Hilbert's Hotel Paradox: Are some infinities bigger than others? (Part 2)

In this video, we talk about how Georg Cantor proved that some mathematical infinities are bigger than others using the Hilbert's Hotel paradox. He used 2 very simple methods to prove it: 1 to 1 Correspondence, and Diagonal method.

Here is the chapter wise line up of the video:
0:00 recap of part 1
1:30 Hilbert's hotel paradox
2:30 infinite guests
3:47 infinite buses & infinite guests
4:49 Euclid's Prime numbers
8:00 Infinite bus & strange seats
9:30 Cantor's Diagonal
11:30 Cantor's proof
12:03 Outrage against Cantor

[joesmith]

Out of box reasoning has always been considered a bad thing thru out human history. I loved how you intertwined Cantor's personal history and Math.

[ijujeb]

This is GOLD. Never knew Cantor was called 'corruptor of the youth' just like Socrates.

[reptilesspurky]

You cant get more than 1 infinity because infinity has no identity it just is an for expanding expression the moment you add to an ''infinite' ' there is an implication you can identify exactly what that infinite number is when infinity is a concept really an confusing one which doesn' t really make any sense.

[koskovictor]

Free Science 365 0:00 0:00
Your making multiple errors
Set theorists themselves don’t claim cardinal it’s is size, it’s different, your contradicting them

Your using your matching method to prove infinite sets are = in size, but such techniques work as well for < and >
The Cantorists limit them to <=, >=, and = but not < or > to obey the law of trichotomy (x is only one of < = > to y)
If you allow all 3 then all Aleph 0 sets are < and = and > each other in size, likewise in Aleph 2. There is no excuse to limit to not < or >
Since you axiomatically assume only applies to = you cannot prove some infinite sets are larger than others

The diagonal argument does not prove > or ≠
It proves the matching proof is incapable of proving =
Tree proofs prove =
Consider all finite binary trees with an extra node (•) on top linking to only one bellow
All nodes other than this extra node link to none or 2 nodes bellow, tree not network
The number of bottom nodes is the number of paths each path a binary number between 0 and 1
The number of pairs of nodes for each such tree is = to the number of paths
Consider an infinite such tree
Using the same logic as your matching method the number of pairs of nodes each pair matched to consecutive numbers (0, 1, 2, 3, …) left to right then top to bottom =s the number of left right possible paths thus real binary numbers from 0 to 1 and the number of pairs of nodes an exponential of the number of layers
This proves an aleph0 set is = in size to an aleph1 set and exponential of an infinity equals that infinity contradicting 2 conclusions of cantorism using Size in place of Cardinality
Set Theory defines cardinality so proven not-proven-=-size by the matching proof is by definition ≠ cardinality, with the matching method used for >= proves > cardinality

Also their diagonal argument uses infinity (aleph0 never ending) overwhelm all finite so overwhelms itself. But using that with infinity - infinity produces negative infinity but also infinity
Likewise the diagonal argument either has the height = to width so set theorist wins OR height exponential to width so anti-set-theorist wins