> On 28 Aug 2019, at 19:15, Jason Resch <[email protected]> wrote: > > > > On Wed, Aug 28, 2019, 11:36 AM John Clark <[email protected] > <mailto:[email protected]>> wrote: > On Tue, Aug 27, 2019 at 7:45 PM Jason Resch <[email protected] > <mailto:[email protected]>> wrote: > > > You can write a program that outputs the string "2 + 2 = 5", but you'll > > never find a program that outputs a proof of 2 + 2 = 5 in any consistent > > and sound system of axioms. > > Even if your system is consistent there is no way you can prove its > consistent while remaining within the system, and if you go outside the > system to prove it then you've just kicked the problem upstairs and you can't > prove the meta system is consistent. And soundness means any formula that you > can derive from axioms, that is to say prove, is true. So if you've got a > error free proof that 2+2=4 how do you know it's true, how do you know 2+2 > isn't 5? You're going to need a independent method of determining the truth > of that and there is only one way to do it, with physics. You put 2 hydrogen > atoms on a scale and note it reads about 2, you put 2 more on and it reads > about 4. You never get exact integers but physics tells us that 2+2=4 is a > good approximation of the truth, so we make sure our axioms and rules of > inference produce that. > > From day one when we started to construct our mathematics we've tried to make > it consistent with physics but there have been a few bumps in the road. It > turns out that is some places (where spacetime is flat) Euclid's fifth axiom > is true but in other places (where spacetime is curved) it's not true. And > both the Axiom Of Choice and the Continuum Axiom involve infinity and physics > has no use for infinity so physics doesn't care if those axioms are true or > not, so there is no way to independently determine their truth, so stuff > based on them are the equivalent to mathematical Harry Potter stories. > > Is this to say you agree? If you disagree with anything I said that missed > it. > > It is true we never access truth,
OK. I guess you mean we never access public 3p truth, through theories, experiments, etc. We have only theories, and means to refute or improved them. We do access truth in the 1p sense, as I guess you agree. That plays some important role to get that we do address the “consciousness” issue. > but the same dilemma haunts us in physics. Physics never tells us our > theories are true, only that they haven't been refuted so far. That's how > the bgs work with building axioms. We can hope they approximate the truth, > and we hope to find better axioms in the future. > > Have I changed your mind regarding what Minsky said regarding possible > programs? > > Regarding axiom of choice, I believe it's independent of ZFC, Of ZF I guess. It is a typo error. Obviously ZFC proves C, as ZFC is ZF plus C, the Choice axiom (or any of its mathematically equivalent formulations). > but that doesn't imply it's independent of another more powerful system. … or that it is a natural way to axiomatise our conception of sets. I am not set theoretically realist, but I am not enough non-realist either so as to doubt the axiom of choice! For a long time I took my notion of sets from Gödel constructible notion of sets, so my theory was V = L, in which there is few doubt on the consistency of Choice and the Continuum Hypothesis. Cohen technic to build models contradicting choice and CH seemed to me both wonderful semantical (even modal) method in logic, but that the results where leading to ad hoc theories too much easily like with 2^(aleph_0) = aleph_122. But the work and book by Patrick Dehornoy and the discovery of self-distributivity, both in braids and in the discovery of (possible) *very* large cardinal, has changed my mind on this. Now, I have reason to suspect that ZFC + Projective Determinacy might be … true, or at least saying something non trivial about some machine/number. That book by Patrick Dehornoy(*) has revived my interest for set theory. It helps a lot to see the relation between the arithmetical hierarchy in theoretical computer science, and the analytical hierarchy and the so called descriptive set theory (the study of the set N^N, identified with the reals, with variate topologies). Patrick Dehornoy, La théorie des ensembles, Calvage & Mounet, 2017. > > Jason > > -- > You received this message because you are subscribed to the Google Groups > "Everything List" group. > To unsubscribe from this group and stop receiving emails from it, send an > email to [email protected] > <mailto:[email protected]>. > To view this discussion on the web visit > https://groups.google.com/d/msgid/everything-list/CA%2BBCJUjdjRHS2w9VFywCAE%2BAwT3HShm52zXTMA3%2BH4Recg%2Brxg%40mail.gmail.com > > <https://groups.google.com/d/msgid/everything-list/CA%2BBCJUjdjRHS2w9VFywCAE%2BAwT3HShm52zXTMA3%2BH4Recg%2Brxg%40mail.gmail.com?utm_medium=email&utm_source=footer>. -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion on the web visit https://groups.google.com/d/msgid/everything-list/D917E871-BDC7-4FB9-BEAF-6E90A08E6E12%40ulb.ac.be.
