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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by Henry Baker 01 May '18

01 May '18

It's pretty clear: if there are consistent models with CH and consistent models w/o CH, then that makes CH *independent* -- to choose whichever way you please. Just like the Euclid's parallel postulate: if you like it, you get Euclid's plane geometry, whereas if you don't like it, you get one of the non-flat geometries -- e.g., Einstein. That's mathematics/mathematical logic. Now consider physics. I find CH completely antithetical to quantum theory -- e.g., "ultraviolet catastrophe" and all that. So while CH is really nice for a lot of math, I don't think that it describes post-19th C. physics very well. Curiously, much of the math of QM seems to make liberal use of CH, so something's wrong with the math at the basis of QM. I believe that this was the essence of Fredkin's, Bennett's, Feynman's (?) etc., arguments. BTW, I seriously doubt that TPC is independent, but I don't have any proofs. At 04:29 PM 4/29/2018, Dan Asimov wrote: >The twin prime conjecture (TPC) says there are infinitely many pairs of prime numbers separated only by 2, like 3 and 5, 5 an 7, 11 and 13, etc. > >No one knows if it's true or false. > >But whichever is the case, it might not be possible to *prove* this fact. > >A similar situation is the Continuum Hypothesis (CH), which states that there is no infinity strictly larger than the size of the integers and strictly smaller than the size of the real numbers. > >Kurt Goedel proved about 1940 that CH was consistent with the axioms of set theory (ZF, for Zermelo-Frankel), and Paul Cohen proved in 1963 that the negation of CH was *also* consistent with ZF. > >So in this case there is *no doubt*: There exists neither a proof of CH, nor a proof of its negation ~CH, purely based on the axioms ZF+AC of set theory. > >Question: >--------- > >So, does that mean that CH is neither really true nor really false, but some third option? > >Or what? > >I'm curious what people think about this. > >ÂDan

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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by hbaker1 01 May '18

01 May '18

Doesn't it get worse? Aren't there *particular* TM's whose haltingness can be arbitrarily chosen? I.e., "TM XYZ halts" becomes an axiom *independent* of the rest of arithmetic? -----Original Message----- >From: Cris Moore <moore(a)santafe.edu> >Sent: May 1, 2018 6:19 AM >To: hbaker1 <hbaker1(a)pipeline.com>, math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? > >I agree with all this: indeed my favorite way to prove the first Incompleteness Theorem is the undecidability of the Halting Problem. For any set of axioms, there is a Turing machine which 1) never halts and 2) that set of axioms cannot prove that it never halts. Otherwise the Halting Problem would be decidable. > >But don’t you agree that the Halting Problem has a definite truth value? In other words, that a given Turing machine (with a given input) either runs forever or doesn’t, regardless of our ability to prove it? Or does the unprovability of non-halting make you a pluralist on this topic — that if we can’t prove non-halting in ZFC (say), then we can have it either way? > >- Cris > >> On May 1, 2018, at 7:02 AM, hbaker1 <hbaker1(a)pipeline.com> wrote: >> But that's the heart of the incompleteness theorem: *any* theory strong enough to describe number theory or Turing Machines has certain true statements that have no (finite) proofs. >> >> Recall that proofs are finite sequences generated by non-deterministic machines from axioms and rules of inference in an attempt to list all theorems. >> >> By a strong analogy to diagonalization of an attempt to list all of the numeral sequences of the real numbers, we can construct a new real number not on the list; and hence we can construct a new truth not on the list of provable theorems. >> >> -----Original Message----- >>> From: Cris Moore <moore(a)santafe.edu> >>> Sent: Apr 30, 2018 10:42 PM >>> To: hbaker1 <hbaker1(a)pipeline.com>, math-fun <math-fun(a)mailman.xmission.com> >>> Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? >>> >>> Right, the Halting Problem is asymmetric: there’s a finite proof of halting if it halts, but not necessarily a finite proof that it runs forever if it runs forever. >>> >>> Cris >>> >>>> On Apr 30, 2018, at 11:02 PM, hbaker1 <hbaker1(a)pipeline.com> wrote: >>>> >>>> Turing Machines which halt: "recursively enumerable" == "RE" >>>> Turing Machines which don't halt: complement of RE. >>>> >>>> Turing proved that they aren't the same sets. >>>> >>>> Thus, RE sets are kind of "fractals", in that their boundaries are a little difficult to determine -- at least from one direction. >>>> >>>> -----Original Message----- >>>>> From: Cris Moore <moore(a)santafe.edu> >>>>> Sent: Apr 30, 2018 2:28 PM >>>>> To: math-fun <math-fun(a)mailman.xmission.com> >>>>> Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? >>>>> >>>>>> On Apr 30, 2018, at 3:21 PM, Cris Moore <moore(a)santafe.edu> wrote: >>>>>> >>>>>> However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>> >>>>> >>>>> Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult. >>>>> >>>>> But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here. >>>>> >>>>> - Cris

