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A problem with the Axiomatic Foundations of Mathematics

  1. Jul 1, 2011 #1
    After looking at the axiomatic systems of modern mathematics and asking myself what proves they are self consistent I went looking for an explanation and so far I have found only that they have not been proven self consistent nor likely will a proof ever exist. So the possibility of a contradiction being present within the system exists. Therefore when using the system you assume it has no contradictions present within it.

    I find this incredibly disturbing as it means every proof I have written would become worthless should the axiomatic system it is written in be demonstrated to be inconsistent.

    I find this so disturbing that I question if I want to pursue my studies in mathematics, one thing I always liked about mathematics is that I considered it built on unshakable ground but this appears not to be the case.

    Has anyone else experienced this revelation? How do you deal with it?
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  3. Jul 2, 2011 #2
    There's a Fields medalist named Vladimir Vovoedsky who's currently working on an alternative foundation for mathematics based on homotopy theory. He suspects that Peano Arithmetic (PA) might be inconsistent.


    His work's generated a lot of controversy. Some mathematicians think he's totally wrong and misguided.


    There's another guy named Ed Nelson at Princeton. He thinks PA might be inconsistent too.


    Here's a good overview discussion of the subject.


    One answer to your question is: If you are interested in the problem, study foundations and work on it. If you discover PA or ZFC to be inconsistent, you will become famous. Or if they are inconsistent, you can study the implications of that.

    The other answer is that if you're not in foundations, you wouldn't care about the problem at all. If set theory blows up they'll just use category theory. If that blows up they'll find something else. None of this applies to "real" math. Physics would still work, for example. In any event, renormalization still hasn't been put on a rigorous footing, so how do you know physics isn't inconsistent too? [Physicists, please check me on that. I don't know anything more about it than what I've gleaned from Feynman's QED book.]

    Another very interesting foundations-related topic getting some Internet buzz lately is whether Wiles's proof of Fermat's Last Theorem relies on the existence of an inaccessible cardinal, which is a type of large set whose existence is independent of ZFC. The question is whether Wiles's proof could in theory be fixed up to not use an inaccessible cardinal. You can google around and get a lot of interesting commentary on that question.


    So to sum all this up ... if math were found to be inconsistent ... well, that would be very interesting!
    Last edited: Jul 2, 2011
  4. Jul 2, 2011 #3
    That's interesting, but as a direct example consider the complete ordered field axioms.

    They have not been shown to be consistent, thus every theorem in analysis will come into question if they are shown to be inconsistent. The possibility of that inconsistency exists.

    So I'm not sure the statement: "None of this applies to "real" math." is accurate.

    In general I find this very disturbing, at least in the sciences I accept I will be using inductive reasoning from the start, but with mathematics I thought I could avoid that. Unfortunately that appears not to be the case. I don't like the idea of spending years of my life learning something I thought was rigorous to in the future being shown that it was wrong and needs a complete reformalization.
  5. Jul 2, 2011 #4
    What I meant by that was: If the current foundations fail, mathematicians will find new foundations. Math is in a certain sense independent of foundations. For example Newton discovered calculus even though he struggled to provide a logically rigorous explanation of infinitesimals. But he was still doing valid math. Eventually people arrived at the modern understanding of limit. But even if they didn't, or if our understanding of the real numbers turned out to be logically problematic, the calculus would still be valid math.

    Now this is of course a philosophical point of view and subject to debate. I'm just trying to point out that even if we had no foundations, we could still do math; just as Newton did; and just like a lot of people do today who aren't concerned about foundational issues.

    Do uncountable sets really exist? Is the axiom of choice really true in any meaningful sense? Who knows. Mathematicians don't care. They use AC when they need to, and they don't worry too much about the exact details of their axiomatic systems. That's why I mentioned the example of Wiles's proof. It's not done in ZFC, it's done in a stronger system. Most specialists feel that it "could" be done in ZFC if they had to do it, but nobody's taken the trouble.

    The point I'm making is that SOME people don't worry about these things too much.

