Am I a crank?
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From: Tony Orlow on 7 Sep 2006 13:09 MoeBlee wrote: > Tony Orlow wrote: >> Hi Aatu - >> You say here that induction follows from our mathematical picture of the >> naturals, > > He's talking about a particular axiom of induction in PA. I agree that > it essential to our understanding of the naturals. But you should > understand that induction is even more general. Inductive sets are a > certain kind of set. Roughly, an inductive set is a set that is the > intersection of a class of sets each of which is closed under a given > relation. So the naturals with the relation of successor is just an > example of an inductive set since the set of naturals is the > intersection of the class of sets that are closed under the sucessor > relation (the class is not a set in Z set theories, so the formulation > in Z set theories must find a workaround that). (Also, if you read > anything about this, you'd understand that the intersection approach > has its counterpart as a certain kind of union of sets. See Enderton's > 'A Mathematical Introduction To Logic', in which his explanation of > this excels.) Yes, that's what I'm saying. Thanks again for the reference. > >> but isn't our mathematical picture of the naturals based on >> the inductively defined set given by the Peano axioms? > > I don't speak for Koskensilta, but this may work in many directions - > first order PA may be thought of as a formalization of our mathematical > understanding, so that it is not required to reverse this as you are > doing. But also, first order PA is embedded in set theory so that all > of the axioms of first order PA are theorems but not axioms of set > theory. Okay. > >> It seems to me >> that inductive proof is one with a loop of logical implication, such >> that one fact implies another, which in turn implies another, ad >> infinitum. > > That's why it's not a loop. A loop comes back around to the beginning. > Not so with induction on the naturals. x=0; while(finite(x)) { add_to_list(x); x++; } That's a loop generating the naturals. > >> The existence of a natural implying the existence of a next >> natural is but an example of this kind of logical construction. Does >> that seem like a wrong perspective to you? > > In what context? Generally, I don't see existence of naturals as > arrived upon inductively. In first order PA, the natural numbers are > not mentioned in the theory itself. The natural numbers and the system > of them are a model of the theory, and each natural number is a member > of the universe of that model, but it is not the theory itself that > proves the existence of each natural number. Then, in set theory, any > given natural number can be proven to exist without recourse to > induction. We use induction to prove that certain PROPERTIES hold of > every natural number, but I think looking at existence itself as proven > that way is odd at best. I don't see it as prof of existence as much as definition out of thin air, which is fine for number systems. > >> Secondly, I would like your opinion on inductive proof in the infinite >> case. > > How many times have I already posted to you that there IS transfinite > induction? Many, but do you think that's what I'm talking about? > >> I am aware that this concept is not compatible with transfinite >> cardinalities or limit ordinals, > > No, that's incorrect. I've been telling you that for months now. There > is transfinite induction. Yes, but is that what I'm talking about? If it were allowable to prove that 2x>x for all x>0, and omega or aleph_0 were considered greater than 0, then 2*aleph_0>aleph_0 would be a fact. However, it's not in the standard construction. > >> but independent of that, does the >> following make any sense to you? I see any set size as a count, >> something in the same sequence as the naturals, albeit possibly >> infinitely far beyond them. It's a quantity. When we prove that, for >> instance, x^2>2x for all x>2, it seems to me that any infinite quantity >> is a number greater than 2, and that this inequality would hold for any >> infinite x as well. > > Then get a set of axioms that prove that. Yessir! > >> If we put the infinite numbers on the same line as >> the finites in this way, and extend inductive proof to also hold for the >> infinite case, then not only can we say that x < 2x < x^2 < 2^x < x^x >> for finite x>2, but also for all infinite x. > > We do have transfinite induction in Z set theories. It just does not > prove what you want it to prove here. So get some axioms that prove > what you want them to prove. > > MoeBlee > ToeKnee
From: Tony Orlow on 7 Sep 2006 13:21 Dik T. Winter wrote: > In article <%ZpLg.5265$%t1.3057(a)reader1.news.jippii.net> Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> writes: > > Dik T. Winter wrote: > ... > > > > So you would balk at asserting that it's true that whatever mathematical > > > > property P is we have that > > > > > > > > if P(0) and for every natural n, P(n) implies P(n+1), then for every > > > > natural n, P(n) > > > > > > > > is true, for example? > ... > > > There is no *a priori* reason to either accept or > > > reject it. > > > > Sure there is. It follows immediately from our mathematical picture of > > the naturals. > > That is not an *a priori* reason. You have already a mathematical picture. > And I think the above statement is part of that picture. The statement > follows from Peano's induction axioma, or from PA's induction schema. > > But what I missed was the "natural" in your statement. Of course, using > that word