[FOM] Re: FOM: set theory

[Sandy Hodges]

Fred Richman wrote:

I think it reasonable to attack set theory on the grounds that it does
not capture our ordinary intuitions.

SH:
I need a more concrete example to be able to think at all; take the
difference between Peter Aczel's hyperset theory and ordinary ZF, based
on the difference between the Foundation axiom and Aczel's
Anti-Foundation axiom.    Hyperset theory seems more minimal (in the
sense of Matt Insall's post) in that ordinary sets are just those
hypersets which happen to be well-founded.    I don't understand what it
could mean to say that hypersets do or do not not exist, so I don't have
an intuition that says they exist, nor one that says they do not.

I don't accept either of the Barwise & Etchemendy accounts that are
based on hypersets.   But suppose I did.   Then I might say that the
Barwise and Etchemendy accounts, provide a truth theory with this
property:

(1)  Sentences in the object language can call other sentences in the
object language true.   The truth theory calls certain sentences true
and others not true.   Object language sentences expressing the theory's
conclusions, are called true by theory.

Truth theories not based on hypersets often lack property (1).    For
example, the Liar sentence may be called "paradoxical," or "neither true
nor false," or "meaningless."     Now a sentence can't be paradoxical
and also true, or meaningless and also true.   So
  (2) "The Liar sentence is not true"
expresses in the object language just what these theories say about the
Liar sentence in the meta-language.    Nonetheless, (2) may not be
called true, but meaningless or paradoxical.

So the claim might be made:
 (3) No language can talk about its own truth, this can only be done in
a meta-language.
But (3) seems to be self-refuting: in what language is it spoken?

So a situation might arise where hypersets allow for a consistent truth
theory, while ordinary sets only a self-refuting one.   I would consider
it absurd, if this ever does happen, to say that the consistent truth
theory is nonetheless not right, because the Foundation axiom has to be
true.

But I would not agree that this counts as the Foundation axiom being
rejected because it conflicts with our intuitions about the meaning of
English words such as "collection" or "true".

Some truth theories fail because they can't handle situations such as
the following:  A person A hears a person B speak a sentence, and A says
to C (who did not hear B), "B's sentence was true."    If a theory is
rejected for not handling this, then it is rejected for not handling an
everyday situation, but not for disagreeing with everyday usage.   A, B,
and C in the above story might be computers, speaking a language of
absolute precision.   However precise the language used, there are truth
theories that work fine as long as every individual is assumed to do
nothing but state mathematical proofs, but which fail as theories about
individuals who have different information, and need to communicate with
one another.   This is a way for a theory to fail that is not a case of
disagreeing with intuition about the meaning of some natural language
word.

--
On the other hand the Foundation axiom is more elegant than the AFA.
- ------- -- ---- - --- -- --------- -----
Sandy Hodges / Alameda,  California,   USA
mail to SandyHodges at attbi.com will reach me.