    Mathematics underwent a complete reformalization only 120 years ago. Cantor published his diagonal proof in 1891. That didn't destroy math, it just gave mathematicians a lot of interesting work to do.

    I can't really answer your concerns about this; but I don't see how these foundational issues really bear on learning differential geometry or abstract algebra or probability theory or any other branch of math. Those fields are interesting in and of themselves, and if you're interested in them, you should study them.

    There is no guarantee that tomorrow morning the next Cantor won't come along and shock everyone by showing the world a new way to look at the foundations of math. Beyond that, I guess this is something you'll have to work out personally. If you require absolute certainty in math, I think Godel put an end to that in 1931.
    Last edited: Jul 2, 2011
  6. Jul 2, 2011 #5
    I guess as of now I question the point of doing proofs since the proofs themselves don't only depend on the axioms you are working within but on the assumption of consistency of those axioms. If those axioms can never be proved consistent it seems like a futile effort or worse yet a waste of time should an inconsistency be demonstrated.

    Hopefully tomorrow I will have a more positive outlook on this issue.
  7. Jul 2, 2011 #6


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    The proofs don't depend on the consisteny of the axioms. The proofs are true whether the axioms are consistent or not. Furthermore, if the axioms are proven inconsistent, then we'll just look at axioms which are consistent, and the proofs remain valid in that setting...
  8. Jul 2, 2011 #7


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    Renormalization is not the sketchy "subtract infinities and claim it's zero" that popular-science particle physics books make it sound to be. Renormalization is well defined. The problems with infinities arise when one formulates a model of something and takes the continuum limit, removing the fundamental length scale of the system, which results in infinities when you try to coarse grain the system.

    It is the coarse-graining procedure which is the heart of renormalization - how does your system look as you zoom out? In continuum models the loss of a length scale causes some divergences, so some systematic adjustments must be performed to remove the infinities you introduced by taking the continuum limit and removing the smallest length scale. (The removal of the smallest length scale leads to integrals diverging, so one typically has to put in some artificial cutoff which is removed later). In lattice models one derives recursion relations for the couplings in the system to determine the fixed points of the recursion relations, which correspond to critical points in the language of phase transitions. The coarse graining procedure does not generate infinities in these models because there is a natural cutoff in the model.
  9. Jul 2, 2011 #8
    Suppose that axioms result in a contradiction though, that a theorem is proven true using the axioms and at the same time proven false using the axioms.

    What do you do?

    Furthermore if the contradiction is dependent on a specific axiom and you modify it, then every theorem which uses the axiom will need to be modified.

    Ultimately I will have to trust my intuitive reasoning that something is self consistent(integers) and thus the things based on it will be self consistent.
  10. Jul 2, 2011 #9


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    Doesn't mean that the proof is wrong. It just means that the theory is trivial.

    Not necessarily. The theorems might hold and the proof might hold as well (with minor adjustments).
  11. Jul 2, 2011 #10

    Stephen Tashi

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    What's unshakable about a pile of assumptions and undefined terms? That is what mathematics must be built upon. The little problem of what these assumptions imply is what you're worrying about.
  12. Jul 2, 2011 #11
    Can you answer two questions I'm curious about?

    * What do you make of the point I made about Newton? He discovered calculus in 1687 -- that's the date of publication of the Principia, though he had the theory well in hand a lot longer than that. And Cantor gave the first rigorous definition of the real numbers in 1871. That's almost 200 years that people were using and developing calculus, multivariable calculus, and differential geometry, without benefit of a fully rigorous theory of what they were doing.

    Do you at least acknowledge the point that a lot of extremely good mathematics can be and often is done, in the complete absence of a logical foundation for it? Often the math comes first and the rigor comes later. And therefore even if the foundations collapsed completely, the math would still be there, and someone would eventually fix up the foundations.

    * What is it you read or heard about that's got you concerned? Was it the discovery of non-Euclidean geometry, which showed that mathematical reality is not absolute, but is rather a function of the axioms you choose? Or was it Godel's incompleteness theorem, which simply asserts that a sufficiently rich axiom system can't prove its own consistency unless it's inconsistent? Or was it something else?

