The tluth
Aatu Koskensilta
theoretical statement amounts to nothing but its provability in, say,
ZFC. Presumably such people interprete such statements as "it is
provable in ZFC that there exists an uncountable set" and "it is not
provable in ZFC that ZFC is consistent" realistically and do not wish to
claim that their meaning has anything to do with their provability or
non-provability in ZFC. For the purpose of this post we can just assume
that these people take all arithmetical statements at face value.
We can now try to make sense of what these people are trying to say by
defining a predicate Tlue by setting
Tlue("P") <=> True("P") and P is arithmetical or ZFC|-P and P is not
arithmetical
Here we take "P is arithmetical" to mean "P is provably equivalent in
ZFC to an arithmetical statement".
Tluth has various curious properties. For example, while it's tlue that
the existence of an inaccessible cardinal implies consistency of ZFC,
the tluth of the existence of an inaccessible cardinal implies that ZFC
is inconsistent. Of course, if we take seriously the idea that "there
exists an inaccessible cardinal --> Con(ZFC)" is meaningless there's
nothing particularly problematic about this. It does show, however, that
in order to avoid all sorts of confusions, tluth and (meaningless)
mathematical truth of set theoretic sentences must be clearly
distinguished.
Another property of Tluth is that it's Delta^1_1 and hence most claims
about Tluth of this or that are, by the doctrine hypothetically put
forward, meaningless, since they're not equivalent (in ZFC or otherwise)
to arithmetical statements. This also has the consequence that for many
sentences the tluth of the tluth of the sentence is not equivalent to
the tluth of the sentence. Of course, this phenomenon is familiar from
the study of ordinary truth predicates, and we could try to define Tlue
informally based on eg. a predicative understanding by allowing
iterations of Tluth along predicative well-orderings and so forth.
I'd like to hear from people who say that the meaning of set theoretical
statements amounts to their provability in ZFC (or any particular formal
theory) whether Tluth is in fact an accurate (partial) rendition of
their ideas about meaning of mathematical statements.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, darüber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus
Aatu Koskensilta
> Another property of Tluth is that it's Delta^1_1 and hence most claims
> about Tluth of this or that are, by the doctrine hypothetically put
> forward, meaningless, since they're not equivalent (in ZFC or otherwise)
> to arithmetical statements. This also has the consequence that for many
> sentences the tluth of the tluth of the sentence is not equivalent to
> the tluth of the sentence.
This is not quite correct. This only happens when we're dealing with
general facts about tluth of sentences of unbounded complexity.
I'll post a more refined proposition after I've thought about this a bit.
Aatu Koskensilta
difficulties and ambiguities in the original.
I wish to provide an explicit formulation of the idea that a set
theoretic statement is meaningless when taken at face value, and should
be interpreted as the assertion that it is provable in ZFC, while
arithmetical statements - such a derivability or non-derivability of
some sentence in ZFC - are meaningful as ordinarily interpreted.
The first problem comes with deciding which sentences are arithmetical
and which are not. There two obvious choices:
1) A sentence P (in the language of set theory) is arithmetical if all
of its quantifiers are restricted to V_omega
2) A sentence P is arithmetical if it is equivalent in ZFC to a
sentence the quantifiers of which are all restricted to V_omega
Since we're interested in meaning, and equivalence in ZFC certainly does
not preserve meaning, alternative 1 seems more reasonable.
We define an informal predicate Tlue by setting
Tlue("P") <=> P is arithmetical and true \/ P can be seen to be
meaningful based on meaningfulness of arithmetical
truth and is true \/ ZFC |- P and P is meaningless
Here "P can be seen to be meaningful based on meaningfulness of
arithmetical truth" means something like "the concepts occuring in P can
be defined by iterating arithmetical truth along a well-ordering
recognizable as such by a shorter iteration of arithmetical truth",
where well-orderdedness of < is understood in the sense "whatever
properties of natural numbers there are, it is impossible that any of
them lacks a <-least element". Standard explications of these can be
given along the lines of the analysis of predicative acceptability, but
these need not concern us here.
There is a problem with the above definition, in that "P is meaningless"
is itself meaningless, since predicative meaningfulness is not a
predicative property. Of course, some sentences can be readily seen to
be meaningless, such as "there exists an inaccessible cardinal". I'll
just ignore this problem.
As mentioned in the earlier version, Tlue has various curious
properties. Perhaps the most striking of these is that Tlue is not
compositional, that is Tluth of a sentence is not defined in terms of
Tluth of the components of the sentence. Thus we can have eg. that it's
tlue that the existence of an inaccessible cardinal implies the
consistency of ZFC while the tluth of there existing an inaccessible
implies the *in*consistency of ZFC. Since usually we understand an
assertion such as "P implies Q" to be essentially equivalent to "what
'P' means implies what 'Q' means", this seems odd. Of course, since eg.
"there exists an inaccessible cardinal" is taken to be meaningless, this
doesn't mean that there's anything contradictory about this. It does
mean that this interpretation of set theoretical statements is at odds
with the way they are usually employed in mathematical contexts,
however. All sorts of queer features of Tlue could be listed, such as
that it is not tlue that the tluth of a statement implies that statement
and so forth, following from the fact that for (meaningless) set
theoretic sentences tluth is just the provability predicate for ZFC.
(This explains an earlier comment of mine to David Ullrich that while
"there exists an uncountable set" is not equivalent (in the mathematical
sense) to "ZFC |- there is an uncountable set", we can't conclude from
this that "there exists an uncountable set" can't be taken to mean that
"ZFC |- there is an uncoutable set" but only that any theory of meaning
according to which this is so must be rather strange, and in particular
must equate the meaning of statements that are not mathematically
equivalent).
Now, one can conceiveably think that a set theoretic statement P means
Tlue("P"), although this implies giving up all sorts of properties
usually associated with truth and meaning. Personally I utterly fail to
understand *why* one would do so, though; it is a perfectly reasonable
to hold that set theoretic sentences are not meaningful, but assigning
to them a strange and wildly ad hoc interpretation, on the other hand,
seems just silly.
I'd like to know whether people who assert that the meaning of, say,
there existing an uncountable set is that the formalization of this is
provable in ZFC, consider Tluth to be a faithful (partial) rendition of
their position - and if so, what makes them adopt this rather queer
position.
--
Aatu Koskensilta (aatu.kos...@xortec.fi)
"Wovon man nicht sprechen kann, daruber muss man schweigen"