[FOM] Non-distributive predication

Tom McKay tjmckay at syr.edu
Mon Jul 14 09:40:00 EDT 2003

On July 12, Dean Bruckner wrote (in response to my posting of July 11):

>Where I may disagree with you is in the possibility of a "set of everything"
     (E x) (y) [ y among x ]

I don't propose a set of everything. The theory has as a theorem that 
some things are such that everything is among them (if there are any 
things at all). But the theory does not propose that there are any 
sets at all.

>Only true, I think, when there are finitely many things.  The idea of an
>infinite set depends on that distinction between "member of" and "subset of"
>that among theory obliterates.  Turning to your plural induction axiom, you
>need to state somewhere that the successor of any x is not within the set of
>     (X among P) (E y among P) [ y succeeds X & ~ y among X ]
>But is P among P or not?  If it is, there is an object among P that is not
>among P.  There may be a way around this, but I haven't see one.

I don't understand this. I don't see the connection between what I am 
saying and any issues that have to do with whether there are 
infinitely many things.

>I had a long interchange with Harvey about this, his view was
>that the onus is on "among" theory to show it can handle "ordinary
>mathematics".  I don't know enough ordinary mathematics to do this.

I have made the more definite proposal that this handles all of the 
mathematics that second-order arithmetic handles, in essentially the 
same way. I too feel a bit inadequate mathematically as far as 
carrying out that development, and that is part of why I sent the 
message to fom.

Thanks for the comments.

Tom McKay
Philosophy Department
Syracuse University
Syracuse  NY  13244
315 443 4501
tjmckay at syr.edu

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