Simple as ABC

The (really really) big news in the math world today is that Shin Mochizuki has (plausibly) claimed to have solved the ABC problem, which in turn suffices to settle many of the most vexing outstanding problems in arithmetic. Mochizuki’s work rests on so many radically new ideas that it will take the experts a long time to digest. I, who am not an expert, will surely die with only a vague sense of the argument. But based on my extremely limited (and possibly mistaken) understanding, it appears that Mochizuki’s breakthrough depends at least partly on his willingness to abandon the usual axioms for the foundations of mathematics and replace them with new axioms. (See, for example, the first page of these notes from one of Mochizuki’s lectures. You can find other related notes here.)

That’s interesting for a lot of reasons, but the one that’s most topical for The Big Questions is this: No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth.

There’s a widespread misconception that arithmetic is about “what can be derived from the axioms”, which is a lot like saying that astronomy is about “what can be discovered through telescopes”. Axiomatic systems, like telescopes, are investigative tools, which we are free to jettison when better tools come along. The blather of thoughtless imbeciles notwithstanding, what really matters is the fundamental object of study, whether it’s the system of natural numbers or the planet Jupiter.

Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.

That in turn implies that there is such a thing as mathematical truth, independent of what we can prove. There are respectable thinkers who deny this, but they are extremely rare among practicing mathematicians, who encounter mathematical truths every day.

Mochizuki might or might not have settled the ABC problem, and if so he might or might not have been forced to replace old axioms with new ones; it’s possible he’s not sure of this himself (in the sense that he might have adopted new axioms for convenience without being sure whether they’re really necessary). But the key point is that this is not a major issue. Mathematicians are opportunists. The goal is to understand the natural numbers, and we’ll pursue that goal by any means necessary.

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45 Responses to “Simple as ABC”


  1. 1 1 Martin

    Steve,

    “No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians (or at least the sort of mathematicians who study arithmetic) don’t particularly care about axioms; they care about truth.”

    Can you think of any parallel in economics, where old axioms are abandoned for new ones, because economist care about the truth? The closest I can think of is the marginal revolution of 1871.

  2. 2 2 Ken B

    “Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.”

    This is absolutely true, and amply illustrated in the history of mathematics, even before the severe formalization of the foundations in Cantor’s era. Look at Weierstrass and the refoundation of calculus for example, or the discovery of the root of -1.

    Someone put it well once I thought http://www.thebigquestions.com/2011/06/08/inconsistency/#comment-27660 :)

    It’s also a point I could not convince many of in my “philawwwwwwwwsophy” of mathematics class, despite being the only mathematician in the class!

    Does Silas actually deny this??

  3. 3 3 CC

    SL, does this mean you have a definite opinion on whether CH is true or not? Does it have a definite truth value? Or is this totally different because it involves real numbers, which are problematic?

    If reals are problematic, what does this say about analytic number theory? Is it on less firm ground than elementary number theory?

  4. 4 4 Ken B

    @CC re 3:
    Good question. Many mathematicians DO have an opinion. Cohen thought CH was ‘obviously false’. I haven’t done any of this stuff in decades but my sense is that most set theorists think CH is false. The older I get the more sympathetic I am to the Brouwerites, who would deny it has a truth value though.

  5. 5 5 James Miller

    “I, who am not an expert, will surely die with only a vague sense of the argument.”

    So sign up for cryonics.

  6. 6 6 Steve Landsburg

    CC: I have no definite opinion on whether CH is true or not, and see no particular reason to believe that it has a truth value. The real numbers, as you say, are much more like a human construction than the natural numbers are.

  7. 7 7 Steve Landsburg

    Ken B:

    Re your link to http://www.thebigquestions.com/2011/06/08/inconsistency/#comment-27660 :

    I had absolutely no conscious memory of this exchange, but obviously it stuck with me. Thanks for planting this way of saying it in my mind.

  8. 8 8 CC

    SL: So is analytic number theory on shaky ground then since it depends on the reals?

