In a Consistent System, Can a True Sentence Imply a False One?
in [Logic]
|
Prev: when math defines the boundary between finite versus infinite at 10^500 #696 Correcting Math
Next: FLT like 4Color Mapping, Poincare C. and Kepler Packing #697 Correcting Math
From: Daryl McCullough on 19 Jul 2010 11:00 George Greene says... > >On Jul 19, 1:13=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >> A system may well be consistent even if some of its axioms are false. > >By definition, if the system is consistent, IT HAS A MODEL. >By definition, IF WE HAVE DESIGNATED SOME PARTICULAR >subset of the statements true in all models of the system AS AXIOMS, >Then it is THOSE models we CARE about! It's sometimes the case that we start with an intended interpretation of the symbols, and then choose axioms that we believe are true in that interpretation. We might be mistaken, though, and we might accidentally add an axiom that is not true in the intended interpretation. Such a situation is especially a possibility if one is trying to reason logically about the physical world. We could write down what we believe to be true about the facts of some criminal case, or properties of elementary particles, for example. What we write down as axioms are not necessarily true of the intended interpretation. Now, you might say that that situation never comes up in mathematics: when we try to characterize integers, or reals, or the Euclidean plane, it's not usually the case that we write down axioms that turn out to be false. What's more likely to be the case is that we write down axioms that are redundant---some axioms can be derived from other axioms. I don't know of any examples where someone competent attempted to axiomatize a mathematical topic and accidentally introduced a false axiom. We can artificially create such a situation, of course, by adding a conjecture such as Riemann's hypothesis as an axiom. Such an axiom may indeed be false. The fact that we added such a conjecture as an axiom in a theory of complex numbers does *not* mean that we are interested in whatever model makes that conjecture true. We already have a good handle on what a "complex number" is, and if we add another axiom, it is because we believe it is true of our pre-existing notion of complex numbers (or because we want to see what follows from the assumption). >It is the theory and not the language that has the intended >interpretations, and the interpretations intended ARE BY DEFINITION >the ones THAT MAKE THE AXIOMS *TRUE*!! I would say that often it is the other way around: the language has an intended interpretation, and we choose axioms that we believe are true in that interpretation. >Choosing your axioms IS HOW you communicate which models you intend! I think that typically, there are two different types of axioms. Some axioms are implicit definitions of the symbols involved, and so I would agree that your choice of axioms is a way of indicating what models you are talking about. For example, the Peano axioms about plus and times amount to a recursive definition of those functions. On the other hand, some statements, such as Goldbach's conjecture, or the twin primes conjecture are not intended as clarification of what we mean by "natural number". If they are added as axioms, it is because we believe (or assume for the sake of exploring the consequences) that they are true of the usual interpretation of the naturals. I don't know why, but I have this feeling that your response to this post will be an angry one, full of capital letters. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 19 Jul 2010 23:00 Daryl McCullough wrote: > George Greene says... >> On Jul 19, 1:13=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: >>> A system may well be consistent even if some of its axioms are false. >> By definition, if the system is consistent, IT HAS A MODEL. >> By definition, IF WE HAVE DESIGNATED SOME PARTICULAR >> subset of the statements true in all models of the system AS AXIOMS, >> Then it is THOSE models we CARE about! > > It's sometimes the case that we start with an intended interpretation > of the symbols, and then choose axioms that we believe are true in that > interpretation. But that's precisely why it's very difficult to understand the position of your side! A set of meaningful words doesn't necessarily make a meaningful sentence right? For example: "What's a country like me doing in a girl like Egypt?" (The Mummy) isn't making sense, naturally. So symbols' interpretation alone isn't sufficient: we need sentences, or axioms in the case of mathematics to really make sense of what we desire to mean. In brief, just like reading a novel, you have to understand the whole collection of paragraphs and sentences, instead of understanding only what words are interpreted to mean! > > Such a situation is especially a possibility if one is trying to > reason logically