Half Man Half Wit is correct about what I was doing with the notational shorthand of “2” for “1 + 1” and so on, but it doesn’t matter; I realized that part of the discussion was getting bogged down unnecessarily by details of my original presentation and so reformulated it into post #452, to which, I am happy to see, you (ch4rl3s) raised no objections.
Well, as I was getting at, how severe are the implications of the mere existence of infinitesimals when this is defined via “An infinitesimal is a number so small that, no matter how many times you add it to itself, the result is never bigger in magnitude than 100,” so that, though we may not realize it at first, even 0 counts as an infinitesimal? What once seemed like a profound statement may turn out rather mundane, once we see the trivialities it allows for. That was the point I was trying to make with this comparison.
Removing axioms from an axiom system will only cause it to be able to prove less. So, what I was saying was, if you start with an incomplete axiom system, and remove axioms (while keeping the language of statements one is dealing with the same), then the system one ends up with will necessarily be incomplete as well [it can’t prove anything the starting system didn’t].
You make oblique reference to the completeness of “impoverished” systems such as Presburger Arithmetic and so forth. But this doesn’t contradict what I was saying. Sure, you can think of Presburger Arithmetic as Peano Arithmetic with all the axioms about multiplication ripped out… and if you think about it that way, then Presburger Arithmetic is incomplete (in the sense that it can neither prove nor disprove certain statements; particularly, those axioms of Peano Arithmetic which dealt with multiplication). In order to think of Presburger Arithmetic as a complete system, we have to also narrow the scope of the language we are concerned with from that of Peano Arithmetic, so that we do not concern ourselves with statements which mention multiplication; in this case, Presburger Arithmetic would not just be Peano Arithmetic with some axioms removed, and all would be alright.
Anyway, it was just a very minor point, but hopefully you can see now what I was saying, which is pretty trivially obvious: removing axioms from an axiom system can at most result in being able to prove less, and thus in being even further incomplete. All the more reason why incompleteness abounds and is not particularly surprising in itself.
As Half Man Half Wit points out, this is already far beyond simply noting that some systems show some incompleteness (and thus why I objected to lumping Goedel’s results in as automatically related to, e.g., the incompleteness of ZFC with respect to the Continuum Hypothesis, which is really a very unrelated example of incompleteness, just as was the example of post #452).
But, sure, I basically agree with what you are saying here, except for one slight nitpick which is not really a vehement disagreement:
It doesn’t really have to do with how “powerful” the axiom system is; it has to do with the balance between how simple the axiom system is to express and how expressive the language of statements dealt with by the axiom system is. That is, the Goedelian incompleteness phenomenon will arise whenever the language dealt with by the axiom system is expressive enough to define the axiom system itself [in suitable fashion]. Thus, we could just as well say, the “weaker” (in the sense of being easier to express) the axiom system, the more vulnerable to incompleteness it is. But it would be best off not to say this either. It would be better to simply say “Given an axiom system for statements in some language, where the language is capable of expressing that axiom system itself, it cannot be both complete and consistent”. To get incompleteness, you want a “powerful” language (in the sense of being expressive) but a very “weak” actual axiom system (in the sense of being easily expressed).
To illustrate what I mean by this: the standard result we all know and love is that there is no computably enumerable axiom system which can interpret natural number addition and multiplication which is both complete and consistent. Great. But there’s two constraints here worth noting: not only that the language be “powerful” enough to interpret statements about <N, +, *>, but also that the axiom system be “weak”/“simple” enough that it can be enumerated by computer. Both of these constraints are necessary; otherwise, we could easily obtain consistent completeness (e.g., the axiom system consisting of all true statements about natural number addition and multiplication is certainly complete and consistent; it just happens to be too complicated to enumerate by computer). And, indeed, the two constraints match, because every computable function is definable (in a tedious way) using only statements about natural number addition and multiplication; so all that is really going on here is the observation that no axiom system can simultaneously be expressed in the language of <N, +, *> and correctly answer all questions about <N, +, *>.
Indeed, even Presburger Arithmetic, complete though it is, illustrates the same result: the axiom system of Presburger Arithmetic deals with the language of <N, +>, but cannot be expressed in the language of <N, +>. As always, no axiom system formulated in language L can correctly answer all questions about language L.
So the phenomenon is not really about the “powerfulness” of the axiom system; it’s a tug-and-pull involving both the “powerfulness” (expressiveness) of the language and the “weakness” (ease to express) of the axiom system. Whenever these match up, we have incompleteness. [Precisely because we could then ask “Does the axiom system being looked at answer ‘No’ to this question?” in the language being dealt with]
But, whatever. We don’t really disagree here. Where we’re going to disagree are on what the philosophical implications we should take from this are.