  9. 9 9 Steve Landsburg

    CC: I don’t know what you mean by “on shaky ground”. If you mean that the results proven about the natural numbers via analytic number theory might be false, then surely not. If you mean that the language of analytic number theory makes it possible to formulate questions that might have no definite truth value, then maybe — depending on what you mean by “the language of analytic number theory”.

  10. 10 10 CC

    Ok, let me make this more concrete. Let’s say that mathematicians get together and decide that CH is false. Then some smart analytical number theorist derives a bunch of results about integers based on this new axiom. How confident are you in his or her new results?

    [Sorry, SL. I'm sure I'm missing something here; I just don't see what it is.]

  11. 11 11 Ken B

    @CC: Can’t speak for analytic number theory but there are precedents in set theory. Pretty much every weird property of the reals seems to be tied to some statement about really big cardinals. People prove theorems about this all the time. I don’t see why, if everyone became convinced Landsburg cardinals existed, or Krugman cardinals did not, that those theorems would be rejected. [But I'd bet it would work the other way, people would pick the cardinals they believe in to get the real number properties they desire.]

  12. 12 12 Silas Barta

    Mature as your reference to be might have been, I don’t remember disputing concern with fundamental reality. And everyone who voiced an opinion on our past exchange on the matter seemed to be equally unconvinced by your arguments to that end. Maybe insult them instead?

    Or maybe ask a certain Computer Science wunderkind to take me out of the acknowledgments for his upcoming paper?

  13. 13 13 Roger

    Steve, I disagree with your attempts to speak for all mathematicians. You have expressed a peculiar philosophical view here, and I very much doubt that more than 5% of mathematicians agree with you. I have never seen any math textbook or paper express such a view. It is not the view of Shin Mochizuki. Can you give any example of a mathematician who agrees with you about this?

  14. 14 14 Ken B

    @Roger: Steve often expresses peculiar phiolosophical views, but on the basic point that mathematicians are happy changing the axioms to get the theory they want I can cite many. Russell showed naive set theory won’t do, and that led to ZFC not the abandonment of mathematics. Zermelo introduced Choice because he needed it to prove a result he believed. I think Robinson’s non-standard analysis is an example, especially in some of its later guises founded on ZFC with non-standard axioms added. I mentioned Weierstrass and Cauchy and the refoundation of calculus. The foundations of mathematics were unclear and in flux then. Cardano changed the definition of number to accomodate the results on solving cubics. Euclid identified the special nature of the parallel axiom.

  15. 15 15 Roger

    @KenB: No, I do not believe for one minute that Russell, Zermelo, Robinson, Weierstrass, or Cauchy would agree with what Steve says here. Not sure about Cardano and Euclid, but I doubt it. Robinson worked within ZFC, but even those adding non-standard axioms would be very unlikely to agree.

    Steve is expressing a view that mathematicians don’t care about what can be proved from axioms. Your examples are mathematicians who spent much of their lives trying to figure out what can be proved from axioms. Find me one who says he does not care what axioms are used in Mochizuki’s proof or any other proof.

  16. 16 16 Ken B

    @Roger: I don’t agree with your reading of what Steve says then. Of course axiomatic mathematicans care about the axioms. Steve is saying there is no obviously privileged set of axioms for most areas of mathematics, so as part of mathematical research we can pick and choose. Finding the best axioms is part of the process, they are not god-given.

  17. 17 17 Roger

    @KenB, I have no idea what you mean by “privileged”. But you are changing what Steve said. I defy you to find one mathematician who agrees with what Steve said above.

  18. 18 18 Keshav Srinivasan

    Steve, concerning your views about the status of the natural numbers vs. the real numbers, I raised an objection in the comment section of a previous post:
    http://www.thebigquestions.com/2012/05/02/the-number-devil/#comment-49702

    But let me restate it here. Basically, what you’re doing is starting with a preformal, intuitive conception of the natural numbers, then you define real numbers in terms of sets of natural numbers, and since there is some ambiguity as to what sets of natural numbers there ARE, you ascribe the same degree of ambiguity concerning what statements about real numbers are true and false.

