At the end of the day all of this infinity stuff is just a waste of time since in the real world it simply does not exist, It;s all on the drawing board. Literally.
@kire271MK At the end of the day, I always like to paraphrase the physicist Richard Feynman.
"Physics is to math what sex is to masturbation."
Physics studies the frontiers of the universe, challenging preconceived notions, and furthering the advancement of our understanding of the universe at large. Many people gather around and do this physics together and experiment.
Math, on the other hand, studies the frontiers of the human mind. This is a more personal experience that the experts like to do alone and are usually too modest to show off the size of their prowess (take Perelman, for example, who didn't accept the Field's medal because he though the results were reward enough). Sometimes, though, people gather together to watch and learn new techniques that we can all use ourselves in our personal time.
and that does not mean that they should make sense in any context or be demonstrable with any basic logic ... there created by man for some purpose ... what your trying to do is drive a car on a cliff, and say it's a paradox that it does'nt fly ... it just was'nt made for it .... You seem to be mathematician ... so I guess you know how useful the notion is !!! To calculus, stats etc !! Just use the thing for what's it's suppose to serve and there's no paradox at all ...
The notion of infinity in maths was invented to solve problem ... it dos'nt mean that you can show physicaly it's existence ... or show it's logic like your trying to do ... It's the same thing for i=sqrt(-1) .... It's ridiculously useful in solving mathematical problems, yet you can't put these numbers on a ruler ... they use cooredonates to try to show it graphicaly but you just have to keep in mind that those are created to help solve problems...
Two sets have the same cardinality (number of elements) if there EXISTS a bijection (1-to-1, onto function) between them.
If you exclude the number 2, both lines still have the same number of elements because a different bijection could be formed between them.
The existence of a bijection is guaranteed by the Cantor-Bernstein-Schröder theorem, since f(x) = x is an injection from line 1 to line 2, and g(x) = x/2 is an injection from line 2 to line 1.
@x1101011x Saying that a bijection exists between two sets of numbers A and B is logically equivalent to the saying that an injection exists from A to B and another from B to A.
why would the 2 segment be less? I see how it's not a one to one correspondence, but wouldn't they both remain infinite, and be equal? So really, why do you need 1-1?
@sherlockfury without 1-1 you cant compare the two. if you could there would potentially be more than one answer for a given number in set one. therefore if one number can correspond to two numbers, the second set is larger
Why is this set correspondence allowed when cantor's diagonal line suggests different (noncorresponding) amounts of infinity? I'm not sure, but I thought proving extra/missing numbers was the proof of different infinite magnitudes.
Ristrict the 2, and there is less numbers on the [0,1] line than on the [0,2) line.
I really like that kind of stuff.
What did you say at the end Sir? You said something like: "That's sort of a Ses De Plate..." I know I must have misheard you, but I must not know what that means, because if I did, I would not have to resort to spelling it phonetically. Please enlighten me.
Oops, sorry! It was just an expression, " . . . that sort of sets the plate for . . ." Nothing profound.
I suspect that Galileo and others were kind of stumped by this phenomenon and didn't know what to do with it. To my limited knowledge, it was Cantor who first took on these issues, two hundred and some years later.
At the end of the day all of this infinity stuff is just a waste of time since in the real world it simply does not exist, It;s all on the drawing board. Literally.
kire271MK 6 months ago
@kire271MK At the end of the day, I always like to paraphrase the physicist Richard Feynman.
"Physics is to math what sex is to masturbation."
Physics studies the frontiers of the universe, challenging preconceived notions, and furthering the advancement of our understanding of the universe at large. Many people gather around and do this physics together and experiment.
MyOverflow 3 months ago
@MyOverflow (cont'd)
Math, on the other hand, studies the frontiers of the human mind. This is a more personal experience that the experts like to do alone and are usually too modest to show off the size of their prowess (take Perelman, for example, who didn't accept the Field's medal because he though the results were reward enough). Sometimes, though, people gather together to watch and learn new techniques that we can all use ourselves in our personal time.
MyOverflow 3 months ago
By not make sens in any context, I meant in every context !!! Of course they should make sens a many context !! But not all !!!
659851 11 months ago
and that does not mean that they should make sense in any context or be demonstrable with any basic logic ... there created by man for some purpose ... what your trying to do is drive a car on a cliff, and say it's a paradox that it does'nt fly ... it just was'nt made for it .... You seem to be mathematician ... so I guess you know how useful the notion is !!! To calculus, stats etc !! Just use the thing for what's it's suppose to serve and there's no paradox at all ...
659851 11 months ago
The notion of infinity in maths was invented to solve problem ... it dos'nt mean that you can show physicaly it's existence ... or show it's logic like your trying to do ... It's the same thing for i=sqrt(-1) .... It's ridiculously useful in solving mathematical problems, yet you can't put these numbers on a ruler ... they use cooredonates to try to show it graphicaly but you just have to keep in mind that those are created to help solve problems...
659851 11 months ago
OHHHHHHHHHHHHHHHHHHHH!!!!!!!!!!!!!!!!!!!! I understood what you meant at 6:09.
thank you! :)
StitchedMusik 1 year ago
Thanks again.
downcard11 2 years ago
This video was fine until 6:33.
Two sets have the same cardinality (number of elements) if there EXISTS a bijection (1-to-1, onto function) between them.
If you exclude the number 2, both lines still have the same number of elements because a different bijection could be formed between them.
The existence of a bijection is guaranteed by the Cantor-Bernstein-Schröder theorem, since f(x) = x is an injection from line 1 to line 2, and g(x) = x/2 is an injection from line 2 to line 1.
x1101011x 2 years ago 7
@x1101011x Saying that a bijection exists between two sets of numbers A and B is logically equivalent to the saying that an injection exists from A to B and another from B to A.
0112471324 1 year ago
why would the 2 segment be less? I see how it's not a one to one correspondence, but wouldn't they both remain infinite, and be equal? So really, why do you need 1-1?
sherlockfury 2 years ago 2
@sherlockfury
two sets can contain both infinite number of elements (say for example the sets of all integers, N, and the set of all real numbers, R.
It can be shown (quite easily) that you cannot map the elements of R to N without overlapping (and infinitely overlapping at least one point).
So it is sais that R is much larger than N, and if i don't say it wring, that the cardinality of R is greater than N.
However, as x1101011x said, the cardinality of the two segments is the same in realty.
Lioobayoyo 2 years ago
@sherlockfury without 1-1 you cant compare the two. if you could there would potentially be more than one answer for a given number in set one. therefore if one number can correspond to two numbers, the second set is larger
bedwarri0r333 8 months ago
Sir, you just blew my mind
JMelo19 2 years ago 2
Why is this set correspondence allowed when cantor's diagonal line suggests different (noncorresponding) amounts of infinity? I'm not sure, but I thought proving extra/missing numbers was the proof of different infinite magnitudes.
Squ34k3rZ 3 years ago
Awesome. That is very cool mind blowing stuff!
Ristrict the 2, and there is less numbers on the [0,1] line than on the [0,2) line.
I really like that kind of stuff.
What did you say at the end Sir? You said something like: "That's sort of a Ses De Plate..." I know I must have misheard you, but I must not know what that means, because if I did, I would not have to resort to spelling it phonetically. Please enlighten me.
Thank you Sir.
nothingstopskings 3 years ago
Oops, sorry! It was just an expression, " . . . that sort of sets the plate for . . ." Nothing profound.
I suspect that Galileo and others were kind of stumped by this phenomenon and didn't know what to do with it. To my limited knowledge, it was Cantor who first took on these issues, two hundred and some years later.
Davidson1956 3 years ago