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From: Ross A. Finlayson on 18 Apr 2005 16:27 Will Twentyman wrote: > He is free to reject Cantor's works. But that is very different from > discrediting it. Rejecting Cantor's work is nothing new, nor is there > anything wrong with it. > Hi, What do you mean by that? I wonder. Basically, it has been shown that the cardinality of the continuum is equivalent to aleph_1, aleph_2, aleph_3, .... You might quickly deny that. I don't! Infinite sets are equivalent. Also, not CH. Will, the set of all sets is its own powerset. There are theories in application today, for example the Quinean "New Foundations with Urelements", where the powerset result does not apply to all sets, there are a variety of other anti-foundational set theories, including all the consistent ones. They, the founders and promoters of those related methods, determined that infinite sets might be equivalent to their powersets because it's untrue to say that they aren't. Cantor was smart, he put forth a variety of progressive statements, but you should realize that there are plenty of mathematicians, with plenty of examination of mathematical infinity, who do not accept what you might see as the gospel of infinity according to Cantor, which it is not. Cantor lived more than a hundred years ago, they barely had toothpaste back then. Do you have any inclination to have a consistent and complete theory? Then, you might want a theory with no paradoxes, and the structural form forcing Goedel into acknowledging that not everything you say is a lie, otherwise, he does. I have one and am quite happy about it. Well, if you want to quantify over sets in your set theory, which is a necessary action for any statement about sets, there is an implicit universal set. Do you happen to know any well-orderings of the real numbers? Consider the infinitesimals. Skolemize: ______. Infinite sets are _____. If you and other members of our little community here who for all visible intents and purposes are Cantor devotees are coming around to realize that there's more to it, then maybe it's time you accept that, and stop following the hypocrite school. I think you might feel better about it when you do. Also, V = L. There's always one more. Ross
From: Will Twentyman on 18 Apr 2005 17:06 Ross A. Finlayson wrote: > Will Twentyman wrote: > > >>He is free to reject Cantor's works. But that is very different from >>discrediting it. Rejecting Cantor's work is nothing new, nor is > there >>anything wrong with it. >> > > Hi, > > What do you mean by that? I mean that anyone is free to use a version of set theory based on Cantor's work, or divergent from Cantor's work. I view the brand of set theory a person works with as a matter of personal taste. However, I do not view rejecting a particular version of set theory for aesthetic reasons as equivalent to showing that it is inferior. > I wonder. Basically, it has been shown that the cardinality of the > continuum is equivalent to aleph_1, aleph_2, aleph_3, .... > > You might quickly deny that. > > I don't! Infinite sets are equivalent. Also, not CH. It depends on what you mean by them being equivalent. > Will, the set of all sets is its own powerset. Not in ZF, where the class of all sets is a proper class and therefor does not have a powerset operation. In other versions, of course, that may be true. > There are theories in > application today, for example the Quinean "New Foundations with > Urelements", where the powerset result does not apply to all sets, > there are a variety of other anti-foundational set theories, including > all the consistent ones. They, the founders and promoters of those > related methods, determined that infinite sets might be equivalent to > their powersets because it's untrue to say that they aren't. Ok. > Cantor was smart, he put forth a variety of progressive statements, but > you should realize that there are plenty of mathematicians, with plenty > of examination of mathematical infinity, who do not accept what you > might see as the gospel of infinity according to Cantor, which it is > not. Cantor lived more than a hundred years ago, they barely had > toothpaste back then. Let me ask you this: does that mean that Cantor's work (with any necessary corrections to make it consistent) is now wrong? Those who make other set theories probably don't claim to have overthrown Cantor, just that they have found a set theory that behaves differently from the one Cantor used. I have no problem with that. > If you and other members of our little community here who for all > visible intents and purposes are Cantor devotees are coming around to > realize that there's more to it, then maybe it's time you accept that, > and stop following the hypocrite school. I think you might feel better > about it when you do. As I've said elsewhere: there is a difference between showing a version of set theory to be inconsistent, and claiming it's "wrong" because you disagree with an axiom or definition. Unfortunately, Eckard seems to fall under the second heading, and I hope he will see that he can simply choose a more palatable set theory without having to overthrow whichever version(s) he doesn't like. My flavor of choice is ZF(C) because it's the one I'm most familiar with. Having said that, I'm open to looking at others, and would not comment on their value without looking into them more. If I come across as a Cantor-Nazi, that is not my intention. -- Will Twentyman email: wtwentyman at copper dot net
From: Chris Menzel on 18 Apr 2005 17:44 On 18 Apr 2005 13:27:43 -0700, Ross A. Finlayson <raf(a)tiki-lounge.com> said: > Basically, it has been shown that the cardinality of the continuum is > equivalent to aleph_1, aleph_2, aleph_3, .... Basically, it hasn't.
From: Ross A. Finlayson on 18 Apr 2005 18:12 step...(a)nomail.com wrote: > > Some people do not understand the concept of 'axiom'. They > seriously believe that things can be proven true, with no > starting assumptions whatsoever. I am not sure why so > many of the usenet philosophers seem to belong to this > group, because it would seem any decent philosopher would > understand the necessity of axioms. > > Stephen Hi, That's funny. I don't know anybody else who's been saying that, except for perhaps Owen, and in a way, perhaps, Mitch, logicians, and some few others addressing the possibility. Personally, that's what I say, I have presented arguments along those lines directly to you for some years. Would you please expand on what you mean by that? Axioms are obviously useful in the specific. If the light is red, stop. Traffic might be proceeding perpendicular to your route. That "axiom" is just a stub for everything else that is true in all of reality. In terms of mathematical logic, logic is to logicism as philosophy is to philosophism. While that's so, logic is to logicism as philosophy is to antiphilosophism. Ross
From: Chris Menzel on 18 Apr 2005 18:39
On 18 Apr 2005 20:54:34 GMT, stephen(a)nomail.com <stephen(a)nomail.com> said: > Some people do not understand the concept of 'axiom'. They > seriously believe that things can be proven true, with no > starting assumptions whatsoever. I am not sure why so > many of the usenet philosophers seem to belong to this > group, because it would seem any decent philosopher would > understand the necessity of axioms. Just so. Any decent philosopher, indeed anyone with half a brain, *does* understand the necessity of axioms in reasoning. |