FOM: Determinacy of statements
Joe Shipman
shipman at savera.com
Fri Jun 23 11:13:15 EDT 2000
There is a logical sign error in my previous posting; I wrote
"indeterminate" in one place where I meant to say "determinate". The
correct paragraph reads
>>
Put another way, I want to set the bar high for indeterminacy, so that I
would like to say, to begin with, "A statement is determinate iff its
negation is determinate" and "All theorems of PA are true (and
therefore determinate)". Under these assumptions, "GC is indeterminate"
implies "~GC is indeterminate" which implies "~GC is not a theorem of
PA" which implies GC, an unsatisfactory situation.
<<
Richman asks
>>Can't we understand the meaning of the statement P(10^(10^10)) without
claiming that it is determinate?<<
Some definitions are in order, because it seems that the participants in
this discussion may be using words in different senses.
For a statement Phi to be "determinate" means, according to me, that
1) it is meaningful, and
2) there is only one possible truth-value for it, whether or not human
investigations could possibly ever discover this value.
For Phi to be "indeterminate" means that it is not determinate, that is,
either there are possible "worlds" (whatever that means) in which it is
true and other possible worlds in which it is false, or Phi is not
meaningful.
A statement that is not meaningful is either vague or meaningless.
"Every Hausdorff quaternion has a covariant NP-hard diffeomorphism" is
meaningless. "Every true pi^0_1 arithmetic sentence is a consequence of
some consistent large-cardinal axiom" is vague. These are not absolute
concepts: "The GCH holds above some cardinal kappa" is meaningful to
committed set-theoretic Platonists, vague to me, and meaningless to
Professor Sazonov.
Note that since my definition of "determinate" excludes statements that
are not meaningful, tautologies like "For all degenerate Gaussian
snozzcumbers, there exists a snozzcumber that is either degenerate or
Gaussian" are indeterminate.
Before you can talk about the determinacy of a statement, you should
agree about its meaningfulness. If you agree about an "intended model"
for the language of the statement (that is, you agree you are talking
about the same thing), then both meaningfulness and determinacy appear
to follow. In the case of the twin prime conjecture, if you accept that
"the" set of integers (together with the operations of addition and
multiplication) exists as a completed whole, then the TPC is determinate
because it is either true or false in that intended model. But even if
you don't accept this, so that you need not agree that TPC is
determinate, you may still agree that TPC is meaningful, because you may
accept that the concepts of integer and primeness are meaningful and use
quantificational language in an ordinary way. (And, while not agreeing
that TPC is determinate, you will probably not claim to know that it is
indeterminate because the possibility exists that someone will present
you with an acceptable proof tomorrow.)
To illustrate, consider the "singleton prime conjecture", that "For all
n there exists p>n with p prime". Before the (constructive) proof of
this statement (factor n!+1 into primes) was discovered, it would have
been possible to deny that the statement was determinate, but
afterwards, even denying the existence of a completed infinity of
integers, one must still admit that SPC is "true" and hence determinate.
Here the constructiveness of the proof is important -- in Harvey's last
posting, where he cites the theorems of Mordell-Weil, Roth, Siegel, and
Faltings which are all of the pi^0_3 logical form "For all m there exist
finitely many n such that [computable predicate of m and n]", some
mathematicians will still refuse to admit the truth of the statements
because they are not constructively proven.
-- Joe Shipman
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