    But you could could do things in the opposite order. You could start with a preformal, intuitive conception of the real numbers, then you define the natural numbers in terms of sets of real numbers as follows: a set X of real numbers is hereditary if x+1 is an element of X whenever x is an element of X, and a natural number is a real number that is an element of every hereditary set containing 0. Then since there is an ambiguity concerning what sets of real numbers there ARE, you can ascribe the same degree of ambiguity as to what statements about the natural numbers are true, such as the ABC conjecture or Goldbach’s conjecture. How would you respond to that?

  19. 19 19 Steve Landsburg

    Roger:

    I defy you to find one mathematician who agrees with what Steve said above.

    Well, let’s see now. For roughly fifty years, nearly every algebraic geometer and a substantial fraction of number theorists has cheerfully invoked the work of Grothendieck, despite a great deal of uncertainty about how much of that work is formalizable in ZFC. Almost none of them has worried about this uncertainty. So: Pick your favorite top-100 math department, go to the web page, and pick out anyone who lists algebraic geometry as a research specialty. There’s your example.

  20. 20 20 Steve Landsburg

    Keshav Srinivasan: Nice try (and I mean that sincerely). But first of all, I think that going from reals to naturals requires higher-order reasoning than going from naturals to reals (sets of sets of sets instead of sets of sets), which makes it more suspect from the get-go. (I’m not sure I’m right about this, though.) Far more importantly, though, there is this:

    Formal logic requires a preformal intuitive concept of the natural numbers (or at least something very like that); for example, a proof is an ordered sequence of statements, and a statement is an ordered sequence of symbols, and the notion of an ordered sequence requires the notion of an initial segment of the natural numbers. It does not require a preformal intuitive concept of the reals.

    Any inquiry has to start somewhere; psychology starts with the assumption that other people have minds; physics starts with the assumption that there is a physical universe. Math has to start somewhere, and I don’t see how it can possibly avoid starting with the natural numbers. It *can* easily avoid starting with the reals. So there’s your fundamental asymmetry right there.

  21. 21 21 Roger

    Steve: Let us know if any of those mathematicians is ever willing to express such a view in writing.

  22. 22 22 Steve Landsburg

    Roger:

    Let us know if any of those mathematicians is ever willing to express such a view in writing.

    Have you not noticed that a lot of those mathematicians have written multiple papers that cite Grothendieck?

  23. 23 23 Steve Landsburg

    Silas (#12):

    The world is full of thoughtless imbeciles, and for the most part there’s nothing to be gained by taking notice of them. But your thoughtless imbecility is in a special class. It’s a cultivated imbecility, one that you’ve chosen to foster and to revel in.

    It is, in other words, a willful sort of thoughtless imbecility — part of the identity you’ve chosen for yourself, as opposed to a handicap you were born with. I infer this from your demonstrated ability to make occasional thougtful comments in areas where you’ve chosen not to play the clown. The foundations of mathematics is not one of those areas.

    Your past comments, both here and elsewhere, betray a fundamental confusion between theories and models. (Peano arithmetic, for example, is a theory; the natural numbers are a model of that theory.) This is a standard rookie mistake, and nothing to be ashamed of. The difference between you and the average rookie is that after this distinction has been explained to you 437,000 times, it still appears to go in one of your ears, pass effortlessly through the area where most of us have brains, and then out the other ear — so that after roughly 436,000 of those 437,000 explanations, you’re still making arguments that depend fundamentally on non sequiturs like “the consistency of the natural numbers”. (Theories are either consistent or inconsistent; the notion of consistency is as inapplicable to a model as it is to a rock.)

    Or (to take one of a kazillion examples almost at random), recall the back-and-forth about whether being able to simulate a squirrel in the structure N of natural numbers implies the ability to simulate a squirrel via a finite set of axioms. It of course does not. Once again, no sin not to understand that. But I and others explained this to you eight kajillion times and you went right on ignoring everyone (while blathering on about how everyone agreed with you, thereby flaunting how far over your head the entire conversation had gone).