already implies (in a sense) acceptance of the induction > axiom/schema. And (I think) implicitly the existence of the set of all > natural numbers. So the truth of the above statement depends on the > presence of some axiom (schemata), that can not be proven from the > other axiom (schemata). And as such, I would state that it is not true > a priory, but only due to the "model" in which we want to work. > > It is quite similar to the parallel postulate in Euclidean geometry. > For centuries it was thought to be true. But that was only due to the > model of geometry in which one wanted to talk. Rejection of that > postulate lead to quite interesting other geometries. So currently > we can no longer state that it is either true or false. In the same > way, rejection of the induction axiom/scheme, or whatever, also may > lead to interesting arithmetic. Different, obviously, but not to > be rejected out of hand. (I do not think it will lead to interesting > things, but I keep my mind open.) > > Look also at synthetic differential geometry where the axiom of the > excluded middle is rejected. Hi Dik - I really have a hard time imagining anything fruitful coming out of mathematics without some form of inductively defined sets or inductive proof. Your point is well taken in general though, that the theory one creates depends entirely on the statements assumed true (axioms). The parallel postulate is a wonderful example, and LEM is another. Each rule included in the theory tends to restrict it in terms of the conclusions it can reach, which is necessary to some extent to ensure consistency. As long as no two axioms contradict each other, directly or indirectly, the theory is consistent. Tony
From: Bob Kolker on 7 Sep 2006 15:04 John Schutkeker wrote: > > Unfortunately, if it is true that Arxiv only accepts the work of the > academically affiliated, then it is an absolute certainty that no > correct work by amateurs, unemployed or underemployed professionals will > ever be published in their venue. :( You can be unaffiliated and publish there. Bob Kolker
From: MoeBlee on 7 Sep 2006 14:32 Tony Orlow wrote: > x=0; > while(finite(x)) > { add_to_list(x); > x++; > } > > That's a loop generating the naturals. If by 'loop' you mean 'eventually returning to the starting value', then NO Peano structure is a loop. > >> The existence of a natural implying the existence of a next > >> natural is but an example of this kind of logical construction. Does > >> that seem like a wrong perspective to you? > > > > In what context? Generally, I don't see existence of naturals as > > arrived upon inductively. In first order PA, the natural numbers are > > not mentioned in the theory itself. The natural numbers and the system > > of them are a model of the theory, and each natural number is a member > > of the universe of that model, but it is not the theory itself that > > proves the existence of each natural number. Then, in set theory, any > > given natural number can be proven to exist without recourse to > > induction. We use induction to prove that certain PROPERTIES hold of > > every natural number, but I think looking at existence itself as proven > > that way is odd at best. > > I don't see it as prof of existence as much as definition out of thin > air, which is fine for number systems. An existence statement is either a theorem of the theory or it is not a theorem of the theory. So when you say, "implying existence", we take that as meaning that there exists a proof of existence (i.e., a proof that there exists an object having the properties). > >> Secondly, I would like your opinion on inductive proof in the infinite > >> case. > > > > How many times have I already posted to you that there IS transfinite > > induction? > > Many, but do you think that's what I'm talking about? You're online "paper" starts out with a SWEEPING claim that in standard mathematics there is no mathematical induction other than "in the finite case" (or whatever particular wording you used). Your paper thus announces the ignorance of its author right from the start. > >> I am aware that this concept is not compatible with transfinite > >> cardinalities or limit ordinals, > > > > No, that's incorrect. I've been telling you that for months now. There > > is transfinite induction. > > Yes, but is that what I'm talking about? If it were allowable to prove > that 2x>x for all x>0, and omega or aleph_0 were considered greater than > 0, then 2*aleph_0>aleph_0 would be a fact. However, it's not in the > standard construction. Then give a logistic system and non-logical axioms in which you prove it. Otherwise, it's just a lot of hot air you're blowing. Not EVERY property of finite sets is a propery of infinite sets. Just to BEGIN with, the property of being FINITE is not going to be a property of infinite sets. So, WHICH properties of finite sets must also be properties of infinite sets? You claim that certain of these cardinal arithmetic properties must carry from the finite to the infinite. But it is arbitrary which ones they are. And it is NOT given inductively, even by transfinite induction. MoeBlee
From: Virgil on 7 Sep 2006 15:26
In article <mah0g29jhf7u65h4um3k1jebid22us331o(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > On Wed, 06 Sep 2006 17:13:32 -0600, Virgil <virgil(a)comcast.net> wrote: > >Zick is the one whose trivia is founded in the trivium. Math is a part > >of the quadrivium. > > And modern math is founded, whatever that means, in the trivium and > not in the quadrivium. And how does someone so self-decaredly ignorant of mathematics know this? |