mersenneforum.org The quantity of integers distributed along the number line is an odd quantity
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2022-02-02, 10:34   #12
R. Gerbicz

"Robert Gerbicz"
Oct 2005
Hungary

112×13 Posts

Quote:
 Originally Posted by Charles Kusniec Excluding the number 0, the cardinality A of positive integers is equal to the cardinality B of negative integers: A=B. Now, can the cardinality of all positive and negative integers (A+B) without the number zero, be considered as an even cardinality?
For your amusement: let f(2*k)=2*k+1 and f(2*k+1)=2*k, this gives a trivial bijection between odd and even integers, for example f(-9)=-8 and f(4)=5.
The union of odd and even integers gives the integers, following your logic then the "quantity" of all integers is even, because that is the (disjoint) union set of two equal size set. But you already "proved" that it should be odd. Where is the cheat?

 2022-02-02, 13:05 #13 Dr Sardonicus     Feb 2017 Nowhere 22×31×47 Posts
 2022-02-10, 13:48 #14 Charles Kusniec     Aug 2020 Guarujá - Brasil 2E16 Posts The quantity of integers distributed along the number line is an odd quantity (continuation) If we were to list all the divisors of the number 0, we would need to list infinitely many numbers. Since for every non-zero number we have its positive and negative number, then 0 has an "even infinite number" of non-zero divisors. The "infinite quantity is even" because for every non-zero number there is always the positive and the negative number in pairs. Both positive and negative numbers are divisors of 0. Although the division 0/0 is of indefinite value, whatever the finite number result is, the remainder will always be 0. If the division 0/0 produces an infinite number, also the remainder will still be 0. This is because we cannot add any remainder other than 0 to a finite or infinite product to get back the number 0. Because of the remainder 0, we may consider that 0 is a divisor of 0 itself with an undefined result. Because it is not possible to have a negative zero different from a positive zero, then 0 will always have an "odd infinite number" of divisors. Just as the other squares have an odd finite number of divisors, 0 has an "odd infinite number" of divisors. Last fiddled with by Charles Kusniec on 2022-02-10 at 14:02
 2022-02-10, 13:50 #15 Charles Kusniec     Aug 2020 Guarujá - Brasil 2E16 Posts Dear Administrator, if it is possible, please join with https://www.mersenneforum.org/showthread.php?t=27545 . Thank you,
 2022-02-10, 13:53 #16 paulunderwood     Sep 2002 Database er0rr 2·11·191 Posts Cantor turns in his grave.
 2022-02-10, 13:54 #17 firejuggler     "Vincent" Apr 2010 Over the rainbow 5×569 Posts the odds of it being even or odd are even.
 2022-02-10, 14:07 #18 Dr Sardonicus     Feb 2017 Nowhere 22·31·47 Posts

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