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Notes by RBJ on

Which Undecidable Mathematical Sentences Have Determinate Truth Values?

by Hartry Field

Section 1.Metaphysical Preamble
Section 2.The Objectivity Issue
Section 3.Putnam's "Models and Reality" and the concepts of finiteness and natural number
Section 4.Extreme Anti-objectivism
RBJ's Conclusions
RBJ conjecture
The Final Word?

Metaphysical Preamble

Field begins by identifying three kinds of ontological stance in relation to mathematics:
fictionalism 
"standard" platonismnumber theory and set theory and the theory of real numbers are each about a determinate mathematical structure
plenitudinous platonisma theory is about its models

[Clearly it is possible to have something between standard and plenitudinous platonism, in which the notion of standard model includes more than one but not all models of the theory. One would also expect this to vary with the theory, e.g. we expect a single model of the natural numbers, multiple (but not all) models of set theory (perhaps the well-founded ones, to eliminate non-standard arithmetic), but all the models of group theory.]

The Objectivity Issue

Now Field puts aside fictionalism and looks at the difference between the two forms of platonism. This he says comes down to asking: "Which undecidable mathematical sentences have determinate truth values?".

The term "objectivist" is used for people who answer this question in relation to arithmetic with the answer "all".

An "extreme anti-objectivist" is someone who answers "none". [and would raise no objections to an -inconsistent extension?]

Field is prepared to accept an anti-objectivist stance for set theory, but finds this implausible for arithmetic. His objections in the case of arithmetic arise from "the feeling that we have a determinate notion of finitude", from which the determinacy of the the concept of natural number flows. [Personally, I have a feeling that we have a determinate notion of natural number, from which the determinateness of the concept of finitude flows.]

Putnam's "Models and Reality" and the concepts of finiteness and natural number

Now Field refers to a paper by Hilary Putnam ("Models and Reality", 1980, Journal of Symbolic Logic 45: 464-482) where Putnam argues against the objectivist position in set theory. Field argues that the scope of this argument is limited and that it does not apply to the concept of finitude.

The argument takes place in relation to a formal theory which consists of

  1. A set theory with urelements (objects which are not sets)
  2. Some extra vocabulary of physical concepts.
Semantically it appears (though he doesn't explicitly talk about semantics) that we are not allowed to pin down the intended interpretation or interpretations of the set theory, but we are allowed to pin down the semantics of the physical vocabulary. What he definitely does say is that the physical vocabulary is determinate, and that from this fact and the following two "cosmological assumptions":
  • time is infinite in extent
  • time is Archimedean
the determinacy of the concept of finiteness follows.

Extreme Anti-objectivism

I think the point of this section is to show that the cosmological hypotheses are not only sufficient but necessary, i.e. that extreme anti-objectivism is tenable in default of metaphysical assumptions (though I don't think he claims to prove that). Field mainly devotes the section to discrediting various arguments which attempt to show determinacy of sentences not decided in PA by using Godel's theorem.

RBJ's Conclusions

I believe that the "determinacy" of mathematical statements is primarily dependent on the precision we can give to the semantics of the language in which they are expressed. (A digression to identify a less ambiguous term than "determinacy" is desirable here.) If we are dealing with mathematics expressed in first order logic, then the semantics of the logic itself are pretty well nailed down. If the theory under consideration concerns a unique structure (up to isomorphism) then we know that each (closed) sentence will have a definite truth value under that interpretation, and there will only be indeterminacy if there is some substantive ambiguity about what this unique intended interpretation is. This is true, even if, as in the case of arithmetic, there can be no complete recursive axiomatisation of the theory, which will normally be the case where there is a unique intended interpretation. So, it seems to me, that there is only indeterminateness in first order arithmetic to the extent that there is some real ambiguity about the intended interpretation, the natural numbers or the arithmetic operators.

This paper appears to be addressing the problem in two steps. First by showing that the determinacy of arithmetic follows from the determinacy of the concept of finiteness, and then by showing that the determinacy of the concept of finiteness follows from the determinacy of certain metaphysical concepts.

