[math](0,1)[/math] is equinumerous with [math]\mathbb{R}[/math],

>>8954553
It is yeah, but I don't think that's a direct consequence of [math](0,1)[/math] having the same cardinality as [math]\mathbf{R}[/math]. I could very well be wrong on the last bit though.

Cardinality is a meme. Measure is were is at

>>8954563
I agree with this man here.

>>8954553
Yes.
>>8954563
Isn't it possible to generalize the counting measure to infinite sets and in that sense make cardinality just into a specific measure?

>>8954553
obviously

>>8954679
prove it

>>8954704
You should've learned it in your first semester analysis class.

>>8954704
Q has the lowest possible infinite cardinality. Subsets have cardinality no larger than their supersets

>>8954716
No larger means they can be smaller you know

>>8954553
yes, it must be that way true.

Suppose not

Then what if you listed all the [math]\mathbb{Q}[/math] as decimals in an infinite list.

then, eventually you would have to cover all of [math]\mathbb{Q}[/math] , because your list is infinity

>>8954553
consider a bijection [math]f \colon \mathbb{R} \to (0,1)[/math] where [math]f[/math] is a rational function. this is possible, just monkey around with [math]\frac{x}{1+|x|}[/math] or something. then [math]f[/math] restricted to [math]\mathbb{Q}[/math] should answer your question.
>>8955406
"obviously" [math](0,1) \cap \mathbb{Q}[/math] is infinite