# Truth is holistic

You already know by now that [truth is undefinable](http://en.wikipedia.org/wiki/Tarski%27s_undefinability_theorem): by a famous result of Tarski, no formal system powerful enough (from now on, just system) can consistently talk about the truth of its own sentences.

You may however not know that Hamkins proved that [truth is holistic](http://jdh.hamkins.org/satisfaction-is-not-absolute/).  
Let me explain: while no system can talk about its own truth, it can nevertheless talk about the truth of its own _substructures_. For example, in every model of ZFC (the standard axioms of set theory) you can consistently define a model of standard arithmetics _and _a predicate that works as arithmetics' truth predicate. This can happen because ZFC is strictly more powerful than PA (the axioms of standard arithmetics).  
Intuitively, one could think that if you have the same substructure in two different models, what they believe is the truth about that substructure is the same in both. Along this line, two models of ZFC ought to believe the same things about standard arithmetics.  
However, it turns out this is not the case. Two different models extending ZFC may very well agree on which entities are standard natural numbers, and yet still disagree about which arithmetic sentences are true or false. For example, they could agree about the standard numbers, how the successor and addition operator works, and yet disagree on multiplication (corollary 7.1 in Hamkins' paper).  
This means that when you can talk consistently about the truth of a model (that is, when you are in a more powerful formal system), that truth depends not only on the substructure, but on the entire structure you're immersed in. Figuratively speaking, local truth depends on global truth. Truth is holistic.  
There's more: suppose that two model agree on the ontology of some common substructure. Suppose also that they agree about the truth predicate on that structure: they could still disagree about the meta-truths. Or the meta-meta-truths, etc., for all the ordinal levels of the definable truth predicates.

Another striking example from the same paper. There are two different extensions of set theory which agree on the structure of standard arithmetics and on the members of a subset A of natural numbers, and yet one thinks that A is first-order definable while the other thinks it's not (theorem 10).

Not even "being a model of ZFC" is an absolute property: there are two models which agree on an initial segment of the set hierarchy, and yet one thinks that the segment is a model of ZFC while the other proves that it's not (theorem 12).

Two concluding remarks: what I wrote was that there are different models which disagrees the truth of standard arithmetics, not that every different model has different arithmetic truths. Indeed, if two models have access one to the truth relation of the other, then they are bound to have the same truths. This is what happens for example when you prove absoluteness results in forcing.  
I'm also remembered of [de Blanc's ontological crises](http://arxiv.org/abs/1105.3821): changing ontology can screw with your utility function. It's interesting to note that updating (that is, changing model of reality) can change what you believe even if you don't change ontology.