[FOM] Concerning Logicism

[Harvey Friedman]

One way of looking at logicism is the effort to set up foundations for
mathematics using only concepts that have fundamental significance
throughout the entire intellectual landscape - rather than only in
situations pertaining to mathematics.

It is obvious that FOL= does have this kind of universality. However,
it is normally used in conjunction with the iterative concept of set
and axioms of set theory that arguably are very tied to mathematics.

So a reasonable objective for the logicist would be to extend FOL= in
such a way that it

1. Maintains the intellectual universality of FOL=.
2. Interprets standard f.o.m. formalisms like ZFC, fragments, and extensions.

An attractive system for the logicist is the two sorted system with
objects and sets of objects, equality between objects only, and a
binary function symbol F from objects to objects. The axioms are

i. The usual first order logic axioms for this setup.
ii. F is one-one and not onto.
iii. Full comprehension for sets of objects.

The justification for ii, that it is not tied to mathematics is this.
Given any two objects, there is the mental construct of putting the
two together in order. And mental constructs are particular kinds of
objects. So we have a one-one and not onto F.

THEOREM. The above system is mutually interpretable with Z_2.

We can use higher types to get to Z_n, each n >= 1. These interpret
vast swaths of mathematics.

Of course, we would like to get to ZFC in this way.

There is a way of doing this in the section on equivalence relations
in https://u.osu.edu/friedman.8/files/2014/01/Mental090514-29co41m.pdf.
However, it is more focused on the idea of flat foundations, and not
no such in the direction of logicism.

So this really needs to be reworked from the logicist point of view.
The basic idea is to consider the series of attributes

large
very large
very very large
very very very large
etcetera

on sets of objects. The idea is that this series is very robust in
that it is indistinguishable from starting at the second term:

very large
very very large
very very very large
very very very very large
etcetera

There are a few kinds of reasonable logicist friendly ways of
formalizing this. After I see some FOM comments, and/or develop this
further, I will move it to my numbered postings.

One way of setting this up is this.

1. Use Russell's type theory, at least for a few types. Use
extensionality and the choice version.
2. Each item in the above list is a set of sets of objects of type 0.
I.e., each item is an object of type 2.
3. The entire list is therefore an object of type 3, which is given by
a constant symbol of type 3.
4. Obvious axioms representing the idea that the elements of the list
represent higher and higher cardinalities of objects of type 1, of
order type omega.
5. Predicates for light faced definable sets at each type, with the
obvious axiom schemes (in fact, only single axioms are needed).
6. MAIN EVENT. The list, and the list with the first (or any) item
removed, are indistinguishable.

Of course, note that the list is VAGUE. And so this development falls
within a theory of VAGUENESS, establishing that basic intuitive
vagueness is extremely strong, logically.

The idea here is that the resulting system approaches, but does not
reach, serious large cardinals like a measurable cardinal. You get way
beyond measurables if you naturally continue with order type omega +
1:

large
very large
very very large
very very very large
etcetera
hyperlarge

And of course you can continue

large
very large
very very large
very very very large
etcetera
hyperlarge
very hyperlarge
very very hyperlarge
...

and so forth, with the obvious indistinguishability axioms of various
kinds pushing up into fancy kinds of measurable cardinals -
concentrators.

It would be interesting to hear from FOM Logicists, to see if this
sort of thing speaks to them.

Harvey Friedman