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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by hbaker1 30 Apr '18

30 Apr '18

Turing Machines which halt: "recursively enumerable" == "RE" Turing Machines which don't halt: complement of RE. Turing proved that they aren't the same sets. Thus, RE sets are kind of "fractals", in that their boundaries are a little difficult to determine -- at least from one direction. -----Original Message----- >From: Cris Moore <moore(a)santafe.edu> >Sent: Apr 30, 2018 2:28 PM >To: math-fun <math-fun(a)mailman.xmission.com> >Subject: Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ? > >> On Apr 30, 2018, at 3:21 PM, Cris Moore <moore(a)santafe.edu> wrote: >> >> However, we should be careful, since even questions about the halting of Turing machines — which are certainly first-order claims about the integers — can be independent of ZFC: https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf> <https://www.scottaaronson.com/busybeaver.pdf <https://www.scottaaronson.com/busybeaver.pdf>> > >Sorry, continuing: note that if a Turing machine halts in finite time, there is a finite proof of that fact — namely, run it and see what happens. But if it runs forever, proving that it does so might be difficult. > >But surely there is a truth of the matter here! The Turing machine will either run forever, or it won’t. I find it hard to be a pluralist here. > >- Cris

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Re: [math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by Keith F. Lynch 30 Apr '18

30 Apr '18

The problem is that "the number line" is ambiguous. Different people mean different things by it. Even any one individual's concept of it is inherently fuzzy, conflating innumerable slightly different meanings. For example, does it only contain real numbers? Or is every real number orbited by an infinite retinue of planet-like infinitesimals, each of which is accompanied by its own second-order satellites, the squares of the infinitesimals, at infinitum? If there are infinitesimals there must also be reciprocals of them, which are numbers which can't be reached by counting. Since I'm a computer person, the opposite approach is more attractive to me: Only computable numbers exist. Unless a finite-sized program with some finite amount of memory can, before the end of time, calculate the number to any desired finite precision, the number has a rather ghostly existence. The real numbers that make Cantor's diagonal argument work can only be created by flipping a coin infinitely many times to decide what each of its binary digits will be. And the resulting number can't be stored anywhere. We can talk about such numbers as a class, but there's no way to point to any one of them. Hence my post here one April Fools' Day when I made the absurd claim that I would be maintaining an online database of all uncomputable numbers. Computable numbers have pretty much all properties you'd want from reals. They're dense, they form a field, etc. If you have a task that requires a number than can never be calculated, you'd better find a different task. That's not quite right. There are numbers which are well-defined but uncomputable, e.g. the Chaitin constants and ... well, actually, those are the only counterexample I can think of. And since we can never know their values, we can live without them. If there are no infinities that can't be put into one-to-one correspondence with the integers, the continuum hypothesis is moot. Computable numbers can be put into one-to-one correspondence with the integers since computer programs basically *are* integers. I don't know of any model of quantum mechanics, or of anything else physical, which depends on CH being true of false. As for whether math is invented or discovered, obviously the latter, otherwise different people wouldn't get the same answers, any more than different artists produce identical paintings.

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[math-fun] Nearly circular curve
by David Wilson 30 Apr '18

30 Apr '18

cos(x) + cos(y) = 1 + cos(1). The plot of this equation is, I believe, an infinite grid of near-circles. I plotted this on Replot, but I couldn't visually distinguish the curves from circles. I'd be curious to know how different the curve centered at the origin is from a unit circle. I suspect it is circumscribed by the unit circle centered at the origin. What is its circumference and area? I suspect these curves are related to the dots used in halftone printing. Do they have a name?

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Re: [math-fun] Continuum Hypothesis
by Dan Asimov 30 Apr '18

30 Apr '18

I've known others with views similar to Adam's. I'm not sure I understand the reason for treating (say) the set of real numbers differently from the integers, though. The reason I believe that [the Twin Prime Conjecture is either really true or really false] is that I can imagine a way of verifying the truth status of TPC by a census of all numbers of the form p+2 where p is prime — at least in principle. But likewise, I can imagine in principle a verification of the truth status of the Continuum Hypothesis. Suppose it's true. Then for every subset X of the reals: we need to check, for every function f : X —> Z *and* for every function g : X —> R, that *at least one of these* is a bijection. (Which itself can be described as a kind of transfinite induction.) Because I can imagine such a verification (whether of CH or not-CH), I find it impossible to imagine that the truth status of CH is indeterminate. And the concept of a "subset of the reals" seems quite well-defined to me. Flip a coin once independently for each real number, and you get some subset; every subset is obtainable in that way. Anyhow, that's my story and I'm sticking to it. —Dan Adam Goucher wrote: ----- My own philosophical position is Platonist/realist about first-order properties of the natural numbers, and formalist about everything else. So I believe Con(ZFC) has a definite truth-value (namely 'true'), but CH does not have a definite truth-value. This position means that, unlike a formalist, I reject the perfectly consistent theory "ZFC + ¬Con(ZFC)" as nonsensical on the basis that it proves false statements about the naturals (such as ¬Con(ZFC)). But I'll happily work in either "ZFC + CH" and "ZFC + ¬CH", as they are both (as far as we know) arithmetically sound. If you translate this position into an invented/discovered dichotomy, and then into German, then it simplifies to Kronecker's quotation "Die ganzen Zahlen hat der liebe Gott gemacht, alles andere ist Menschenwerk". -----