Now I haven't checked the detail of the arguments for these connections, (and this would be tricky, even if I thought it desirable, because the electronic copy I have been reading omits the mathematical symbols!), but this doesn't seem to me controversial and I'm quite prepared to take that on trust.

The problem I have with this tack, is, that the concepts which are being used to render determinate the concept of natural number are themselves, in my subjective opinion, not so clear as that of the natural numbers.

So I don't see, even if Field is wholly successful in establishing the connections, that they add anything to my confidence in the determinacy of arithmetic (which is in fact, very high). Most especially, it seems to me odd, to base conclusions about the natural numbers on suppositions about the nature of time.

Field's main objective however seems to be not to argue that the cosmological assumptions suffice, but to argue that objectivism depends upon them, and perhaps that anything weaker than extreme anti-objectivism depends upon them.

He does this mainly by dismantling objectivist arguments based on Gödel's incompleteness results. Gödel's incompleteness results seem to me an implausible approach to supporting objectivism.

In my opinion the best approach is to argue the uniqueness of the intended interpretation and explain how from that and from the well definedness of the semantics of first order logic determinateness follows.

Alternatively an argument may be based on the requirement for -consistency. To do this we first observe that to regard all sentences not decided in PA as indeterminate is incorrect, since any sentence which is -inconsistent with PA is clearly false (in the intended model) whether or not it can be disproven in PA (assuming PA sound).

This seems to me to provide a conclusive argument against extreme anti-objectivism. The extreme anti-objectivist (as defined by Field) is surely involved in the wholly implausible hypothesis that an -inconsistent extension of PA is just as acceptable a theory of arithmetic as any other.

If it is accepted that consideration of the need for -consistency eliminates extreme anti-objectivism we can then ask whether it leaves room for any anti-objectivism at all, i.e. are there any choices among PA undecidables which are not determined by the requirement for -consistency?

Conjecture/Question

Consider the theory WPA which is obtained by adding to Peano Arithmetic (PA) the negation of every closed formula which is undecidable in PA and -inconsistent with PA, and then taking the closure under derivability in PA. Assuming that PA is sound (axioms true, rules preserve truth) then:

Facts:
  1. WPA is not recursive, since it includes all true (and only true) consistency claims for recursive axiom systems (whose negations are -inconsistent with PA because they assert the existence of a derivation of a contradiction though no such derivation exists)
  2. WPA is consistent, since it was obtained by adding only true sentences of arithmetic.
Conjecture:
  • WPA is "true arithmetic", the set of sentences of PA which are true in the standard model.

Corollary:

  • There is only one -consistent decidable extension of PA.
The odds are that if this is true it is known to be true. And probably easy to prove.
Flawed Proof of Conjecture:
  1. Suppose that there is a statement s of true arithmetic which is not provable in WPA.
  2. Consider the theory PA+s
  3. PA+s is consistent (assuming PA is sound)
  4. The consistency of PA+s is provable in WPA.
  5. By a result of Feferman ([Feferman60] Theorem 6.2) if PA+s proves t then PA+Con(PA+s) proves t
  6. WPA proves s
  7. by reductio absurdum there is no theorem of true arithmetic which is not a member of WPA
Nice try, but when I actually look at [Feferman60] Theorem 6.2 doesn't say what I thought it did. It says that PA+s is interpretable in PA+Con(PA+s), which makes WPA very strong but doesn't quite establish that it is "true arithmetic".

What we need here is a real logician, this half baked amateur philosopher can't cut it.

I posted the problem to FOM and got an answer from Bob Solovay who said that my conjecture is false, because WPA is definable in arithmetic and if my conjecture were true then truth in first order arithmetic would be definable in first order arithmetic (which it isn't, since if it were diagonalisation would yield the liar paradox).

The Final Word?

OK, so my conjecture is false. But what I needed was the corollary, and it looks like the corollary is true. Rob Arthan has generously provided me with a proof that the only -consistent complete extension of PA is "true arithmetic".
Arithmetic Truth IS Determinate

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