[FOM] Logic/Syntax versus Arithmetic

[katzmik at macs.biu.ac.il]

Tim,

I have the impression again that you are talking about the metalanguage three
here.  Developing a formal object theory for "the" natural numbers necessarily
takes you beyond the metanaturals.

MK

On Thu, February 27, 2020 03:44, Timothy Y. Chow wrote:
> Alan Weir wrote:
>
>> I discuss (in my book chapter six) the concept of equiform tokens,
>> noting how complex and resistant to definition it is (a sound and an ink
>> mark in some font can be equiform) but assume it is acceptable to take
>> it as primitive for the purposes of syntax. Then with, for example, a
>> bit of mereology, a 'tipe' can be taken to be the mereological sum of
>> all (actual) tokens equiform to a given actual token. This gives us
>> finitely many tipes but nothing like numbers because the tipes aren't
>> structured in the right sort of way.
>
> So what's wrong with this?  Introduce the concept of equicardinal tokens
> (in effect, strings of symbols that contain the same number of symbols).
> Note how complex and resistant to definition the word "equicardinal" is
> but assume it is acceptable to take it as primitive.  A "natural number
> tipe" is then a mereological sum of all actual tokens equicardinal to a
> given actual token.  This gives us finite many natural number tipes.
>
> Even if this particular plan has some subtle flaw---if you're going to
> talk about wffs then somewhere along the way you're going to have to talk
> about "strings of 3 symbols" and if you've made some sense of that then
> you've made sense of the number 3.  Haven't you?
>
> Tim
>