Sme question about irrational numbers

[Hurkyl]

Anything wrong with the φ I gave in my post?

 

[hello3719]

ok ill be more clear, Could you find a way to map EVERY irrationnal number to some real number or the other way around.

Sorry but hadn't took any analysis courses yet, so I don't know the terminology well.

 

[Moo Of Doom]

Could you find a way to map EVERY irrationnal number to some real number or the other way around.

I believe Hurkyl's method was quite successful in doing that.

 

[Hurkyl]

I repeat, what's wrong with φ?

 

[hello3719]

How can we be sure that every irrationnal number is of the form q + (n+1) \sqrt{2}

 

[Hurkyl]

They're not. That's why I added a clause specifying what happens to things not of that form: they're left unchanged.

 

[hello3719]

so those which are not changed are not "associated" to some real number ?

sorry maybe I am skipping something obvious

 

[Hurkyl]

If x is not of the form q + m √2, then φ(x) = x.

 

[Moo Of Doom]

Those that are not changed are associated with themselves. Irrational numbers are real numbers. So something like \pi, which is not in the form q+n\sqrt{2}, is mapped to \pi.

All Hurkyl is doing is eliminating rational numbers by mapping them to irrational numbers, and then mapping those irrationals which they are mapped to to other irrationals, etc.

 

[hello3719]

tx

ok, so they are are mapped to themselves
sorry my logical flaw was that i wanted inconsciously a mapping of every irrationnal number to a general number not described in function of itself

 

[matt grime]

I can't believe you've (tann) asked me to explain why cardinality and connectedness are not related in any meaningful way.

Look at the definitions. Not that you seem to show any inclination to do this.

Where do you see any requirement for connectedness or any mention of it in cardinals?

Connectedness is a property of topological spaces. So you really think i have to define the machinery of topology to talk about cardinals?

Moreover given any set there is the trivial topology on it. You cannot disconnect this. And there is the discrete topology, easily disconnected except in one case (which is when? hint, you can describe it using cardinal numbers if you wish).

 

[Tann]

Those that are not changed are associated with themselves. Irrational numbers are real numbers. So something like \pi, which is not in the form q+n\sqrt{2}, is mapped to \pi.

All Hurkyl is doing is eliminating rational numbers by mapping them to irrational numbers, and then mapping those irrationals which they are mapped to to other irrationals, etc.

If I understand it correctly, then we get a 1-1 mapping between Q and a partial collection of irrational numbers, and then we can conclude that since some irrational number (like \pi for example) is out of the domain of this map, then the cardinality of the irrational numbers is 2^aleph0.

It is as if N and Q are in the "shadow" of the irrational numbers.


But what if we do the same trick between Q and a partial collection of N (for example all odd numbers).

Now we can say that all even numbers are mapped into themselves and number 2, for example, is out of the domain of Q <--> Set_of_all_odds.

In both cases we deal with partial collections, but now Q is under the "shadow" of N.

I can't believe you've (tann) asked me to explain why cardinality and connectedness are not related in any meaningful way.

Look at the definitions. Not that you seem to show any inclination to do this.

You are a funny fallow Matt with a lot of emotions, as much as they are related to Math.

But I like to think my own thoughts together with other people, and I hope it is ok in this forum.

I thought about it and I came up with some points:

1) A set is a collection of elements where each element in it can be clearly distinguished from the entire elements that exist in this collection.

2) a 1-1 mapping can work only if (1) is true.

3) In the case of N collection we have a minimal unit (called one) that can help us to clearly separate each n from the entire N elements.

4) This is not the case in Q elements where a minimal unit does not exist, but there is a bijection between N and Q because each Q member is actually the ratio between any two N members. So in this case we still have something in common between N and Q that gives us the ability to use the 1-1 mapping technique.

5) Irrational numbers have no minimal unit and no ratio between N pairs (that’s why they are called irrational numbers) but each element can be expressed as x/1 where x is any irrational number.

6) Please pay attention that we use 1 in 1-1 mapping expression, in order to say that each element in both sides of the mapping expression, can clearly be distinguished form the entire members that shares with it the same collection (and this is the notion behind (5) ).

