Why are there more irrational numbers than rational numbers?

[pnaj]

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]

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

Let's take a look at your original post:

Post #2:

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:

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 existence 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.

"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.

[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?

 

[pnaj]

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]

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.

 

[pnaj]

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]

Paul, I am sorry to have offended you in this thread, it certainly wasn't my intention.

Matt

 

[pnaj]

Ok, Matt,

See you round ... and NateTG.

P.

 

[Elliot.AT]

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.