Consistent extrapolated beliefs about math?

I think we can make this more precise. Second-order arithmetic theoretically covers what I believe about the natural numbers. We then need a theory of sets or collections to interpret it. Gaifman, in his discussion of non-standard models, suggests that we go meta instead, and say that whenever we learn a new formula we expand the principle of induction to include this formula. In both cases, the open-ended part comes less from arithmetic than from some larger framework.

So: I don't think my larger 'theory' is necessarily self-consistent. I do think there exists some consistent intermediate theory which includes the consistency of second-order PA, and the uniqueness of its model up to isomorphism.

If we define phi+1 as the union of the statement(s) phi and the assertion of phi's consistency, someone might possibly doubt the consistency of PA+omega, since the concept "omega" need not appear in first-order PA. I find myself hesitating slightly about PA+the Church-Kleene ordinal, although ultimately I think many large cardinals are consistent with(in) set theory. And yes, I think at this point we should start assigning probabilities to mathematical statements.

Okay. If I want to know something about marbles and someone begins invoking infinities, that raises a yellow flag for me.

If I want to know about the asymptotic behavior of an algorithm, then no such flag is raised.

I think you meant to say "We don't believe that there are infinite different amounts of marbles", since the amount of marbles in an infinite amount of marbles isn't even described by an integer.

No, you just broke your model by doing something not covered by it.

Ah, I see. I misunderstood your point. You don't worry about your current beliefs, you worry about the extrapolated ones. How do you see this represented? An algorithm, which, when fed a theorem, outputs a proof/disproof/proof of undecidability? Or something that spits all possible true statements, unprompted, never shutting up?

if you don't mean any ontological import you might want to phare it as "integers is system that works". The negation would be things that dont' work. For example a triangle with a right corner with angles summing over 180 degrees. But even then you have to specify the background assumptions as those kinds of triangles actually work out. Usually a mind defaults to a euclidean mindset while the applied concepts could apply to non-euclidean context too.

That integers exist could mean a number of things. Like that x + y = y + x for every x and y that is a value. However the logic of non-commuting values has been figured out. Therefore that sentence would be false. There are things that don't fall under this rule meaning this is not an universal rule. Having some assumtion that you need not have violated in your life doesn't mean such a violation would be impossible. The only way back would be to explicitly declare the delineation of that context ie the various properties needed. But then "integers exists" becomes just "assuming integers integers is all there is" which isn't very surprising or would need any explanation.

As long as you aren't clear what you mean with "exist" the rest of the argument is build on quicksand.

Landsburg writes:

To me, by far the most satisfying solution is a full-fledged Platonic acknowledgement that numbers are indeed just “out there” and that they are directly accessible to our intuitions. I embrace this view for (at least) three reasons: A. After a lifetime of thinking about numbers, it feels right to me. B. Pretty much every one else who spends his/her life thinking about numbers has come to the same conclusion. C. It seems enormouosly more plausible to me that numbes are “just out there” than that physical objects are “just out there”, partly because there is in fact a unique system of (standard) natural numbers, whereas the properties of the physical universe appear to be far more contingent and therefore unnecessary.

But the set of all possible minds is so vast that the fact that numbers feel real to us feels to my mind as extremely weak evidence that numbers are real.

I don't doubt that just about anything can be formalized in ZFC or some extension of it. I am aware that a Turing machine can print any recursively axiomatizable theory.

all sets of axioms are countable, because they are subsets of the set of all finite strings

The set of all finite strings is clearly order-able. Anything constructed from subsets of this set is countable in that it has cardinality aleph_1 or less (even if it contains the set).

I read this book on something called language theory (I think it's now called "formal language theory"), an attempt to apply the idea that all mathematics is represented in the language of finite strings. According to the text as I remember it, the set of all finite strings is equivalent in size to the set of all the statements that can be made in closed languages.

My question is, treating math as an open language, is it possible to axiomatize in a semantically meaningful way, consistent with the bulk of constructive mathematics? I believe the answer is yes, but I would genuinely like to hear your thoughts on the subject.

The reason I think this question is worth asking for three reasons. 1) from a purely structuralist/historical perspective, new concepts enter math all the time and they often challenge the consistency of some portion if not all of mathematics. True they are explained in terms of old concepts, but from a purely observational point of view, the language of math behaves much more like an open language than a closed one. 2) I believe all theories have axioms whether overtly stated or hidden deep within nomenclature. If any set of axioms is both incomplete and inconsistent, then the only way of evaluating competing theories is to compare them. But we can play the stronger weaker logic game all day without knowing if we're forming a closed loop. From that point of view, it becomes even more important to consider the possibility of a theory that explains why some theories work for some things and not other. So I close my eyes and try to imagine the parameters of a theory that is complete and consistent. I think Godel is right -- so it has to have uncountably many axioms otherwise paradox. 3) This is the part I don't know how to explain in mathematical terms. Which axioms to use ... I mean if Zorn's Lemma and the axiom of choice are the same thing, then the axioms we see must be as much a consequence of the language as they are a reflection of whatever is the core of mathematics. When I read a textbook in number theory, I'm seeing the axioms of algebra transformed to fit a different way of thinking of numbers. The concepts are conserved, but the form they take is just a mask and I know that there are questions we don't know how to answer yet. But there is a general pattern that all branches of mathematics follow -- all try to eliminate the extraneous and unnecessary, to streamline axioms to fit the demands of the language ... If we are to devise a self consistent theory of sets, the first axiom (after the definition of a set, of addition, of inequality, of the null-set, of infinity) would be the axiom of incompleteness. After all, if the list of axioms never terminates, the Turing machine can't halt. :-)

4) I don't like the idea of questions that cannot be answered or at least outlawed for the sake of sanity.

With that in mind, I think it's okay to have unanswered questions about integers.