    Your inability to grasp this point would be less infuriating if you didn’t a) keep repeating it; b) completely ignore every goodnatured and careful attempt to educate you; c) take each of these attempts as an opportunity to shout the moral equivalent of “Ha-ha”; d) keep announcing that everyone agrees with you because you are choosing to be too damn dumb to see the difference between what you are saying and what everyone else is talking about, and e) attempting to hijack a lively and thoughtful discussion. Arguing with you is like arguing with Nelson Muntz.

    You do the same damned thing whenever we talk about the Coase theorem, with your junior-high-school example of the guy who revs the motorcycle outside someone else’s window. Either the motorcycle-revving should be permitted or it shouldn’t be. Coase says that one of those rules is likely to be better than the other. You point out that one of those rules is better than the other and follow it up with another Muntzian “Haha”.

    Sometimes you appear to be “Haha-ing” over the fact that you didn’t need Coase to tell you that one rule is better than the other. This willfully overlooks the fact that deep ideas sometimes can have trivial applications. Other times you appear to be “Haha-ing” over your ignorant belief that Coase said some other thing which in fact you made up out of whole cloth.

    Again, misunderstanding the Coase paper is no sin. The sin is ignoring 17,000 good faith attempts to point out your misunderstanding — including pointers to textbook discussions of the motorcycle-revving problem (or perfect analogues thereof). The appropriate response to those good faith attempts is “Gee, thanks, I had that wrong. I appreciate your taking the time to set me straight”. Instead, your response is another “Haha”.

    More than once, you’ve hijacked good discussions about mathematical logic and/or the Coase theorem, where many informed and intelligent viewpoints were being expressed and much was being learned on all sides. Your juvenile repetition of the same stupid errors over and over and over and over and over and over and over and over is not cute, it’s not funny, and for people who haven’t yet figured out what a dick you are and therefore waste their time reading you, extremely distracting. All of this has earned you a place on my extremely short list (under half a dozen people) whose comments are always put in the queue for moderator approval. And at this point, even when your comments are reasonable, I don’t plan to feel any particular hurry about approving them.

    If you want to engage in intellectual discourse, a good starting point would be to learn something. A good way to do that is to ask questions and listen to the answers, as opposed to making up your own answers and then putting your fingers in your ears lest anybody correct them. In other words, Silas, grow the hell up.

  24. 24 24 Bearce

    Whoa! Epic…

  25. 25 25 Keshav Srinivasan

    Steve, responding to your comment:

    1. Going from real numbers to natural numbers does not require higher-order reasoning compared to the other way around. All you need is a single quantification over sets of real numbers. In terms of real numbers, “x is a natural number” just means “For all sets of real numbers X, if X is hereditary and 0 is an element of X, then x is an element of X”, where “X is hereditary” means “For all real numbers y, if y is an element of X then y+1 is an element of X.” That’s it, so every first order statement about natural numbers just becomes a second-order statement about real numbers, i.e. a statement about sets of real numbers.

    2. With Dedekind cuts, first-order statements about real numbers become at least second-order statements about natural numbers, and depending on how much encoding you do they can even be higher-order. (For instance, you can choose to encode rational numbers, which are ordered pairs of natural numbers, as single natural numbers, but if you don’t do that encoding then statements will be pushed into third order.) And the Cauchy sequence construction is even worse: you’re dealing with equivalence classes of Cauchy sequences of rational numbers, which are themselves ordered pairs of natural numbers. (Again, you can do away with ordered pairs by encoding.) So if anything, going from real numbers to natural numbers is more natural.

    3. I certainly agree with you that the practice of mathematics involves writing down ordered sequences of symbols, but as you’re fond of saying, the map is not the territory. For instance, just because mathematicians use axiomatic formal systems to find mathematical truths does not mean that mathematics is the study of consequences of axiomatic formal systems. In the same way, just because mathematicians communicate mathematical truths using languages (formal or informal) that consist of ordered sequences of symbols does not mean that mathematics itself is founded upon ordered sequences. Even the English language consists of ordered sequences of symbols; that doesn’t tell us much about what role natural numbers play in the subjects we communicate about using the English language.