7) If (6) is not true, then 1-1 mapping cannot work.

8) Meaning, any pair of elements in the collection of all irrational numbers must be in the form of {x/1, y/1} where x not= y.

9) if x = y, then (6) is not true and we get (7)

10) So if we use the 1-1 mapping technique and also wish to get some results, we realize that we actually have to ignore the universe that exists between x/1 and y/1.

11) If we refuse to do that then x=y and we get (7)

12) Since we use the 1-1 mapping technique as one of the main tools of modern mathematics, then we have to admit that no set, which is based on {x/1 ,y/1} where x not= y, has the power of the continuum.

13) It has to be clearly understood that the whole idea of the set concept (please look at (1) ), actually limit our ability to explore the universe that may exist between any R elements,
And we can clearly see that our results are no more than the outcome of our current agreed methods.

14) By using connectedness we actually say that each explored element on the real-line must be included in N, Z, Q or R (where any element is limited by (1) ) , but no one of these collections has the power of the continuum, because if we get the power of the continuum, the 1-1 mapping technique simply does not work.

I cannot avoid this analogy, but I thing that the future of the language of mathematics is some how related to the discovery of the relations between sets and metric space, where sets are the particle-like side of these relations, and the metric space is the wave-like side of these relations.

So, the connection between metric space and sets is up to us.

 

[matt grime]

My God, you are Doron. But the standard of English was so much better...
Andif you aren't the similarity is eerie: all that guff abuot wanting to learn from other people then completely ignoring them and waffling on with your own rubbish.

So, that's the end of answering your posts.

 

[Tann]

My God, you are Doron. But the standard of English was so much better...
Andif you aren't the similarity is eerie: all that guff abuot wanting to learn from other people then completely ignoring them and waffling on with your own rubbish.

So, that's the end of answering your posts.

Excuse me sir, but what are you talking about?, I am going to send this post to the moderator of this forum if you are not apologize right now!.

 

[chronon]

All I see here is a collection of symbols, that does not prove anything.

I'm wondering whether the problem here is the distinction between 'rigorous proof' and 'a collection of symbols'.

In the case of finite arithmetic there can be a proof which convinces you of a certain fact about numbers, without having to base this on a certain set of axioms.

For transfinite arithmetic this is not the case. Any proof must be based on axioms, and so can just be considered as a collection of symbols. You can accept such a proof as formally correct, even if you don't believe in uncountable sets (and some people don't).

Remember Skolem's paradox: Every theory has a countable model.

 

[matt grime]

For comparison here is one of Dorons many posts under many different pseudonyms. It's style and content are eeriely similar to yours, Tann. Though the standard of English isn't. In particular Doron repeatedly talkde about "exploring" and "between", not to mention "members" .

Oh, and one of the reasons I called it guff and rubbish (I think those were the terms; I have little tolerance for cranks - there are some very deep issues that are of great philosophical importance in mathematics, almost all of which are completeyl misinterpreted or not even known to those who write about them so authoritatively as to include the phrase "please pay attention", which is also a Doronism) was, say, rule 4 in your last post

"4) This is not the case in Q elements where a minimal unit does not exist, but there is a bijection between N and Q because each Q member is actually the ratio between any two N members. So in this case we still have something in common between N and Q that gives us the ability to use the 1-1 mapping technique."

The parts of this that I can even make sense of are not true on many levels. Write any negative rational as a ratio of two naturals. Even allowing for that you don't explain why this is a bijection from N to Q, it isn't even a map from N to Q. Moroever each element in Q is a ratio of any two N members? What? 1/2 is the ratio of any two N members? Even 3 and 7?


Here's Doron minus some web links..

Quote:
Let us go straight to main points, where my theory is different from the Standard point of View.

1) Through My new point of view, any number is first of all an information-form which is based on at least {._.}_AND_{.}, where {._.}_AND_{.} is the minimal existence of any number which is not 0.

2) In any given quantity which is > 1, each number can be ordered by several internal symmetrical degrees that can be clearly shown here: <url deleted>].

3) {._.}_AND_{.} of set N, cannot be put in a bijection with proper subsets of themselves, because of {._.} that exists as an internal building-block of each N member, and I clearly show it in page 3 of <url deleted>

4) Standard Math ignores {._.} because through its point of view, any given {._.} can be defined by {.}.

