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**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])?

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