Re: Replacement of Cardinality

On 8/15/2024 7:16 PM, Moebius wrote:

Am 16.08.2024 um 00:51 schrieb Jim Burns:

Half or more of my proofs to WM say
"Assume otherwise... However... Contradiction."

>
Yeah, to be precise a proof by contradiction
assumes a STATEMENT/CLAIM.

Many times, a false existence claim.

But maybe I'm crazy.

>
No, you aren't, I guess.
>

[nonsense deleted]

Well.known, ancient proof undeleted (below).
If that is nonsense, everything is nonsense.
That proof uses properties such as
well.order and unique.prime.factorization
which p₂ and q₂ would have if they existed
in order to show that they don't exist.
Yes,
reasoning from those properties of p₂ and q₂
generates nonsense, contradiction.
However,
the nonsense is the point,
the reason that we conclude that p₂ and q₂ don't exist.
Another example.
ZFC+"An inaccessible ordinal exists"  proves
ZFC
In order to make that proof,
we need the _definition_ of 'inaccessible cardinal'.
https://en.wikipedia.org/wiki/Inaccessible_cardinal
You will not prove that an inaccessible cardinal _exists_
not from ZFC+"An inaccessible ordinal exists" --
or, if you do, Gödel shows that proves _inconsistency_
----
√2 is irrational.
⎛ Assume otherwise.
⎜ Assume p₃,q₃ ∈ ℕ₁:  p₃⋅p₃ = 2⋅q₃⋅q₃
⎜
⎜ p₃ ∈ {p ∈ ℕ₁: ∃q ∈ ℕ₁: p⋅p = 2⋅q⋅q}
⎜ p₂ = min.{p ∈ ℕ₁: ∃q ∈ ℕ₁: p⋅p = 2⋅q⋅q}
⎜ ∃q₂ ∈ ℕ₁: p₂⋅p₂ = 2⋅q₂⋅q₂
⎜ ¬∃p₁ < p₂: ∃q₁ ∈ ℕ₁: p₁⋅p₁ = 2⋅q₁⋅q₁
⎜
⎜ However,
⎜ p₂⋅p₂ = 2⋅q₂⋅q₂
⎜ 2 is prime.
⎜ 2|p₂ or 2|p₂
⎜ p₂ = 2⋅p₁
⎜ 2⋅p₁⋅2⋅p₁ = 2⋅q₂⋅q₂
⎜ 2⋅p₁⋅p₁ = q₂⋅q₂
⎜ 2|q₂ or 2|q₂
⎜ q₂ = 2⋅q₁
⎜ 2⋅p₁⋅p₁ = 2⋅q₁⋅2⋅q₁
⎜ p₁⋅p₁ = 2⋅q₁⋅q₁  and p₁ < p₂
⎝ Contradiction.