[FOM] Functions from sets to ordinals

[Richard Kimberly Heck]

On 12/27/19 12:00 AM, Joe Shipman wrote:
> The following functions from hereditarily finite sets to natural numbers have natural extensions to functions from arbitrary sets to ordinals:
>
> 1) cardinality 
> 2) rank
> 3) nim-value (the nim-value of a set is the first ordinal not the nim-value of any of its members)
> 4) cardinality of V_rank
>
> These functions all have the property that if A is a subset of B, then f(A) <= f(B).
>
> A fifth function satisfying this property is the Ackermann enumeration of HF sets:
> 4) f(A) is the sum over all x in A of 2^f(x).
>
> The finite ordinals themselves map as follows in these cases:
> 1) 0,1,2,3,...
> 2) 0,1,2,3,...
> 3) 0,1,2,3,...
> 4) 0,1,2,4,16,65536,...
> 5) 0,1,3,11,2059,...
>
> How far can this last function be generalized? If you have a global choice function or a global well-ordering given in a class theory like NBG, can you define a global well-ordering which reduces to 5) for HF sets and has the monotonicity property?

Hi, Joe,

Hope the holidays have been enjoyable and refreshing!

There's some recent work by Paolo Mancosu that I think bears upon this.
He's interested more in the case where if A⊂B, then f(A)<f(B) (in large
part, because of an interest in non-Cantorian notions of size), but it's
obviously a closely related issue. There are quite a few results here,
many of which originate with Zermelo.

Riki


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Richard Kimberly (Riki) Heck
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Brown University

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