    4. In your previous post “The Number Devil”, you talk about the devil taking out real numbers from the real number system, so that the real numbers become countable as viewed from the outside even though they still seem uncountable as viewed from the inside. Well, the same ambiguity extends to the natural numbers: the devil can add natural numbers to the natural number system, which are infinitely large when viewed from the outside, but seem finite when viewed from the inside. So just like you say that maybe we don’t all have the same real number system in mind, couldn’t you equally say we may not have the same natural number system in mind?

  26. 26 26 Steve Landsburg

    Keshav Srinivasan:

    Re points 1) and 2), thanks for setting me straight.

    Re point 3): My point is that we have to start *somewhere*. It’s often said that we can start with formal systems and thereby dispense with the natural numbers as a starting place; I’m arguing that it’s not so clear this works.

    Re point 4): So just like you say that maybe we don’t all have the same real number system in mind, couldn’t you equally say we may not have the same natural number system in mind? Yes and no. Second order Peano arithmetic has a unique model (up to isomorpism of course) and I think it’s safe to assume that’s the model we all have in mind.

  27. 27 27 Keshav Srinivasan

    Steve, the second-order theory of real numbers is also categorical, i.e. it has a unique model up to isomorphism. This theory is the same as the first-order theory, except we replace the axiom schema with a single axiom that says that every set of real numbers with an upper bound has a least upper bound. (Of course, just like second-order arithmetic it’s not a recursive theory.) So we are equally justified in saying that there is a standard model of the reals.

    And in fact, the second-order theory of the reals actually implies all of second-order arithmetic, i.e. the standard model of the reals contains the standard model of the natural numbers.

  28. 28 28 Bob Murphy

    Steve @23: Holy crap.

  29. 29 29 Ken B

    @Steve re 23: You know when I called you St Anslem of Rochester I was just kidding right?

    Squirrels??

  30. 30 30 Nick

    Steve @23

    Please point me in the direction of online education in maths. I’m not belligerent like the subject of your ire but I sometimes do skip a bit of your technical points.

  31. 31 31 Gene Callahan

    @Roger: “Steve is expressing a view that mathematicians don’t care about what can be proved from axioms.”

    No. Get some reading skills!

  32. 32 32 Martin-2

    Here’s where I first read about this big news.

    http://news.yahoo.com/mathematician-claims-proof-connection-between-prime-numbers-131737044.html

    The author makes some very insightful points such as – “the square-free part of the product A x B x C, denoted by sqp(ABC), divided by C is always greater than 0. Meanwhile, sqp(ABC) raised to any power greater than 1 and divided by C is always greater than 1.”

    Indeed! I remember an interesting thing you once said about journalists. Something to the effect of “at least we don’t let them design bridges”.

  33. 33 33 Joseph Fetz

    Steve @23,

    Wow! Just, wow.

    I took a little hit to the ego last x-mas over that hoarding thing, but I got over it (you were correct, I was confused). However, I don’t know that anybody can still have an ego left after what you just unleashed.

  34. 34 34 Ken B

    @Joseph Fetz: You are very young I see.

  35. 35 35 Joseph Fetz

    Relatively, yes. What are you getting at?

  36. 36 36 Ken B

    Being unable to sleep …
    @Roger: You have really missed the boat on this one. Steve never said anything like ‘mathematicians don’t care what can be proved from axioms.’ He said they were flexible — ‘opportunistic’ — about which axioms to use. Let’s look (my emphasis):

    No mathematician would consider rejecting Mochizuki’s proof just because it relies on new axiomatic foundations. That’s because mathematicians … don’t particularly care about axioms; …. [axioms] are investigative tools, which we are free to jettison when better tools come along … if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.”

    It’s clear Steve is saying most mathematicians do not care which particular set of axioms are in vogue with set theorists this week. If we need to adjust these “foundations” then we will. I cited a litany of examples.

