Re: Antonelli’s Logicism

[Sam Sanders]

Dear Joe and FOM,

I have some comments on the paper you mention.
I am also curious what other people would think.  

> I strongly recommend this paper by the late Professor Aldo Antonelli:
> 
> https://projecteuclid.org/download/pdf_1/euclid.ndjfl/1276284780
> 
> He provides the most satisfactory version of logicism that I know of, by using a “Frege quantifier” F that provides a logical representation of a notion that could also be described as “There are at least as many B as A” or “There is an injection of A into B” or “|A|<=|B|”.

So the framework is based on injections.  As shown by myself and Dag Normann ( https://arxiv.org/abs/2007.07560 ), it is consistent 
with rather strong systems (namely the third-order version of second-order arithmetic by Sieg-Feferman) that there is an injection 
from [0,1] to N (or 2^N to N, of course).  The same holds for bijections, btw. 

For some questions:

Does an injection from [0,1] to N pose any problems for logicists or this particular account of logicism?

If so, what are the logicists views on Kleene’s 3E, the comprehension functional needed to prove that there is no injection from [0,1] to N?

Best,

Sam