Imagine a collections of dots connected by one-way arrows, starting at one dot (the left-most) as here:

o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> ...

Each dot has exactly one SUCCESSOR as indicated by a one-way arrow in this diagram. Each dot has at most one PREDECESSOR, also indicated by a one-way arrow. The diagram represents an endless succession of dots. The left-most dot has no predecessor.

We can formalize this structure using the language of ordinary set theory. Let D be a non-empty set (the collection of "dots"). Let S be a function on D, i.e.

ALL(a):[a in D => S(a) in D]

S(x)=y means the successor of x is y, and that the predecessor of y is x.

The functionality of S guarantees that every element of D has a unique successor, as required.

To ensure that every element of D has at most one predecessor, we have:

ALL(a):[a in D => ALL(b):ALL(c):[b in D & c in D => [S(b)=a & S(c)=a => b=c]]]

Or equivalently:

ALL(a):ALL(b):[a in D & b in D => [S(a)=S(b) => a=b] (i.e. s is injective on D)

To ensure that an element of D has no predecessor, we have:

EXIST(a):[a in D & ALL(b):[b in D => ~f(b)=a] (i.e. s is not surjective)

Thus, if we have set D and function S: D --> D where S is injective and not surjective, then we can say that D is INFINITE.

Comments?

Dan

Download my DC Proof 2.0 freeware at http://www.dcproof.com
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On Thursday, December 23, 2021 at 12:26:29 PM UTC-8, Dan Christensen wrote:
> Imagine a collections of dots connected by one-way arrows, starting at one dot (the left-most) as here:
>
> o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> o--> ...
>
> Each dot has exactly one SUCCESSOR as indicated by a one-way arrow in this diagram. Each dot has at most one PREDECESSOR, also indicated by a one-way arrow. The diagram represents an endless succession of dots. The left-most dot has no predecessor.
>
> We can formalize this structure using the language of ordinary set theory. Let D be a non-empty set (the collection of "dots"). Let S be a function on D, i.e.
>
> ALL(a):[a in D => S(a) in D]
>
> S(x)=y means the successor of x is y, and that the predecessor of y is x.
>
> The functionality of S guarantees that every element of D has a unique successor, as required.
>
> To ensure that every element of D has at most one predecessor, we have:
>
> ALL(a):[a in D => ALL(b):ALL(c):[b in D & c in D => [S(b)=a & S(c)=a => b=c]]]
>
> Or equivalently:
>
> ALL(a):ALL(b):[a in D & b in D => [S(a)=S(b) => a=b] (i.e. s is injective on D)
>
> To ensure that an element of D has no predecessor, we have:
>
> EXIST(a):[a in D & ALL(b):[b in D => ~f(b)=a] (i.e. s is not surjective)
>
> Thus, if we have set D and function S: D --> D where S is injective and not surjective, then we can say that D is INFINITE.
>
> Comments?
>
> Dan
>
> Download my DC Proof 2.0 freeware at http://www.dcproof.com
> Visit my Math Blog at http://www.dcproof.wordpress.com

Put endless finite together in concept and it suggests the "infinity" completed...

Mitchell Raemsch