First of all, let me issue a warning: model-theoretic truth is a mathematical notion, which (a priori) doesn't have anything to do with the real-world sense of truth!
A short introduction to model theory follows. It is not LW quality, but hopefully it's good enough to answer some questions about MrMind's post. Prerequisites: merely some familiarity with formal reasoning, but I guess knowing the Mental Concepts of Model Theory doesn't hurt.
The 1st part is the general introduction to model theory, the examples about non-standard models are in the 2nd and 3rd parts.
1 Models explained
Axioms are the starting points of formal reasoning. A collection (system) of axioms is inconsistent if it is possible to prove a contradiction using them. E.g. consider the following system of three axioms.
This system is inconsistent, because the contradiction "Socrates is mortal and Socrates is not mortal" is a consequence of the axioms. On the other hand, the following system (from now on referred to as the example system) is not inconsistent:
Inconsistent systems are liars: the conclusions derived from them cannot be trusted. ^1 Axiomatic systems can be defined for many specific purposes (mathematics, ethics, et c.). Hopefully I don't have to explain why an inconsistent system of ethics would be disastrous. We would like some assurance that our frameworks are not inconsistent: proofs of consistency!
Proofs of consistency are possible because mathematicians have agreed upon a powerful axiomatic system, the Set Theory ZFC that they believe to be consistent. ^3
Take any axiomatic theory S. Sometimes, you can re-label the axioms of S to be about mathematical objects. This is possible if
All quantifiers that occur in the axioms can be restricted to range over some given set M (in the example system, you would replace "All men" with "All men belonging to the set M*").
Each symbol occuring in the axioms can be identified with an element of the set M (in the example system you would interpret the word "Socrates" to refer to some specific element of the set M).
Each predicate occuring in the axioms can be identified with a subset of the set M (in the example system you would interpret "is a man" by a subset of M, and "is mortal" by another subset of M).
The sentences of S, when interpreted this way, are consequences of the axioms of ZFC.
Such sets M, whenever they exist, are called the models of S. The set of numbers less than 5 {0,1,2,3,4} is a model of the example system, because you can
Interpret the symbol "Socrates" as referring to the number 1.
Interpret the predicate "is a man" as the subset of odd numbers; this means that you consider 1 and 3 men, but not 0, 2 and 4.
Interpret the predicate "is mortal" as the subset of numbers less than 4; this means that you consider 0, 1, 2 and 3 to be mortals, but not 4.
Now, the sentence "All men are mortal" means "All odd numbers in M are less than 3", which is a true mathematical statement that you can prove using the axioms of ZFC.
The sentence "Socrates is a man" means "The number 1 is odd", which is again a true mathematical statement that can be proved from the axioms of ZFC.
The relabeling interprets the axioms and consequences of S as true mathematical statements (the form of the statements is preserved, even if meaning is not). If a contradiction follows from the axioms of S, then it can be relabeled into a contradiction in mathematics (ZFC). Therefore, every axiomatic system that has a model is at least as consistent as mathematics itself: giving a model amounts to giving a consistency proof. We say that this consistency proof is relative to ZFC.
It can be demonstrated that Model Theory is the most general method of giving consistency proofs (relative to ZFC): if ZFC proves that a system is consistent then the system has a model, and vice versa.
^1 There are also consistent liars. Observing an inconsistency is sufficient to conclude that an axiom system is a liar, but it is not necessary.
^2 Observe that we still have no assurance about the consistency of this general-purpose system.
^3 This is not entirely true, but it is a reasonable non-technical explanation.
^4 Unfortunately, the satisfaction relation "satisfied in a model" is commonly referred to as true in a model. Worst of all, "X is satisfied in the standard model" is sometimes abbreivated to X is true, giving these results a false aura of deep philosophical relevance.
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.
0 < K
1 < K
2 < K
...
65534 < K
65535 < K
...
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.
For any non-zero natural numbers n, k, m we can find a natural number N such that if we color each of the n-element subsets of S = {1, 2, 3,..., N} with one of k colors, then we can find a subset Y of S with at least m elements, such that all n element subsets of Y have the same color, and the number of elements of Y is at least the smallest element of Y.
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.
Got really much me thinking. Why are we regarding non-standard natural numbers as "junk"? I guess the identification of standard natural number as the simplest construciton that is a natural number system.
The thing is a know a perfectly legimate construction for a number that is non-standard and not deliberately a "wrench in the machinery". The surreal number {1,2,3,4...|}=ω I have sometimes seen characterised as a integer and it's construction is of the same shape as other integers with lower birthdays (althought they use finite sets, ω is the first to use infinite sets).
The hydra problem seems natural as you can't have ω-(n1) with finite n that reaches 0, and in fact ω-(n1) is still bigger than any finite number. I can also see how the successor of ω is ω+1 which I guess is the property that successor and addition play nice together.
