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Experiment: Mapping real numbers between 0 to 1 to natural numbers

smacpherson64.github.io

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Re: Experiment: Mapping real numbers between 0 to 1 to natural numbers

#11

Post by smac__ »

gus_massa wrote: post #5

You mapped the numbers between 0 and 1 that have a finite number of decimals, like 0.123123425736289 But there are many more real numbers between 0 and 1 that have an infinite number of decimals, like 0.1231234736876578632786278365782635879281987087420... As jgreen says in a sibling comment, which number do you assign to 1/3 = 0.333333333333333333333333333333333333333333... ? The result you got is well known but inte…

Again, excellent point! These numbers clearly show why* my reasoning is missing a rather large portion of this set.

[edit] - fixed typo

Re: Experiment: Mapping real numbers between 0 to 1 to natural numbers

#12

Post by neonskies »

smac__ wrote: post #9
neonskies wrote: post #6

Not sure what's on the vid. Didn't watch it. But if I understand you correctly, the following is impossible as shown in Cantor's simple diagonalization argument: > Follow each branch as deeply as possible counting all sub branches until there are no sub branches left.

Good point, the depth first search algorithm is a bit lack luster for this case: 1 -> .1 2 -> .01 3 -> .001 4 -> .0001 5 -> .00001 6 -> .000001 . -> .0∞1 I would definitely recommend not judging the quality of the video based upon the quality of my reasoning in the experiment.

> 1 -> .1 2 -> .01 3 -> .001 4 -> .0001 5 -> .00001 6 -> .000001 . -> .0∞1

That would only count the rationals which you can definitely count off a carefully arranged matrix of rationals. But reals = rationals and irrationals and the irrationals are uncountable.

Re: Experiment: Mapping real numbers between 0 to 1 to natural numbers

#13

Post by smac__ »

neonskies wrote: post #12
smac__ wrote: post #9

Earlier quoted context omitted.

Good point, the depth first search algorithm is a bit lack luster for this case: 1 -> .1 2 -> .01 3 -> .001 4 -> .0001 5 -> .00001 6 -> .000001 . -> .0∞1 I would definitely recommend not judging the quality of the video based upon the quality of my reasoning in the experiment.

> 1 -> .1 2 -> .01 3 -> .001 4 -> .0001 5 -> .00001 6 -> .000001 . -> .0∞1 That would only count the rationals which you can definitely count off a carefully arranged matrix of rationals. But reals = rationals and irrationals and the irrationals are uncountable.

You are definitely right, I missed quite a few numbers in this set. (Thank you!)

Re: Experiment: Mapping real numbers between 0 to 1 to natural numbers

#17

Post by gus_massa »

smac__ wrote: post #11
gus_massa wrote: post #5

You mapped the numbers between 0 and 1 that have a finite number of decimals, like 0.123123425736289 But there are many more real numbers between 0 and 1 that have an infinite number of decimals, like 0.1231234736876578632786278365782635879281987087420... As jgreen says in a sibling comment, which number do you assign to 1/3 = 0.333333333333333333333333333333333333333333... ? The result you got is well known but inte…

Again, excellent point! These numbers clearly show why* my reasoning is missing a rather large portion of this set. [edit] - fixed typo

Too late, but I hope you see it. I only want to add a highlight.

One you fix a finite set of symbols, let's say ascii [1], the amount of numbers you can describe with it with strings of finite length is different from the amount you can describe if you allow also infinite length.

The difference between finite length and infinite length in this problem is very important.

[1] I'm not sure about unicode /s

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