I think you’re imagining an exit that isn’t there. For example, ZFC can formalize Gentzen’s proof of consistency of PA, and supports quite a bit of transfinite induction. Yet it still has to obey Gödel’s theorem, like any other recursively axiomatizable theory.
(Also, all sets of axioms are countable, because they are subsets of the set of all finite strings, which is countable. I assume you meant to say something else, but I can’t guess what.)
The amazing thing about Gödel’s theorem is how general it is. I mentioned that in the post. Any Turing machine that prints theorems (regardless of the internal mechanism) must obey Gödelian limitations, as must any Turing machine that receives proofs and checks them for correctness by any method. The only way to escape these limits is by hypercomputation, but I wouldn’t hold my breath.
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.
This might be expressible with a finite axiomatisation (e.g. by building functions and arithmetic in ZFC), and indeed I’ve given a finite schema, but I’m not sure it’s ‘fair’ to ask for an example of a theory that cannot be compressed beyond uncountably many axioms; that would be a hypertask, right? I think that’s what Joshua’s getting at in the sibling to this comment.
I think you’re imagining an exit that isn’t there. For example, ZFC can formalize Gentzen’s proof of consistency of PA, and supports quite a bit of transfinite induction. Yet it still has to obey Gödel’s theorem, like any other recursively axiomatizable theory.
(Also, all sets of axioms are countable, because they are subsets of the set of all finite strings, which is countable. I assume you meant to say something else, but I can’t guess what.)
The amazing thing about Gödel’s theorem is how general it is. I mentioned that in the post. Any Turing machine that prints theorems (regardless of the internal mechanism) must obey Gödelian limitations, as must any Turing machine that receives proofs and checks them for correctness by any method. The only way to escape these limits is by hypercomputation, but I wouldn’t hold my breath.
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.
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.
Sorry, I don’t understand what you’re talking about. Can you give an example of a theory with uncountably many axioms?
(A): There exists a function f:R->R
and the axioms, for all r in R:
(A_r): f(r)=0
(The graph of f is just the x-axis.)
This might be expressible with a finite axiomatisation (e.g. by building functions and arithmetic in ZFC), and indeed I’ve given a finite schema, but I’m not sure it’s ‘fair’ to ask for an example of a theory that cannot be compressed beyond uncountably many axioms; that would be a hypertask, right? I think that’s what Joshua’s getting at in the sibling to this comment.
Example infers more than one representation could exist, which for an object this large would be absurd.