Irrational/rational numbers

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Hi Matt,

Intention or not, you've been hoist by your own petard on this one.
You're right there!

But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals.

He has misquoted me on a number of occasions (and you have too, once). Now he assigns implications to me that I have NOT made and heads off on his flight of 'post hoc' fancy, trying to imagine my thought processes. I have acknowledged valid criticisms ... please refer to my earlier posts.

Paul.

PS.
You are forgetting your own advice. According to you, I have to prove every statement I make ...
Your claim that:

" The distinction is that naturals, and thus rationals, are countably infinite and irrationals aren't."


is not fine on it its own since you have not proven that the reals are not countable, and that the finite union of two countable sets is countable.
... but you don't have to!
Though I've not proven you may list the rationals, and not list the reals, though they are proofs found in many places.
Best make up your mind, Matt.
 
matt grime
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"But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals."
i didn't insist that. clearly an injection from Q to N is sufficient to show they are countable (for several reasons, not least the obvious schroeder bernstein theorem, or is that showing off?)

and the last bit, yes, hyposcrisy abounds doesn't it! but you did accuse me (probably me) of not answering the question properly, and i'm pointing out you perhaps didn't either by your own standards.
 
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Matt,

I have acknowledged the mistakes I have made. I just will not acknowledge mistakes that have been made on my behalf.

So far, according to you, I've been 'wrong', 'badly wrong' and then 'oh, perhaps not wrong'. Not once have you acknowledged that YOU misunderstood my first statement and were WRONG in your assumption that I needed to show equal cardinality in order to list the rationals. Nor has NateTG.

It might have helped the situation somewhat if you and he had decided in the first place to just correct my mistakes, rather than throw out terse, rude little asides.

I am not sorry for insisting that you or NateTG be precise in your criticism. In fact, that was YOUR original criticism of me.

If you call that hyposcrisy (sic), that's up to you.

Paul.
 
matt grime
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Could have sworn I posted this but: understood, my apologies.
 
matt grime
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Could have sworn I posted this but: understood, my apologies for being terse and rude and misrepresenting you. However, if you may point out where I said you (and I presume that you is the specific "you", rather than a general "you") need to show equal cardinality in order to list the rationals? Listing the rationals clearly shows they are of equal cardinality, indeed is a proof they are countable. I have read back and can't see what it is makes you think that, though perhaps I misunderstand what you're saying.

Is it perhaps that you think I wanted you to prove them countable? No, that wasn't it, it really was the assertion you made that "clearly there are more irrationals than naturals" without any substantiation, as there was no indication of what you meant by "more" or "size" (post 2), and what led to your out burst about "jeez some people in this forum amaze me..."

When I say "wrong" etc, perhaps I should make it clear I meant, oh, not false, but, in some loose sense, not mathematical enough, though that is very subjective. I suppose I was just mystified by post 6 which apparently starts off with a premise and a has conclusion, but they aren't linked. That is how it appears, though we've clarified that isn't what was meant.

The reference to hypocrisy was aimed at me, not you.
 
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Matt,

Thanks for your last post. I accept everything you say.

As I said before, you've taught me a really good lesson in how NOT to answer questions. Hopefully, I won't be making the same mistake again.

Paul.
 
NateTG
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Pnaj, I may have misinterpreted your posts, and I apologize for that.

Let's take a look at your original post:

Post #2:
pnaj said:
Somewhat counter-intuitively, the rationals have a one-to-one relationship with the natural numbers (that is, you can list the rationals as you can the naturals.)
Post#22:
pnaj said:
The trouble is, I most certainly did NOT imply equality of cardinality ... you have just wrongly assumed that, without actually carefully considering EXACTLY what I wrote.
The paranthetical remark indicates that it's possible to list the rationals in the same way that it's possible to list the naturals - a statement that's equivalent to saying that there's a bijection between the two sets - which is stronger than stating that there is a 1-1 relationship.

