- #1
phoenixthoth
- 1,605
- 2
infinities := means "is defined to equal."
:= means "is defined to equal."
x_y means y is a subscript of x.
N:={0, 1, 2, ...}
[P,0](S):=S
[P,1](S):={x : x is a subset of S}
for n >= 1,
[P,n+1](S):={x: x is a subset of [P,n](S)}
card(S) is the cardinal number of S.
1_n:=[P,n](N).
aleph_n := card(1_n).
2_1:={x: x is an element of 1_n for some n in N}; i.e., 2_1 is the union of {[P,0](N), [P,1](N), [P,2](N), ...}.
what is card(2_1)?
for n>=1,
2_(n+1):={x: x is an element of [P,k](2_n) for some k in N}; i.e., 2_(n+1) is the union of
{[P,0](2_n), [P,1](2_n), [P,2](2_n), ...}.
what is card(2_n)?
3_1:={x: x is an element of 2_n for some n in N}.
for n>=1,
3_(n+1):={x: x is an element of [P,k](3_n) for some k in N}.
what is card(3_n)?
for m>=1,
(m+1)_1:={x: x is an element of m_n for some n in N}
for n>=1,
(m+1)_(n+1):={x: x is an element of [P,k](m+1_n) for some k in N}.
what is card(m_n)?
[1]:={x: x is an element of m_n for some m,n in N}.
for n>=1,
[n+1]:={x: x is an element of [P,k]([n]) for some k in N}.
what is card([n])?
[N,1]:={[n]: n is an element of N}.
for k>=1,
[N,k+1]:={[n]: n is an element of [P,j][N,k] for some j in N}.
what is card([N,k])?
:= means "is defined to equal."
x_y means y is a subscript of x.
N:={0, 1, 2, ...}
[P,0](S):=S
[P,1](S):={x : x is a subset of S}
for n >= 1,
[P,n+1](S):={x: x is a subset of [P,n](S)}
card(S) is the cardinal number of S.
1_n:=[P,n](N).
aleph_n := card(1_n).
2_1:={x: x is an element of 1_n for some n in N}; i.e., 2_1 is the union of {[P,0](N), [P,1](N), [P,2](N), ...}.
what is card(2_1)?
for n>=1,
2_(n+1):={x: x is an element of [P,k](2_n) for some k in N}; i.e., 2_(n+1) is the union of
{[P,0](2_n), [P,1](2_n), [P,2](2_n), ...}.
what is card(2_n)?
3_1:={x: x is an element of 2_n for some n in N}.
for n>=1,
3_(n+1):={x: x is an element of [P,k](3_n) for some k in N}.
what is card(3_n)?
for m>=1,
(m+1)_1:={x: x is an element of m_n for some n in N}
for n>=1,
(m+1)_(n+1):={x: x is an element of [P,k](m+1_n) for some k in N}.
what is card(m_n)?
[1]:={x: x is an element of m_n for some m,n in N}.
for n>=1,
[n+1]:={x: x is an element of [P,k]([n]) for some k in N}.
what is card([n])?
[N,1]:={[n]: n is an element of N}.
for k>=1,
[N,k+1]:={[n]: n is an element of [P,j][N,k] for some j in N}.
what is card([N,k])?
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