Is Principia Mathematica outdated?

[disregardthat]

All (quantified) statements of ZFC involve unbound quantifiers, so I really can't figure out what you're thinking.

If undecidability is not unprovability, but merely the lack of a specific context, then I wouldn't treat statements involving unbound quantifiers as equivalent in type to statements involving bound quantifiers or no quantifiers at all, because of our inability to judge their truth value. I agree with that they are meaningful, but I can not see how they can possesses any truth value in general independent of context - and is thus merely statements, not theorems.

I interpreted undecidable as equivalent to unprovable in the beginning, but I see now that it does not make sense.

 

[Hurkyl]

If undecidability is not unprovability

The very definition of "P is undecidable" is that neither "P" nor "not P" is provable.

I can not see how they can possesses any truth value in general independent of context - and is thus merely statements, not theorems.

Statements don't have truth values, not even tautologies. "Truth" is a matter of semantics -- e.g. each set-theoretic interpretation yields a truth valuation: a function that maps the set of statements to the set {true, false}. (and a model of a theory is one in which each of its theorems map to true)

 

[disregardthat]

The very definition of "P is undecidable" is that neither "P" nor "not P" is provable.

Then, as I previously asked: is it true that any unprovable statement is unprovable merely because of the lack of context? Is, for example, CH undecidable because of the lack of some specific theory in which CH is treated?

 

[Hurkyl]

Oh! It just struck me that maybe you don't know that semantics is a technical term.

In formal logic, the syntax, heuristically speaking, deals with form -- languages, grammar, provability, theories, and the like.

Semantics, heuristically speaking, deals with meaning -- studies models of theories, truth valuations on languages, and the like.

 

[Hurkyl]

Then, as I previously asked: is it true that any unprovable statement is unprovable merely because of the lack of context? Is, for example, CH undecidable because of the lack of some specific theory in which CH is treated?

What precisely do you mean by context?

CH is undecided in ZFC because it is neither provable nor disprovable. CH is, of course, provable in ZFC+CH.

For any particular (classical) model of ZFC, the truth value of CH is either true or false.

 

[disregardthat]

Statements don't have truth values, not even tautologies. "Truth" is a matter of semantics -- e.g. each set-theoretic interpretation yields a truth valuation: a function that maps the set of statements to the set {true, false}. (and a model of a theory is one in which each of its theorems map to true)

Interesting, it makes sense that only a set-theoretic interpretation determines the truth of any well-defined statement in set theory. But still, I have thought of the undecidability of CH as something different. I have thought of (the interpretation of) CH as a statement not bearing a truth value regardless of any set-theoretic interpretation (in which CH is interpreted), thus differentiating this "type" of undecidability from the pure syntactical statements, which are not bearing any truth value.

EDIT: By context I meant a set-theoretic interpretation.

CH is undecided in ZFC because it is neither provable nor disprovable. CH is, of course, provable in ZFC+CH.

For any particular (classical) model of ZFC, the truth value of CH is either true or false.

Oh, so it does make sense to call a statement provable if it doesn't bear a truth value. Using your definitions; the syntax determines the provability of a statement, but CH is not provable (in ZFC).

If so, how does it make sense to call CH semantically true (or false) in a set-theoretic interpretation of ZFC if it is syntactically unprovable (in ZFC)?

 

[CRGreathouse]

If so, how does it make sense to call CH semantically true (or false) in a set-theoretic interpretation of ZFC if it is syntactically unprovable (in ZFC)?

The statement E = "'Elephant' is a natural number" is unprovable in Peano arithmetic. In the model

"0" = 0
"S(x)" = x + 1
"N" = {0, 1, 2, ...}

E is false, because Elephant is not a member of {0, 1, 2, ...}. In the model

"0" = 1
"S(x)" = 2x
"N" = {1, 2, 4, 8, 16, ...}

E is also false, because Elephant is not a member of {1, 2, 4, 8, 16, ...}. In the model

"0" = Elephant
"S(x)" = -3, if x = Elephant, and x + 2 otherwise
"N" = {Elephant, -3, -1, 1, 3, 5, ...}

E is true, because Elephant is a member of {Elephant, -3, -1, 1, 3, 5, ...}.

 

[disregardthat]

The statement E = "'Elephant' is a natural number" is unprovable in Peano arithmetic. In the model

I see your point, but this means that proof (like that of the non-existence of the elephant) also is relevant in the semantical realm. I thought provability, in Hurkyls definitions, was restricted to the syntax of the formal language and the semantics merely brought a truth function to all well-formed statements.

 

[CRGreathouse]

I have thought of (the interpretation of) CH as a statement not bearing a truth value regardless of any set-theoretic interpretation (in which CH is interpreted), thus differentiating this "type" of undecidability from the pure syntactical statements, which are not bearing any truth value.

EDIT: By context I meant a set-theoretic interpretation.

No, CH will be true or false in a given model/interpretation. For example, in the constructable universe L the CH is true.

 

[disregardthat]

To be honest, set theory seems to me quite superfluous to mathematics in general. It can be interesting in itself, but it (or any similar theory) does not (in my opinion) deserve the status as any kind of "foundation of mathematics".