Post by Jim BurnsPost by WMPost by FromTheRaftersNot showing a bijection is not the same as
not having a bijection.
Showing a not bijection proves
different sizes of sets.
I'll call that a Mückenheim.set.
Mückenheim.sets are finiteⁿᵒᵗᐧᵂᴹ sets.
No.
The set {1, 2, 3, 4, 5, ..., ω} is infinite,
although there is a last element.
The set {1, 2, 3, 4, 5, ..., ω} is not.Mückenheim.
Consider not.bijection
g: {1,2,3,4,5,...,ω} ⇉ {1,2,3,4,5,...,ω}
g(ω) = 1
g(n) = n+1 otherwise
{1,2,3,4,5,...,ω} is not a different size than
{1,2,3,4,5,...,ω}
finiteⁿᵒᵗᐧᵂᴹ does not mean what you think it means.
Under f(x) = 2x
we get the image {2, 4, 6, 8, 10, ..., 2ω].
The mapping restricted to the natural numbers shows
less evens than naturals.
s/less/fewer
No.
f(x) = 2x
not.bijection f: ℕ ⇉ ℕ
f(ℕ) = 𝔼 ≠ ℕ
If Mückenheim ℕ then |ℕ| ≠ |ℕ|
not.Mückenheim ℕ
One can't draw valid conclusions from
the assertion that ℕ is Mückenheim.
Post by Jim BurnsPost by WMBut we can assume that
very simple mappings like f(x) = x are true
even for dark elements.
Therefore
between the rational numbers and the natural numbers
f(n) = n/1 can be accepted,
also f(n) = 1/n,
but not f(n) = 2n.
Where are darkᵂᴹ numbers introduced?
They are the places where
Bob rests when the mapping is finished.
Where in these definitions
did I introduce darkᵂᴹ numbers?
For k,m ∈ ℕᴶᵛᴺ define addition such that
k+m = nₖₘ ⟺
exists Mückenheim.3.tuple.sequence ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩
such that
⟨k,0,k⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
⟨k,m,nₖₘ⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
∀i ∈ m: ∃!⟨k,i,j⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∧
∀⟨k,i,j⟩ ∈ ⟨⟨k,0,k⟩,…,⟨k,m,nₖₘ⟩⟩ ∋ ⟨k,i⁺¹,j⁺¹⟩
lemma. nₖₘ ∈ ℕᴶᵛᴺ
For k,m ∈ ℕᴶᵛᴺ define multiplication such that
k⋅m = n′ₖₘ ⟺
exists Mückenheim.3.tuple.sequence ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩
such that
⟨k,0,0⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
⟨k,m,n′ₖₘ⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
∀i ∈ m: ∃!⟨k,i,j⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∧
∀⟨k,i,j⟩ ∈ ⟨⟨k,0,0⟩,…,⟨k,m,n′ₖₘ⟩⟩ ∋ ⟨k,i⁺¹,j+k⟩
lemma. n′ₖₘ ∈ ℕᴶᵛᴺ
For k ∈ ℕᴶᵛᴺ define fraction iₖ/jₖ such that
sₖ = max{h: (h-1)(h-2)/2 < k }
iₖ = k-(sₖ-1)(sₖ-2)/2
jₖ = sₖ-iₖ
lemma. iₖ,jₖ ∈ ℕᴶᵛᴺ
For i,j ∈ ℕᴶᵛᴺ define index kᵢⱼ such that
sᵢⱼ = i+j
kᵢⱼ = (sᵢⱼ-1)(sᵢⱼ)-2)/2+i
lemma. kᵢⱼ ∈ ℕᴶᵛᴺ
lemma. kᵢⱼ = k ⟺ ⟨i,j⟩ = ⟨iₖ,jₖ⟩