On 17 Dec 2012, at 22:02, meekerdb wrote:

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On 12/17/2012 11:47 AM, Bruno Marchal wrote:On 16 Dec 2012, at 20:28, meekerdb wrote:On 12/16/2012 2:31 AM, Bruno Marchal wrote:No. With the CTM the ultimate truth is arithmetical truth, and wecannot really define it (with the CTM). We can approximate it inless obvious ontologies, like second order logic, set theory,etc. But with CTM this does not really define it.Don't confuse truth, and the words pointing to it. Truth isalways beyond words, even the ultimate 3p truth.What would it mean to 'define truth'? We can define 'true' as aproperty of sentence that indicates a fact.That's the best definition of some useful local truth. But whendoing metaphysics, you have to replace facts by "facts in somemodel/reality".OK. But then it's "True relative to the model." and it's notnecessarily The Truth.

`Indeed. But for arithmetic (or Turing equivalent), "The Truth" = true`

`in the standard model (learned in high school).`

But I'm not sure how to conceive of defining mathematical 'true'.It is the object of model theory. You always need to add more axiomin a theory to handle its model. You cannot define the notion oftruth-about-set in ZF, but you can define truth-about-set in ZF inthe theory ZF +kappa (existence of inaccessible cardinals).PA can define all the notion of truth for the formula with abounded restriction of the quantification.So what is that definition?

`It is long and has to be defined by induction on the complexity of`

`formula. Like "ExP(x)" is true if it exists a n such that P(n), etc.`

Does it just mean consistent with a set of axioms,No. That means only having a model. true in some reality. But forarithmetic "true" means satisfied by the usual structure (N, +, *).i.e. not provably false?How is not provably false different from 'satisfied by the usualstructure'? Can you give an example?

`Well the most famous example is "provable "0=1". This is not provably`

`false (as PA cannot prove ~Bf), but is false in the standard model.`

That just consistent.I would think it was incompleteness. Consistency means not beingable to prove every proposition.

or ~Bf

But in a consistent system there can be propositions that areneither provable nor disprovable. Are those true?

`Some are, some are not. Bf is not provable and false. Dt is not`

`provable and true. All arithmetical interpretation of any formula of`

`G* minus G are true but not provable. Their negations are false and`

`not provable.`

Bruno

BrentTrue entails consistency, but consistency does not entail truth. Bruno--You received this message because you are subscribed to the GoogleGroups "Everything List" group.To post to this group, send email to everything-list@googlegroups.com.To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com.For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

http://iridia.ulb.ac.be/~marchal/ -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.