> On 18 Dec 2018, at 23:44, Bruce Kellett <[email protected]> wrote: > > From: Bruno Marchal <[email protected] <mailto:[email protected]>> >>> On 18 Dec 2018, at 01:14, Bruce Kellett <[email protected] >>> <mailto:[email protected]>> wrote: >>> >>> On Tue, Dec 18, 2018 at 6:15 AM Bruno Marchal <[email protected] >>> <mailto:[email protected]>> wrote: >>> On 17 Dec 2018, at 08:50, Bruce Kellett <[email protected] >>> <mailto:[email protected]>> wrote: >>>> >>>> What I am curious to know is how how many of these statements you agree >>>> with: >>>> >>>> "2+2 = 4" was true: >>>> 1. Before I was born >>>> 2. Before humans formalized axioms and found a proof of it >>>> 3. Before there were humans >>>> 4. Before there was any conscious life in this universe >>>> 5. As soon as there were 4 physical things to count >>>> 6. Before the big bang / before there were 4 physical things >>>> >>>> "2+2=4" is a tautology, true because of the meanings of the terms >>>> involved. So its truth is not independent of the formulation of the >>>> question and the definition of the terms involved. >>> >>> What about ExEyEz (x^3 + y^3 +z^3 = 33) ? >>> >>> What about it? Not all syntactically correct formulae are either true or >>> false; some are undecidable. >> >> A is undecidable means ~[]A & ~[]~A. >> >> It does not mean ~A & ~~A. >> >> That A (= ExEyEx(x^3 + y^3 + z^3)) is either true or false is a direct >> consequence of the excluded middle. Either such numbers >> exists or they don’t. If it is undecidable in PA, it means that PA cannot >> decide it, but it does not mean that ZF or ZF+kappa could not decide it. And >> even if they couldn’t, it is still does not mean that A is not true, or >> false, which it is, certainly. > > So you have no idea whether it is true or not, because you do not know in > which system the Turing program designed to search all possible combinations > of integers to find an (x,y,z) combination with the required properties would > halt, or in which systems the similar Turing machine would not halt, and you > could not prove halting or not in any system. Then the problem becomes > totally trivial, we could define a system in which your A above is an axiom, > or one in which ~A is an axiom. In either case the question is trivially > decidable, but not in any useful sense -- such systems still do not produce > the required triple of integers
No problem, don’t see the point though. This does no make A undeciadbale, but illustrate that you do believe that A is either true or false, independently of our ability to solve the problem. (To be sure, if it is shown undecidable in some theory like RA or PA, or ZF …, it would automatically mean that there is no solution, as if there is a solution, even RA can find it (all universal machine can find it). > > >>> Unless and until you find some x,y,z that satisfy this relationship, the >>> statement is neither true nor false. >> >> In intuitionist logic. I have made clear that I use classical logic (indeed, >> in mathematical theology, we can expect many statement to be undecidable by >> the finite creatures/theories). Same already in theoretical computer >> science. The notion of totality is non constructive, “halting” is non >> constructive, in fact all attribute of programs can be shown to be >> necessarily non constructive. I recursion theory, constructive is equivalent >> to limiting research in the security zone of some RE subset of TOT. Machines >> have no right to search for a number which might not exist. > > Who gets to say what rights machine may or may not have? Nobody. But in constructive logic, or in intuitionist logic, or with some operator system, some action are forbid, because it would make you going out of the security zone. I give the right to my kids/machines to search for numbers, but the price is that I cannot be sure they will ever come back, or even that they will perhaps meet dangerous strangers, etc.. Al universal machine has the choice between security and limber (universality). No machine can have both. > I can write a program to search all possible integer combinations for a > solution to (x^3 + y^3 +z^3 = 33). But I cannot determine whether this > program will halt or not. You cannot claim that I do not have a right to > write such a program. Yes, but you make my point. You cannot know in advance if you machine will stop. But you know in advance that either it will stop, or not. > >>> Unperformed experiments have no results! >> >> Even in physics, this is doubtful and indeed contradict Einstein’s physical >> realism. > > Einsteinian realism is already contradicted by experiment. The rejection of > counterfactual definiteness is intrinsic to quantum mechanics, even in MWI. > >> No problem, as we know it has to be tampered with the Mechanist assumption. > > You might assume this, but you cannot know it! In science, nobody know anything, but some beliefs are more plausible than others, but still needs the continuation of the verification, forever. Bruno > > Bruce > > -- > 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 post to this group, send email to [email protected] > <mailto:[email protected]>. > Visit this group at https://groups.google.com/group/everything-list > <https://groups.google.com/group/everything-list>. > For more options, visit https://groups.google.com/d/optout > <https://groups.google.com/d/optout>. -- 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 post to this group, send email to [email protected]. Visit this group at https://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/d/optout.