about the physical world. Again, to adequately describew physical word, we need coherent, consistent interpretation of many observations each of which is a sentence, not just symbols! > We could write down > what we believe to be true about the facts of some criminal case, > or properties of elementary particles, for example. What we write > down as axioms are not necessarily true of the intended interpretation. I think you started it wrong to begin with. The word "kill" has a very clear interpretation, meaning. But If ones writes a crime report then "He killed" or "He didn't kill" could be true or false but that has no bearing on whether or not the "interpretation" of the word "killed" is intended, broken, or otherwise! > > Now, you might say that that situation never comes up in mathematics: > when we try to characterize integers, or reals, or the Euclidean plane, > it's not usually the case that we write down axioms that turn out to > be false. What's more likely to be the case is that we write down axioms > that are redundant---some axioms can be derived from other axioms. > > I don't know of any examples where someone competent attempted to > axiomatize a mathematical topic and accidentally introduced a false > axiom. But your side is already wrong in one respect: you were assuming something absolute and that if one comes up with axioms counterintuitive to that something then one is incompetent! That's a very bizarre - and incorrect - notion! People do come up with different axioms, depending the contexts they're working at! > >> It is the theory and not the language that has the intended >> interpretations, and the interpretations intended ARE BY DEFINITION >> the ones THAT MAKE THE AXIOMS *TRUE*!! > > I would say that often it is the other way around: the language > has an intended interpretation, and we choose axioms that we believe > are true in that interpretation. No. GG is correct here and you aren't. Language interpretation is actually useless. At minimum you have to have model-interpretation! But if that's the case it's virtually a theory interpretation! > >> Choosing your axioms IS HOW you communicate which models you intend! > > I think that typically, there are two different types of axioms. > Some axioms are implicit definitions of the symbols involved, and > so I would agree that your choice of axioms is a way of indicating > what models you are talking about. For example, the Peano axioms > about plus and times amount to a recursive definition of those > functions. Axioms are just equally the same w.r.t. to being wff with semantics and so your 2-different-type-classification of axioms are obscured. > On the other hand, some statements, such as Goldbach's > conjecture, or the twin primes conjecture are not intended as > clarification of what we mean by "natural number". Why are these 2 'not intended as ... what we mean by "natural number"'? -- --------------------------------------------------- Time passes, there is no way we can hold it back. Why, then, do thoughts linger long after everything else is gone? Ryokan ---------------------------------------------------
From: herbzet on 19 Jul 2010 23:34 Daryl McCullough wrote: > George Greene says... > >It is the theory and not the language that has the intended > >interpretations, and the interpretations intended ARE BY DEFINITION > >the ones THAT MAKE THE AXIOMS *TRUE*!! > > I would say that often it is the other way around: the language > has an intended interpretation, and we choose axioms that we believe > are true in that interpretation. Or we could, for various reasons, choose axioms we believe false in that interpretation (e.g. variations on the parallel postulate) -- perhaps we are just seized by the imp of the perverse! .... > I don't know why, but I have this feeling that your response to > this post will be an angry one, full of capital letters. Well, how else? It wouldn't be genuine George Greene otherwise. -- hz
From: George Greene on 20 Jul 2010 00:29 On Jul 19, 11:34 pm, herbzet <herb...(a)gmail.com> wrote: > Or we could, for various reasons, choose axioms we believe false in > that interpretation (e.g. variations on the parallel postulate) -- > perhaps we are just seized by the imp of the perverse! Are you really NOT now going to Google Lobachevsky et al?? That should be your sentence (in the punitive sense) for having dared to say this.
From: herbzet on 20 Jul 2010 01:27
George Greene wrote: > herbzet wrote: > > Or we could, for various reasons, choose axioms we believe false in > > that interpretation (e.g. variations on the parallel postulate) -- > > perhaps we are just seized by the imp of the perverse! > > Are you really NOT now going to Google Lobachevsky et al?? > That should be your sentence (in the punitive sense) for having dared > to say this. Not getting your drift -- of course I'm aware of the history of non-Euclidean geometries. -- hz |