinion. — TonesInDeepFreeze

I'm currently reading a book on mathematical philosophy. I'm no good at math although I'm fascinated by the subject. I noticed that math, its various branches, start life more as vague intuitions rather than crystal-clear concepts/ideas. Rigor comes much, much later if I'm not mistaken.

Too, the definitions in math give me the impression that true understanding is being sacrificed for logical formalism.

Rigor comes much, much later if I'm not mistaken. — Agent Smith

To know, we would have to have access to the mental states of mathematicians. We would have to know how long was the time between their first pre-formal musings and then putting them down in concrete formulations. There is no reason to believe that for many mathematicians that time might be very brief.

Anyway, if cranks said, "Here are my pre-formal musings, maybe something could come of them", then that would be one thing, but instead cranks insist that their view and only their view is correct; that ordinary mathematics (and even the alternative systems that the crank is ignorant of) are wrong. It is the crank, not the mathematician, who is dogmatic and exclusionary.

the definitions in math give me the impression that true understanding is being sacrificed for logical rigor. — Agent Smith

I have never had any such impression. Very much to the contrary.

start life more as vague intuitions — Agent Smith

Example?

Take the square {(x,y):0<x<1,0<y<1} and map it one-to-one to the line {r:0<r<1) by using the procedure implied by the simple example (.329576914..., .925318623...) <-> .39229537168961243... — jgill

Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many. (same for (.0329576914..., .0925318623...).

That is not a real number, you understand, right?
11m — TonesInDeepFreeze

It's a pair of numbers. You must quote the whole line I wrote.

Sounds good mr. Gill. Almost convincing. But you construct a new number from the both. Giving them both different decimal places. The diagonal proof of Cantor says you leave numbers out. Infinitely many. — AgentTangarine

Give me an example of a number you think is left out. I bet it's not. Avoid the .999... =1.0 thing. Have you figured out what the algorithm is?

Too, the definitions in math give me the impression that true understanding is being sacrificed for logical formalism. — Agent Smith

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.

Yes, a pair of numbers. Not a number as you wrote.

You keep resorting to saying that I must consider the rest of what you posted. But each time it turns out that the rest of what you posted doesn't actually qualify into correctness the initially incorrect statements you make.

But funny, <x y> actually is a number. It's a complex number.

a number (x,y) say (0.678567, 0,98678) is contained in a single number 0.65456456. — AgentTangarine

Whatever you mean by an ordered pair being "contained" in a number, what we have is each number mapped to an ordered pair. The claim of the prover is that the whole mapping is 1-1 and onto RxR. All it takes then is to see that no ordered pair is mapped to by two different numbers, and that each ordered pair is mapped to by a number.

The diagonal proof of Cantor says you leave numbers out. — AgentTangarine

The diagonal proof shows that any map from N to R is not onto R. That is, there are real numbers not mapped to.

I wrote:

Well, the point made is that a pair of numbers (x,y)...

You keep resorting to saying that I must consider the rest of what you posted. — TonesInDeepFreeze

Where did I do that?

It is the crank, not the mathematician who is dogmatic and exclusionary. — TonesInDeepFreeze

It's you who is the crank. You are exclusionary and dogmatic. And you have no sense of humor. Sense of rigor, maybe. Jgill knew to convince me (well, almost...) in one comment. But he's a real mathematician.

Cantor diagonal set

The diagonal proof shows that any map from N to R is not onto R. That is, there are real numbers not mapped to.
28m — TonesInDeepFreeze

That's not what the proof is about. It just shows that [0-1]is uncountable. Every time you think you counted a new number shows up. After infinity.

Where did I do that? — AgentTangarine

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