Re: Contradiction of bijections as a measure for infinite sets

On 3/25/2024 5:04 PM, Jim Burns wrote:

On 3/25/2024 6:11 PM, WM wrote:

Le 25/03/2024 à 20:38, Jim Burns a écrit :

 

[...]

>
You cannot add a natural number to ℕ.

 You cannot add a natural number to ℕ ==
ℕ holds all sizes of sets for which
changing by one element changes the set's size.

The set of odd numbers is infinite. There are an infinite number of natural numbers. infinity = infinity. Density is another matter.
.01->.001 can represent infinity... Its denser, so to speak.

 ℕ doesn't have any of those sizes.
 ℕ isn't a set for which
changing by one element changes the set's size.
 

But a bijection of ℕ with |E = {2, 4, 6, ...}
would prove that both sets have
the same number of elements.

 ℕ and 𝔼 aren't sets for which
changing by one element changes the set's size.
 

Adding an element to |E destroys this state

 1 ⟼ 2
n ⟼ n+2
𝔼∪{1} ⇉ 𝔼
 

and shows ℕ is larger than ℕ.
Contradiction!

 | CAESAR (recovering his self-possession):
| Pardon him, Theodotus:
| he is a barbarian, and thinks that
| the customs of his tribe and island are
| the laws of nature.
|
George Bernard Shaw, "Caesar and Cleopatra" (1898)