    This is because just what the best set of axioms is is unknown. That’s what I meant by no set of axioms being privileged. Mathematics has no infallible Pope to tell us the foundational truths. The “foundations” of mathematics are NOT the simple truths we find most evident. Let me repeat that. The “foundations” of mathematics are not the simple truths we find most evident. They are a construction used to clarify, order, and simplify the core parts of mathematics used day to day. If we find day to day mathematics needs a broader “foundation”, we (set theorists, logicians) oblige.

  37. 37 37 Roger

    @Ken B: Steve certainly did deny that mathematicians care about what can be proved from axioms. He said, “Mathematicians care about what’s true, not about what’s provable; if a truth isn’t provable, we’re fine with changing the rules of the game to make it provable.”

    As you say, Steve often expresses peculiar philosophical views. He often posts his own opinions about provability from axioms. My quarrel here is with his charactization of mathematicians.

    Sure, mathematicians like to specialize, and most of them do not study foundations. Algebraic geometers especially have a cult-like reputation of ignoring others. The Italian school of algebraic geometry is famous for making sloppy assumptions that are sometimes wrong. But even still, today’s algebraic geometers do not condone changing the rules of the game because some (alleged) truth is not provable.

    Saying that algebraic geometers cite Grothendieck without worrying about foundations is about like saying that they don’t worry about differential equations. Yeah, they would rather let someone else worry about the differential equations. But that does not mean that they don’t care about what is provable.

  38. 38 38 Martin-2

    When I read “unprovable” on TBQ I think of three things:

    Something can be true, but not provable in first-order logic. An example is the hydra game and its guarantee of victory. There is a recursive function that gives the number of steps it will take for Hercules (playing badly) to defeat the hydra. First-order logic can’t tell you if this function terminates (or maybe doesn’t help you with any recursive functions?), second-order logic can.

    Something can be true but not provable at all. A possible example is the conjecture that there are infinitely many primes p such that p + 2 is also prime. It might literally be impossible to know if this is true or false, but still it must be one or the other.

    The ABC conjecture and Fermat’s Last Theorem are true but not provable in any established axiomatic system. However, new axioms were created that lead to proofs of these theorems. Of course, the truth of these new axioms can’t be proven using the established axioms either, so I’m guessing we know they’re true because they don’t lead to contradictions or are intuitive on some level.

  39. 39 39 Steve Landsburg

    Roger:

    Saying that algebraic geometers cite Grothendieck without worrying about foundations is about like saying that they don’t worry about differential equations. Yeah, they would rather let someone else worry about the differential equations. But that does not mean that they don’t care about what is provable.

    This is a very poor analogy. Nobody has ever suggested that the major theorems in differential equations might not be provable in ZFC. It has been widely suggested and believed that the major theorems of modern algebraic geometry are not provable in ZFC. To say that algebraic geometers don’t care about differential equations is not remotely comparable (in this context) to saying that algebraic geometers don’t care whether their own theorems are provable in ZFC.

  40. 40 40 Steve Landsburg

    Martin-2:

    Something can be true, but not provable in first-order logic.

    You need to say this a little more carefully. Everything is provable in first-order logic, because you can take anything you want as an axiom, whereupon it acquires a one-line proof. What you mean to say is “not provable in first-order logic from standard axioms“.

    The ABC conjecture and Fermat’s Last Theorem are true but not provable in any established axiomatic system.

    We do not know this; my understanding from the people who have thought hardest about this is that the Wiles/Ribet proof of FLT can indeed be formalized in ZFC and quite probably (with a lot more work) in Peano Arithmetic. Part of the point, though, is that many people were very unsure of this for a substantial time, and essentially none of those people suggested that this was a reason to reject the theorem.