5) Through my point of view {._.} cannot be defined by {.} .

6) If any number is at least {._.}_AND_{.}, then a bijection between Q or R sets to proper subsets of themselves, can be defined if and only if any Q or R member is both some unique element, and a scale factor of the entire set (in N or Z the scale factor result is always out of the domain of the original pairs, when no one of the pairs is 1,-1 or 0).

7) My new system is consistent and well-defined, and cannot be understood by Standard point of view (yellow glasses).

8) Standard Math has no answer to the question: "What is a number"?

My theory gives rigor answer to this question by using the information concept, where redundancy and uncertainty are fundamental ("first-order") properties of it.

Standard Math uses only the on_redundancy no_uncertainty information form
as "first-order" property.

9) Furthermore, my theory includes our cognition's ability to define 'a number', as a legal part of it.

 

[Tann]

Matt,

I wrote a private message to Hurkyl about you.

The parts of this that I can even make sense of are not true on many levels. Write any negative rational as a ratio of two naturals. Even allowing for that you don't explain why this is a bijection from N to Q, it isn't even a map from N to Q. Moroever each element in Q is a ratio of any two N members? What? 1/2 is the ratio of any two N members? Even 3 and 7?

1 is in N, 2 is in N and 1/2 is the ration between 1 and 2.

3/1 is a Q member, 7/1 is a Q member.

 

[Tann]

According to what I wrote at the top of post #62, is there some logical proof that prevents from us to conclude that Cantor's second diagonal method actually shows that we cannot use the 1-1 mapping technique with the collection of all R members?

 

[Hurkyl]

Remember Skolem's paradox: Every theory has a countable model.

And remember the resolution -- that measure of countability is external. While we, "superobservers" playing with that countable model in our mathematics laboratory may be able to produce a bijection from the model's naturals to that model's reals, there is no such internal mapping. Someone living in the theory will never be able to produce a bijection between its naturals and reals.

 

[arildno]

Post 62:

It is as if N and Q are in the "shadow" of the irrational numbers.

This shows only that you have a "shadow" of intelligence.
Stop posting gibberish.

 

[Tann]

This shows only that you have a "shadow" of intelligence.
Stop posting gibberish.

MAMA'MIA !

This is no more than an analogy.

Hurkyl,

Can you please explain to me why your advisors are so violent?

Why do you allow such a behavior?

 

[arildno]

If you want to speak maths, speak maths, not crappy analogies.
If you do that, and it isn't the first time, it will be pointed out to you. BLUNTLY.

 

[matt grime]

Violent? Oh, please, have you even read any sci.math?

Anyway, the simple explanation is that despite claiming that you wished to learn from professionals you've shown no absolutely no sign of doing so and have ignored our advice and just posted more of the same unmathematical musings.

For instance, to repeat myself yet again, why is the map f(s)=s defined for all s in some set S not a bijection? You have implied that it cannot be. Instead of answering this you went off on some bizarre tangent about "holds" which we find means "is true", though how a function "is true" is a mystery.


And as for your "number 4", the reals are expressed as limits of sequences of elements ni Q, so why isn't R countable? Using your "logic"?

There are several delicate areas of constructability and definability that (some) mathematicians care about. This isn't really getting close to it.

 

[Tann]

Violent? Oh, please, have you even read any sci.math?

There is no moderator in sci.math.

If you want to speak maths, speak maths, not crappy analogies.
If you do that, and it isn't the first time, it will be pointed out to you. BLUNTLY.

I simply do not believe it. Hurkyl what is going on are you a moderator or not?

And as for your "number 4", ...

What number 4, what are you talking about, Hurkyl why do you allow this madness?

 

[Hurkyl]

Because yesterday there was a productive side discussion going on, and today I'm sick and was hoping I wouldn't have to deal with this. It's clear this isn't going anywhere now, though, so the decision is easy.

 

[chroot]

For future reference, Tann was indeed another alias for Doron. His new alias has been banned, as will any more aliases he creates. Please let a mentor know if he pops up here again.

- Warren