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[math-fun] Is the Continuum Hypothesis a) really true or really false, or b) something else ?
by Dan Asimov 30 Apr '18

30 Apr '18

The twin prime conjecture (TPC) says there are infinitely many pairs of prime numbers separated only by 2, like 3 and 5, 5 an 7, 11 and 13, etc. No one knows if it's true or false. But whichever is the case, it might not be possible to *prove* this fact. A similar situation is the Continuum Hypothesis (CH), which states that there is no infinity strictly larger than the size of the integers and strictly smaller than the size of the real numbers. Kurt Goedel proved about 1940 that CH was consistent with the axioms of set theory (ZF, for Zermelo-Frankel), and Paul Cohen proved in 1963 that the negation of CH was *also* consistent with ZF. So in this case there is *no doubt*: There exists neither a proof of CH, nor a proof of its negation ~CH, purely based on the axioms ZF+AC of set theory. Question: --------- So, does that mean that CH is neither really true nor really false, but some third option? Or what? I'm curious what people think about this. —Dan

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Re: [math-fun] Continuum Hypothesis
by John Golden 30 Apr '18

30 Apr '18

I’ve been loving this discussion. With students (in a capstone course) we talk about Cantor and Gödel in sequence, and the CH comes off as an example of incompleteness. I.e. it could be true or false but unverifiable. So you have to change the axioms if you want it to be one or the other and provable. Am I propagating a misconception? On the other hand, most mathematicians with whom I’ve discussed this have a personal belief that it’s true or not. It often segues into a discussion on invented or discovered. I remember a great weeklong lecture from Alain Connes on why math is discovered, in which I think he identified as a CH believer. Hardcore invented types seem to stick to the CH being unverifiable, nothing more to say.

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[math-fun] When is base 2 not binary
by Lucas, Stephen K - lucassk 30 Apr '18

30 Apr '18

New to the list, so I hope this will go in the right place. But replying to Joerg Arndt’s listing of the first few natural numbers in base 2+i using the digits {0,+1,-1,+i,-i}: The problem with this digit set is that it is not immediately obvious what the rule is to go from number to number adding one. Here is another idea: use the digits {-1,0,1,2,3}. And base 2+i has the carry rule [-1,+4,-5], which says if a digit is 4 or greater, subtract 5 from it, add 4 to the next most significant, subtract one from the next next most significant. One way of verifying this is to see that (-1)(2+i)^{k+2}+4(2+i)^{k+1}-5(2+i)^k=0. For example, with least significant digits on the right, 1 = 1, 2 = 2 3 = 3 4 = (-1)4(-1) = (-1)3(-1)(-1) 5 = (-1)3(-1)0 6 = (-1)3(-1)1 7 = (-1)3(-1)2 8 = (-1)3(-1)3 9 = (-1)3(-1)4 = (-1)23(-1) 10 = (-1)230 11 = (-1)231 12 = (-1)232 13 = (-1)233 14 = (-1)234 = (-1)17(-1) = (-2)52(-1) = (-1)202(-1) and so on. A couple of notes: these digits are exactly the same representing natural numbers in base 2-i as well. This is current (!) joint work with Jim Propp on digits representing numbers in different bases simultaneously. And in base 2+i, you can extend the classic work of Katai and Szabo and represent Gaussian integers in base 2+i. If you limit yourself to nonnegative digits, they proved only -n+i works, not n+i. But a negative digit does the job! Steve

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Re: [math-fun] Continuum Hypothesis
by Guy Haworth 30 Apr '18

30 Apr '18

I met Marcus du Sautoy at the Royal Society in London once: I hasten to add that I was a guest of Fellow Ian Mills (he of the 'New SI') to hear Marcus' talk. Marcus explained to Ian that he was succeeding Richard Dawkins as the Simonyi Professor for the Public Understanding of Science. I was half way through saying "Well, at least there's only one mathematics" when I realised this was not true. The Continuum Hypothesis stands beyond the reach of 'the axioms', it can either be a new axiom or not. That gives you two flavours of mathematics. Marcus said that real mathematics had been done down both tracks. I haven't seen any maths done down the 'assuming a third infinity' track. [ Can we prove that if Infinity1 =/= Infinity2, then 2^Infinity1 =/= 2^Infinity2 ? ] g

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