When geometry was axiomatised it was discovered that there are euclid and non-euclid geometries. They were not called non-standard geometries despite them getting way less attention. In general euclid and non-euclid geometries share some properties (those that stem from aximo not regarding parallel lines) but have different properties in general (ie different parallizaiton rules lead to genuinely different systems). Coudn't it just be that we are using a way too general system where the formal meaning of a integer captures more entities that we have in mind when we are really interested only in certain kinds of natural numbers? That is ω might be a integer as the axioms read out but when people say integers they don't mean entities like ω (like when people say space they usually don't mean minowskian spaces althought those are spaces too).
I do like the rigour that when a mathematician lays out a set of axioms he can know whether all cases are covered without being able to come up with any "viable" exception to them. That is any kind of arithmetic thing that hinges on the differences of finite and infinite numbers is already ambigious based on axioms of arithmetic because finiteness and infiniteness is ortohogonal to the issue (a kind of separation of concerns where you don't even know how many concerns there are).
Wouldn't the holistic nature of the truth be viewed as if you have an ambigious delineation on the universe of discourse then you can't have all properties nailed down. As in if you have a theory of "tallness" that doesn't allow you to determine an objects color.
This is an interesting topic, but your exposition is sorely lacking layperson examples. The closest it comes to one is
they could agree about the standard numbers, how the successor and addition operator works, and yet disagree on multiplication
but it is still unclear what exactly they disagree on. Does one of them state that 2 times 2 is 5? Probably not.
When I explain complicated physics to a non-physicist, I strive to give examples which are as simple as possible. I wish mathematicians did that with math.
All these articles are difficult for me to follow, but I get the feeling that they are approximately about this:
And the articles like this explore the various consequences of "if you are allowed to have finitely many axioms of given type, you cannot really precisely define natural numbers", and find many things that fall apart because of that.
Am I at least approximately correct here?
Here's what I gather:
You can define the natural numbers using certain rules (the Peano axioms). These rules make reference to sets, so they are in turn governed by the rules of set theory (ZFC), which admit several models.
What the paper goes on to talk about is that even if you just use the natural numbers--not some weird system--then there will still be some statements whose truth depends on the ambient model of set theory. In other words Model 1 and Model 2 of ZFC can both admit the natural numbers, but they can still disagree about what's true. Certain things will always be true, but others will be true or false depending on which model you use.
I think so. I'm confused as to whether these results can apply to the actual standard natural numbers, or just partly-weird sets which some model of set theory believes are the standard numbers - but I think it's the second one. The paper itself says that "satisfaction is absolute,"
whenever the formula φ has standard-finite length in the meta-theory (which is probably closer to what was actually meant by those asserting it)
which appears to mean that none of the statements being disagreed about are actual mathematical statements, but rather "weird numbers" that the models think encode statements. (Note that any other way of defining mathematical formulas would probably be at least as ambiguous as the natural numbers.)
My understanding passes the first test which occurs to me, namely, it would not allow a model of set theory to realize that it has the wrong natural numbers. (If my interpretation predicted that a model with weird numbers could look at another model which disagrees about arithmetical truth and realize, 'Wait, these disagreements only arise when we allow weird numbers, so my numbers must be partly weird,' then I would have to be wrong.) Evidently if even one of two models can look at the satisfaction relation that the other uses to define arithmetical truth (for example), then they must agree.
Logic sometimes breeds monsters.
-- Henri Poincare
Philosophical question: do these results give insight into the truth "out there" or are they artifacts of mathematical logic? We may be trained to reason from the axioms, using the established rules of logic. Some theorems are always true and can be proven, some are true but not provable, and others may or may not be true depending on our model. In practice, we already know many important results in a field long before anyone comes up with a list of axioms, and the axioms are chosen so that they allow one to deduce the important results. If ZFC didn't let us prove things we knew to be true, we wouldn't be using it. If the Peano axioms produced a single theorem that conflicted with established number-theoretic results, the axioms would be changed.
It's almost a given that in higher math, we can't have nice things. Much effort goes into coming up with novel concepts to get around the various pathologies: there's the Weierstrass function and the Peano Curve and the Devil's Staircase and the Alexander Horned Sphere, not to mention the good old Banach-Tarski paradox. Our attempts to reduce our intuitions about the outside world into a a small number of obvious statements always end with a horde of monsters showing up and exposing our ignorance. It seems that all our maps are destined to have an area labeled "Here be dragons".
So back to the first question: are these problems of truth a property of the territory, or do they only show up when we try to draw a map?
There isn't just one kind of truth.
Objective truths are objectively true.
But they aren't absolutely true. Objective truths are formed from theories and observations, laden with units of measurement and language.
Objective truth, some feel, should be static and independent of an observer. But that's a needless and problematic assumption.
Absolute truth and objective truth are different.
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.