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• So, I got a lot of messages and comments from people after the video about Cantor's Diagonal

• Proof suggesting the following:

• If you can take an infinite list of rational numbers, and prove it can't ever contain all

• rational numbers because you take one key digit from each number and make a new real

• number that's different in every key digit and is therefore not on the list no matter

• what the list is, why can't you reverse this process, try to make a list of whole numbers,

• and prove that there's more whole numbers than whole numbers by going on the other diagonal,

• a reverse-Cantor?

• I kind of like this line of thinking.

• On the one hand, it's a mathematical way of thinking.

• If a proof seems to prove one thing is weird, you should probably test it to see if it proves

• everything is weird, and if everything ends up weird you might have a flaw in your premise

• or your thinking or maybe everything actually is weird.

• On the other hand, I don't like this line of thinking because it reminds me of the slippery

• slope fallacy.

• That's the one where you're like, well, if that argument works for this case, then why

• wouldn't it work for this more general case, and who knows where it will stop, so we'd

• better not allow it in the first place!

• And that's not actually a logical or mathematical argument if you don't prove it actually does

• apply to the more general case.

• So can you do the reverse cantor?

• Does this argument actually apply?

• If you take one digit from each number on your infinite list of all whole numbers, and

• change them all, shouldn't you get something that can't be on the list, thus proving the

• list is incomplete?

• Well, you do get something not on the list, so, that part of the reasoning is correct.

• Let's try it out: for every list item, we have a decimal place with a number in it.

• There's as many list items as decimal places, and there's an infinite number of each.

• That means that this number we're constructing will have infinite digits going out to the

• left, all of which conflict with the corresponding list number's nth decimal place.

• Now, the list of all whole numbers is infinite and includes arbitrarily large numbers approaching

• infinity, but no real number, no counting number, no whole number is actually infinite.

• They all stop somewhere.

• They all end up with infinite zeroes at some point.

• But this number we're constructing doesn't stop, it has to have infinite digits going

• out to the left in order to conflict with every number in every decimal place.

• No infinite zeroes.

• So no, it's not on the list, but it's also not a badly-named-so-called real number.

• Real numbers can have infinite digits going out to the right, any combination of digits

• going out to the right infinitely, no last digit, .9repeating actually exactly equals

• 1 and pi never ends or repeats and any random roll of infinite dice is a number different

• from the same number with one digit randomly changed, that's how many digits there are,

• etcetera, which is why the proof does work for reals, it's all in the decimals, the inbetween,

• but real numbers can't have infinite digits going out to the left, the counting numbers

• approach infinity but they don't get there and they certainly don't go past it, so the

• reverse-cantor is not a problem.

• But it is interesting to think about what it might mean to extend the real numbers to

• include ones that do have infinite digits to the left.

• digits going to the left but they don't work the same way as reals so this doesn't actually

• represent an infinite number.

• But what if you could represent infinite numbers like this?

• You'd run into some difficult questions.

• How do you define them?

• Is this number bigger or smaller than this number?

• You can't compare the leftmost digit, because there is no leftmost digit.

• Are they all equal, simply infinity?

• How do you add them?

• If you do it the normal way, isn't it weird that adding ...5555555 to ...555555 is ....000,

• there's no leftmost digit for the 1 to get carried to so it's just gone, which is the

• same as if you subtracted them.

• I think it probably all devolves from there, because if ...55555 = -...5555 then add ...555550

• to both sides to get 5=-5 and from there you break all of mathematics for all numbers,

• so you definitely can't do arithmetic the obvious way, if at all. or maybe .....000

• is an infinitely large number that's different from just plain 0, which is a nice concept,

• if you can find a way to make it work.

• Like, if you want ....000 to be different from 0, you might be in trouble if ....0000

• + ....55555 is the same as 0 + .....5555.

• But maybe this is a special case that you can avoid the same way you have to avoid dividing

• by zero for algebra to work.

• I dunno.

• You can play with that if you want, I'm going to bed.

So, I got a lot of messages and comments from people after the video about Cantor's Diagonal

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# The Reverse Cantor

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林宜悉 posted on 2020/03/30
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