[FOM] Functions from sets to ordinals

[Joe Shipman]

Can’t find a paper on what Mancosu did that I don’t have to pay for, what I could glean from abstracts and reviews and math overflow references is that it probably answers my questions in the way I expected (namely, by technical arguments requiring ultrafilters to construct the desired functions). I’ll look deeper next time I’m in a library.

— JS

Sent from my iPhone

> On Dec 28, 2019, at 12:30 AM, Richard Kimberly Heck <richard_heck at brown.edu> wrote:
> 
> 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
> 
> 
> -- 
> 
> ----------------------------
> Richard Kimberly (Riki) Heck
> Professor of Philosophy
> Brown University
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