Re: Contradiction of bijections as a measure for infinite sets

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.
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)