    I think if you let us know what specific thing is troubling you, people's advice can be more specific.

    After all, if you are interested in foundations, why not just study logic and set theory and computability theory and see if you can understand the problem better and perhaps even solve it?

    Just curious to get your take on these two questions.
  13. Jul 4, 2011 #12
    A trivial theory is not worth anything, certainly not worth the ammount of time that would have been put into it.

    Although I would like to know how you can make any statment true from an inconsistency in the axioms as I'm not seeing how to do it immediately.

    Isn't the point of proof to remove the word - might? Isn't the reason you prove anything at all because you want it to be assurdly true?

    I agree that many advances of applied mathematics came before rigor and perhaps the only real basis of proof should be a measure of how well mathematics predicts what it is modeling.

    I disagree that if the foundations collapsed, from a rigorous perspective that the mathematics would still hold. Intuitively it would, but isn't the point of the axiomatic development to rid mathematics of intuition and place it on "rigorous" axiomatic ground?

    After coming to the realization that the axioms I have used throughout my university studies could be inconsistent and hence worthless I have a terrible feeling that every proof I have ever read or written is something which could completely come crashing down. That ultimately it was wasted effort.

    I will spend some time on reading about mathematical logic, it's not something that has much interest to me in of itself but because this issue I feel it is so significant I will spend some time on it.


    Also I have a few more general questions.

    Supposing I construct the real numbers from the rationals using Cauchy sequences, and I know the rationals are consistent does it follow that the reals are consistent? How does this chain of consistency work?

    Also I found a supposed proof of the consistency of Peano Arthmetic:


    Is this proof correct?
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  14. Jul 4, 2011 #13


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    One of the most important points of axiomatic development* is abstraction. Euclidean geometry (using Tarski's axioms), for example, is theory of points, lines, incidence, betweenness, and congruence, satisfying some axioms. Euclidean geometry doesn't care about what points, lines, incidence, betweenness, and congruence "really are". If we want to construct a Euclidean plane out of sets in some fashion, that we did so is completely irrelevant to axiomatic Euclidean geometry.

    Overall, you're making a fallacy of the inverse. There is a theorem:
    If set theory is consistent, then Euclidean geometry is consistent.​
    But, from the hypothesis that set theory is inconsistent, you seem to be trying to infer that Euclidean geometry is inconsistent. But logic doesn't work that way.
    (by "Set theory", I mean Zermelo set theory)

    The scarier problem is if logic itself is found to be inconsistent....

    *: Really, I should say "formal logic" here. Axiomatic development is just a particular way to specify a formal theory and an approach to pedagogy.

    P.S. Math already went through this "crisis" once, when Russel proved that naive set theory is inconsistent. (and the similar fact that the Liar's paradox rendered an old form of formal logic inconsistent) Except for set theorists and logicians, I understand that this crisis had pretty much no effect on mathematics.
    Last edited: Jul 4, 2011
  15. Jul 4, 2011 #14
    I'm not saying that the consistency of geometry depends on the consistency of set theory, but through using the axioms of geometry that geometry itself could be shown inconsistent.

    The same goes for every other theory based on axioms such as analysis, abstract algebra, topology...

    How does one rationalize putting so much effort into something which could break down instantaneously if a contradiction is found?
  16. Jul 4, 2011 #15

    Stephen Tashi

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    Are you talking rationalizing it axiomatically by making some logical argument from a set of assumptions? That would make the whole question rather circular, wouldn't it?

    Are you talking about rationalizing "to civilization", by making some philosophical argument that is not purely logical?

    Or are you talking about rationalizing to yourself (or myself) as a matter of personal taste.

    From your remark:
    I'd guess you are talking about personal taste unless you have written proofs of new mathematical results. (From the viewpoint of civilization, the proofs people write as answers to exercises in textbooks and exams are fairly worthless insofar as they don't contribute anything to existing knowledge.)
  17. Jul 4, 2011 #16
    You are making an assumption that is not true.