In the context of an explanatory answer, it might be good to define, or describe, what a 1-1 relationship is, which is what I assumed the paranthetical remark was supposed to be, and in that context the statement is inaccurate - perhaps it was intended for a different purpose.

The post also assumes that the reader is sufficiently familiar with cardinal numbers to understand that the existance of 1-1 mappings, or bijections is usefull in comparing the 'size' of sets, and it's unclear whether you want to show less than or equal to, or equal to.

pnaj said:
"But I've shown quite clearly that both you and NateTG were wrong to insist that a 1-1 map is not sufficient to make a listing of the rationals."
A 1-1 map from the rationals to the natural numbers is sufficent to make a list of the rationals. As indicated in my prior post, it's not at all clear from the phrasing you initially used ",1-1 relation", whether you're referring to an injection in a specific direction. (Yes, I know that there's always an injection between two sets.) Moreover, invoking implied notions of countability is a really poor way to describe something to anyone who is unfamilar with cardinal numbers in the first place.

Consider, for example that the reals have a 1-1 relationship with the natural numbers - for example, the usual embedding of the naturals in the reals is an injection - but we both already know that the reals are not countable.

pnaj said:
[NateTG] has misquoted me on a number of occasions (and [matt grime has] once).
Where? I cut and pasted the direct quotes.
I may have interpreted what you wrote differently than you, but that because the posts are ambiguous at best, and, in several places, I'm not the only one that finds your interpretation unusual. Could you list examples of what you consider to be me misquoting you?
 
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NateTg,

Well all I can really say to that is: fair comment.

Paul.

P.S.
NateTg ... if you really want me to list the misquotes, I will, but I'd rather not.
 
NateTG
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pnaj said:
NateTg ... if you really want me to list the misquotes, I will, but I'd rather not.
I don't really think that I've misquoted you, so I would really like to see what you consider to be misquotes.
 
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NateTG,

Well I didn't want to open it all up again.

But fair enough.


The 'Jeez' post was actually directed at your previous post. I had already asked Matt to explain his comments and he hadn't responded yet. You interpreted it as if I was further questioning Matt's comments.
Matt Grime's point is entirely valid. In some situations it's very important to realize that what you cheerfully refer to as 'definitely more' is really a rather technical notion.
On the same post ...
Odd, that you're the one that brought up 1-1 relationships (somewhat inaccurately, no less) and then accuse Matt Grime of being a crackpot when what he said is completely correct.
... I have no idea where you got this, and I'm afraid it coloured my judgement of you. It seemed to me that you weren't actually reading what I actually wrote, just what you thought I wrote.


We've cleared this one up already.
Correct, but also that's really irellevant since the inital post was about rational numbers, not natural numbers, and has nothing to do with the conclusion that you reach.
Here's another one ...
You initial statement implies that a 1-1 function is sufficient to demonstrate that two sets have the same cardinality. This is incorrect, since, for example, the natural mapping of the rationals into the reals is 1 to 1.
It didn't, as I showed earlier, but you interpreted it that way.


And you know what I think about this ...
This is an example of the post hoc ergo propter hoc falacy - you assume (probably unintentionally) that because the conclusion you reached is correct that the argument made for it is valid. However, the last sentence has very little, if anything to do prior claims.

It's almost as if it started as something about a finite number of rational numbers in an interval, and someone, upon realizing that that was false, and thoughtlessly substituted natural numbers rather than accepting that the argument did not hold water to begin with.
Paul.
 
matt grime
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Paul, I am sorry to have offended you in this thread, it certainly wasn't my intention.

Matt
 
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Ok, Matt,

See you round ... and NateTG.

P.
 
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I just googled this question and I read this:

Somewhat counter-intuitively, the rationals have a one-to-one relationship with the natural numbers (that is, you can list the rationals as you can the naturals)

It should be clear that there are more irrationals than naturals!
This is what goes through my mind:

"I can understand that on an interval, there are more irrationals than naturals, but surely there would also be more rationals than naturals too?"

I'm confused as to why you don't admit that yours wasn't a helpfull answer.
 

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