Help recalling theorem

[Gokul43201]

This is something of an odd request, I guess.

I have a very foggy recollection of a theorem in some field of applied mathematics - probably information theory. I think it has to do with a seemingly surprising result about the information needed to describe any real in some closed interval compared to the information needed to describe all the reals in that interval.

I can't recall anything more definite about this, and what I've said above may itself be more wrong than right. Hopefully there's just enough correct stuff there to help ring a bell with someone. Does anyone know what I'm rambling about?

 

[CRGreathouse]

The information needed in both cases is uncountably infinite -- in particular, $$\beth_1:=2^{\aleph_0}.$$

 

[maze]

Wouldnt you be able to describe a single real with a sequence that converges to it (which is countable)?

 

[CRGreathouse]

Wouldnt you be able to describe a single real with a sequence that converges to it (which is countable)?

There are $$\beth_1$$ sequences consisting of $$\aleph_0$$ rational numbers. So, you could describe a real number that way, but it wouldn't be countable.

 

[Gokul43201]

Thanks CRG. I was about to ask how you get beth_1 for both cases... but I ought to give it some thought first. Also, this tells me that I probably haven't recalled the theorem correctly.

 

[CRGreathouse]

A closed real interval is of the form $\{x:a\le x\le b\}$ for some real a and b. Thus describing the interval requires only giving two real numbers. Two real numbers can be combined 'for the price of one' in many ways, like interleaving digits:

1.12345 (interleave) 2.24680 = 21.1224364850

Work from the decimal point out, since real numbers can't be infinite.


The second point is only that $$\aleph_0^{\aleph_0}=2^{\aleph_0}:=\beth_1.$$

Also, this tells me that I probably haven't recalled the theorem correctly.

I'm just trying to jog your memory.

 

[CRGreathouse]

Of course an arbitrary subset (rather than interval) of the reals has cardinality $$\beth_2=2^{\beth_1}.$$

If the result was information theory, then we're probably looking at the wrong stuff. Information theory usually deals with finite numbers of bits, right? $$2^{2^{\aleph_0}}$$ doesn't strike me as particularly 'applied'.

 

[maze]

Perhaps they have infinities sitting around that get renormalized away?