  41. 41 41 Keshav Srinivasan

    Steve, I just wanted to prove the two assertions I made in my comment #27. First, the fact that just like second-order Peano arithmetic, the second-order theory of real numbers has a unique model upto isomorphism. Let R and S be models of the second-order theory of real numbers (i.e. R and S are ordered fields such that every set which has an upper bound has a least upper bound.) Each of these real-number systems has its own set of rational numbers, Q_R and Q_S respectively, and these sets are clearly isomorphic, let’s say with the isomorphism g from Q_R to Q_S. Then let us define the map f from R to S as follows: f(x) is the least upper bound of the set g({q in Q_R: q<x}). In other words, f is the least upper bound of the Dedekind cut in Q_S which corresponds to the Dedekind cut of x in Q_R. We can easily check that f is an isomorphism, so any two models of the second-order theory of real numbers are isomorphic.

    Second, I made the claim that the second-order theory of real numbers implies second-order Peano arithmetic (which, as a consequence of the result above, means that the standard model of the real numbers contains the standard model of the natural numbers). Recall that the definition of natural number I'm using is the following: a real number x is a natural number if x belongs to every hereditary set containing 0, and a set of real numbers X is hereditary if it contains y+1 whenever it contains y. So without further ado, let me prove the five Peano axioms:

    1. To prove that 0 is a natural number, we must prove that 0 belongs to every hereditary set containing 0. That's trivial because 0 belongs to *every* set containing 0.

    2. We must prove that whenever x is a natural number, so is x+1. Consider any hereditary set X containing 0. Then by definition of natural number, x must belong to X, and thus by definition of hereditary, x+1 must belong to X. Since X was arbiteary, x+1 belongs to every hereditary set containing 0, and thus x+1 is a natural number.

    3. The fact that x+1=y+1 implies x=y follows directly from the field axioms, which are part of the second-order theory of real numbers.

    4. We must prove that there exists no natural number x such that x+1=0, which in our case is equivalent to the statement that -1 is not a natural number. Well, just consider the set of all natural numbers greater than or equal to 0 (which is of course all of them, but we don't know that yet). Then this is a hereditary set containing 0, and yet -1 doesn't belong to it, so -1 is not a natural number.

    5. The principle of induction says that if X contains 0 and contains x+1 whenever it contains X, then X contains all the natural numbers. But that is just saying that if X is a hereditary set containing 0, all natural numbers belong to it, which is trivial given our definition of natural number.

    Finally, the defining axioms of addition and multiplication in Peano arithmetic follow easily from the field axioms.

  42. 42 42 Steve Landsburg

    Keshav Srinivasan: Thanks for this. How do you pick the set of rational numbers out of a model of the second-order theory of the reals?

  43. 43 43 Keshav Srinivasan

    Steve, I’m not exactly sure what you mean by “pick out”. I already told you the definition of natural number in the second-theory of real number, from which we can easy define the set of rational numbers as Q={x: x=ab^-1 for some a,b in N}, where N. Since we can already define and prove the existence of the set Q in the formal theory, what difficulty is there in “picking it out” in any model?

    By the way, I should mention that this is one of several proofs of the result; you can find others in many analysis texts, but the uniqueness proof I presented above comes from these analysis notes:

    http://math.berkeley.edu/~gbergman/ug.hndts/m104_Rudin_exs.pdf (pg 7-8)

  44. 44 44 Paul T

    SL: “Mochizuki might or might not have settled the ABC problem, and if
    so he might or might not have been forced to replace old axioms with new
    ones; it’s possible he’s not sure of this himself”

    How in the world does one vet a 500 page proof? I’ve never been close to anything like that. Is there a committee, with a chairman doling out
    bits of the proof to various members?

    Anywho, a corollary occurs to me. If the conjecture is true, with
    A + B = C, and finite number of C such that C is greater than factors
    of etc. per some specified exponent, then there must be a largest such
    exponent, such that there is no C which is greater than etc.

    This is remarkable, as there is an infinite number of A, B, C, and
    formulae, yet still there must be a finite, largest such exponent.

  45. 45 45 Paul T

    *** MONDAY PUZZLE ***

    f(n) is a function of any integer n, positive or negative, which produces an integer value, with conditions:

    a) f(f(n)) = n

    b) f(f(n + 2) + 2) = n

    c) f(0) = 1

    1. Determine f(n)
    2. Prove your solution is unique.

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