Your two cardinality vid's are helping me with my set theory review. Thanks ! I got a chuckle hearing you provide techniques for getting control of a set (6:04). They are slippery suckers ! It's like you are teaching us ju jitsu to wrestle those things to the ground ;^)
(Part 2) Cantor's work was not well-received at the time, as the cure seemed worse than the problem. It casts some doubt on the use of proof by contradiction, and an episode like this returns concerning the Axiom of Choice. While we cannot order the binary sequences with tags form N, AoC says we can well-order it. Just don't ask how. Drilling further down into the foundations only results in more uncertainty, culminating in Godel's work. If you doubt CD, you are doing it right. - Bob
You're welcome! Bullseye. (Part 1) The proof by contradiction is a test: a potential one-one correspondence (with built in ordering) is presented, and we show that it misses at least one element. In fact, the element that we produce depends on the ordering given. Loosely, our job isn't to fix the list, just to send it back when it fails to catch everything. In your approach, you produce a new listing to test, but the argument generates a new element not in the new list.
(comment continued) Why, for instance, can't we just dynamically 're-sort' our sequence as we go digit-by-digit into the binary representation? The idea is lexicogical, that every possible diagonal number begins as a sequence already present in our list. If we could order dynamically, as we go, we are 'drilling' down into our pre-existing list by a factor of 2 (or 10 if decimal) every time, we would just be refining the possible proper subsets - which are, of course, infinite. Thanks ;)
To solve the infinite paradox of Galileo, that two different infinite sets are both the same and different in size (for example the natural numbers set and the squares of natural numbers set can be paired up, but some elements lack in one of the sets) you can use "numerosity", which solves this question once and for all, which Cantor's cardinal and ordinal numbers can't. See "Numerosities of labelled sets: a new way of counting", www.dm.unipi.it/~dinasso/papers/13.pdf Following this new sice concept "numerosities", we can now compare infinite sets with perfect precision. If you add or subtract even one element from a fixed infinite set (like the natural numbers and the odd natural numbers) you create a new set with a different size!
A very important video. I take the fallacies #1 and #2 starting at 19:32 to explain why there is no contradiction-proof for uncountability of the Natural Numbers (i.e. what comes out isn't a Natural) but my mind baulks at the diagonalization argument at 12:40 because that 'arbitrary' sequence assertion (13:00) seems unconvincing (or contrived).
great job explaining!
Your two cardinality vid's are helping me with my set theory review. Thanks ! I got a chuckle hearing you provide techniques for getting control of a set (6:04). They are slippery suckers ! It's like you are teaching us ju jitsu to wrestle those things to the ground ;^)
Chris Bedford Glad to be of help! Good analogy - cardinality is like Greco-Roman wrestling (upper body contact only) in oil.
Thanks! This shoot was rough. Mild jiu-jitsu hangover, which wrecks my pacing. And of course you have to bring your A-game for cardinality. - Bob
BOB. I LOVE YOU MAN !!
(Part 2) Cantor's work was not well-received at the time, as the cure seemed worse than the problem. It casts some doubt on the use of proof by contradiction, and an episode like this returns concerning the Axiom of Choice. While we cannot order the binary sequences with tags form N, AoC says we can well-order it. Just don't ask how.
Drilling further down into the foundations only results in more uncertainty, culminating in Godel's work.
If you doubt CD, you are doing it right. - Bob
You're welcome! Bullseye.
(Part 1) The proof by contradiction is a test: a potential one-one correspondence (with built in ordering) is presented, and we show that it misses at least one element. In fact, the element that we produce depends on the ordering given. Loosely, our job isn't to fix the list, just to send it back when it fails to catch everything.
In your approach, you produce a new listing to test, but the argument generates a new element not in the new list.
Can you go into further depth with the 'further results, number 2' please
(comment continued)
Why, for instance, can't we just dynamically 're-sort' our sequence as we go digit-by-digit into the binary representation? The idea is lexicogical, that every possible diagonal number begins as a sequence already present in our list. If we could order dynamically, as we go, we are 'drilling' down into our pre-existing list by a factor of 2 (or 10 if decimal) every time, we would just be refining the possible proper subsets - which are, of course, infinite.
Thanks ;)
what is the caridnality ofv {0,{0},{0,{0}}}?
To get the elements of the set, remove only the outer braces.
To solve the infinite paradox of Galileo, that two different infinite sets are both the same and different in size (for example the natural numbers set and the squares of natural numbers set can be paired up, but some elements lack in one of the sets) you can use "numerosity", which solves this question once and for all, which Cantor's cardinal and ordinal numbers can't. See "Numerosities of labelled sets: a new way of counting", www.dm.unipi.it/~dinasso/papers/13.pdf
Following this new sice concept "numerosities", we can now compare infinite sets with perfect precision. If you add or subtract even one element from a fixed infinite set (like the natural numbers and the odd natural numbers) you create a new set with a different size!
You're welcome!
Thanks. Great video.
A very important video.
I take the fallacies #1 and #2 starting at 19:32 to explain why there is no contradiction-proof for uncountability of the Natural Numbers (i.e. what comes out isn't a Natural) but my mind baulks at the diagonalization argument at 12:40 because that 'arbitrary' sequence assertion (13:00) seems unconvincing (or contrived).