    If Peano Arithmetic (PA) or Zermelo-Fraenkel "You want Choice with that?" set theory ZF(C) were discovered to be inconsistent tomorrow morning, it is NOT true that anything would "break down" or that existing mathematical truths would become invalid. On the contrary, the vast bulk of mathematics would be unaffected.

    In the foundations community, the following would happen:

    * People would search for new axiom systems and foundations; and

    * People would start unpacking existing results to see exactly what fragment of PA of ZF(C) the result actually needs. They would clarify the path from known results back to weaker axiomatic theories that are free from whatever inconsistency was found.

    How do I know these things would happen? Because the foundation community is ALREADY busily engaged in both these areas of research.

    Set theorists study a variety of new axioms, such as large cardinal axioms; and they also study the effect of each of their new axioms on existing math. For example they kick around systems where every set is Lebesgue measurable, etc. In other words they already study all the little variations in standard math that you get from messing around with the axioms.

    And, there's already a subject called "reverse mathematics," in which they work back from known theorems to find the weakest axiom system the theorem can be proved in.

    In other words, foundationalists take the possible inconsistency of PA or ZFC as the starting point of lots of interesting work. They don't despair and give up math. I don't know why you think this would be the case. Foundationalists have no illusions about PA or ZFC. They're just systems to be studied, they're not the last word. Why do you think they are?

    You are correct that one of the general goals of foundations is to find an axiom system for all of mathematics that's plausible and that can be proved consistent in a way that satisfies everyone from the strict constructivists on up. But that's an ongoing project, not a done deal.

    And the rest of mathematicians don't worry about it. Because their work's valid regardless. That's a point you don't get. Let me toss out another example that might help. Maybe the real numbers as constructed in set theory turn out to be horsefeathers [sorry, I seem to have offended the message board software]. It's certainly possible, the real numbers are pretty strange actually.

    But that would NOT affect anything you do starting from the axioms for a complete ordered field. Everything you can prove starting from those axioms is still valid.

    Perhaps our concept of the real numbers as a MODEL for those axioms might be flawed. But that doesn't affect what we prove from the field axioms. So everything from calculus all the way up through differential geometry and beyond, including all of modern physics, would not be affected in the least.

    You are simply wrong that an inconsistency in foundations would collapse math. Foundations are not a finished system; they are a work in progress.

    Can you say exactly what it is that you read that's led you to this intellectual crisis? I get the feeling it's something specific whose implications you are misunderstanding.

    Math is the work of humans. It's fallable.
  18. Jul 4, 2011 #17


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    What does being "based on axioms" have to do with anything?

    In many ways. In no particular order,
    • This line of thought eventually leads to the stance that you should never do anything at all. (because otherwise you're risking failure)
    • A contradiction in some established mathematical theory would be a very deep theorem. Finding one would make you famous, and would represent a significant milestone in the progress of mathematics.
    • A lot of the effort you spend is in the art of problem solving. This skill is applicable to nearly any task you could perform as a human being.
    • Many, most, or possibly even all of the techniques you learn would still be applicable to other fields of mathematics, such as the one that replaces whatever theory was found contradictory.
    • If nobody studies precise formulations of a theory, then you have no chance to discover which techniques give meaningful results and which techniques are invalid. If the theory itself is inconsistent, you might never discover it!
    • It's fun and rewarding.
    • There's very low risk of established mathematics being found inconsistent.
    • Even if a theory is inconsistent, you'll still get a lot of use out of it until then. And possibly even after it was found inconsistent.
  19. Jul 4, 2011 #18


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    Err no. By Godel's second theorem, any axiomatic system containing basic arithmetic that can prove itself being consistent, is inconsistent.
  20. Jul 4, 2011 #19
    Geometry exists independently of it's axiomatization, just as the integers and their properties exist independently of the Peano axioms. It wouldn't be geometry that would be shown inconsistent, it would the the particular axiomatization, just it wouldn't be the integers and their properties that would be shown to be inconsistent, it would be the Peano axioms.
  21. Jul 5, 2011 #20
    Whats an axiom?
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