You already know by now that truth is undefinable: 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.
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: 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.
2. A multitude of models
As a general rule, consistent theories have multiple models. Models have more consequences than the theories they model: for example, our model of the example system proves that there are only 2 men, even though this does not follow from the axioms. A sentence follows from the axioms only if it is satisfied in every possible model of S. ^4
Even the axiomatic theory of natural number arithmetic, which we would think is absolute, has multiple models. Mathematicians have agreed on a standard model (the so-called set of natural numbers), but it is easy to prove that other models exist:
Extend the theory of arithmetic (PA) with a new constant K, and the following (infinitely many) axioms.
Surprisingly, the resulting theory PAK is consistent. Proofs are finite: any proof of a contradiction in PAK would use only finitely many axioms, so there is a largest number n such that n < K is used in the proof. Therefore, K can be replaced in the proof by n + 1, yielding a proof of a contradiction in PA itself! Since arithmetic is consistent, there is no proof of contradiction in PAK.
We have shown that PAK is consistent relative to ZFC. Therefore, it has a model. A model of PAK is a model of arithmetic, but it is clearly not the standard model. Therefore, arithmetic has a non-standard model, which contains the standard integers, as well as non-standard integers (such as the one corresponding to our constant K). In a sense, the non-standard models contain "infinite" numbers that the model cannot distinguish from the real, finite numbers.
The existence of non-standard models is a serious issue: There are situations where the standard model has no counterexamples to a statement, but some non-standard model has. This means that the statement ought to follow from the axioms of arithmetic, but we cannot prove it because it fails in a weird, non-standard model.
For example, some non-standard models disagree with the following statement (the Ramsey theorem), which is satisified by the standard model.
Another, more accessible example is whether you can kill the Hydra or not. You can kill the hydra in the standard model, but many non-standard models disagree. If you would number all the hydras in a non-standard model, the counterexamples would be numbered by non-standard numbers such as K in the proof above.
We need to add new axioms to the axiomatic system of arithmetic, so that it corresponds more faithfully to the standard model. However, our work is never over: as a consequence of Gödel's incompleteness theorem, new axioms can rule out some non-standard models, but never all of them.
3. Generalised models and Hamkins' paper
So far:
Consistency relative to ZFC is a useful notion: giving a model allows us to prove that our theories are as consistent as mathematics itself.
Arithmetic has multiple models. There is a so-called standard model of arithmetic, which is not some real-world or transcendent notion. It is merely a set that mathematicians have agreed to call the standard model. The axioms of arithmetic are unable to exactly describe the standard model: they always describe the standard model plus some other "junk" models.
Do we know that ZFC is consistent? The short answer: we don't and we can't. By Gödel's incompleteness, if ZFC is consistent then it has no models. However, by adding new axioms to ZFC (e. g. large cardinal axioms). we can create set theories that have generalised notions of models. While ZFC has no models, it does have generalised models.
Unlike arithmetic, ZFC itself has no agreed-upon standard generalised model. There is not even a standard system in which we construct generalised models. In all of the above, we have refused to choose a specific model of ZFC (i. e. we did not use the phrase "satisfied in a generalised model of ZFC" or any semantically equivalent sentences). We used the notion of provability in ZFC (which is absolute).
If we replace provability in ZFC with "satisfiability in some specific model", we are suddenly able to prove more properties about the standard model of arithmetic (similarly to how we can prove more theorems about numbers by passing to the standard model of arithmetic from the axioms of arithmetic). Unfortunately, it is well-known (and intuitively obvious) that if you and I choose different generalised models, our conclusions (about these previously undecidable properties) can disagree.
The paper of Hamkins collects some stronger results: our conclusions can disagree even if our chosen generalised models are very similar. For example
There are two generalised models which agree upon the elements that constitute the standard model, yet disagree on the properties of these elements.
There are two generalised models which agree upon the elements that constitute the standard model, agree upon the properties of the addition operation, yet disagree about the properties of the multiplication operations.
and so on... Unfortunately, the proofs of these rely on powerful lemmas, so I can't instantiate them to produce explicit examples.
Anyway, this should be enough to get you started.
You should definitely post it as a top-level post in Main.