CANTOR'S POWER PROOF!
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From: Dan Christensen on 18 Jun 2010 01:57 On Jun 17, 5:26 pm, George Greene <gree...(a)email.unc.edu> wrote: > On Jun 17, 10:42 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> > wrote: > > > On Jun 17, 7:00 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > > CANTOR'S POWER PROOF! > > > Superinfinity is based on the circular reasoning > > > "no box contains the box numbers that don't contain their own box number". > > > I don't know about "boxes" and "box numbers," but it is relatively > > easy to formally prove the power set of any set s is larger than s: > > > Suppose p is the power set of s. Suppose further that f is a > > surjective function mapping s onto p. Select a subset k containing > > those and only those elements of s that are not elements of their > > images under f. (Do you see a problem with that?) > > Of course he does. First of all, you said "those and only those", so > this > subset is UNIQUE -- it is THE subset, NOT *a* subset, of those > elements. True. > Second of all, you are talking about elements of s under THEIR OWN > images, > so he's going to claim THAT'S SELF-referential! It ISN'T, and even if > it WERE > it would NOT be a PROBLEM, BUT THAT'S THE PROBLEM HE CLAIMS TO SEE! > Examples: Suppose f(1)={1, 2, 3} and f(2)={4, 5, 6} Then 1 e f(1) and ~ 2 e f(2) and ~ 1 e k and 2 e k. No self-reference. Not that I see anything inherently wrong with self- reference. In my opinion, banning it just makes for a more complicated system of axioms. There are simpler ways of dealing with so-called paradoxes of self-reference. > Seriously, you have been around here long enough to know better than > this. > This has been going on for close to a decade now and you HAVE been > here > for most of it! > Thanks for noticing! > > k is an element of p, so there must be a pre-image of k under f, say k'. Applying the > > definition of k, you can then obtain the contradiction k' e k and ~k e k. > > > I have generated a formal proof using my DC Proof software (download > > it free atwww.dcproof.com): > > This will be a proof FROM SOME AXIOMS. The only set theoretic axiom used in this proof is the subset axiom. The rest is just FOL. > Herc WILL NOT UNDERSTAND your axioms, let alone think they mean > something real about sets. He will also not understand the point that > THE AXIOM OF INFINITY IS NOT USED in this proof and that therefore, > infinity simply has nothing to do with this. He may be interested to know that, in my set theory, there is no axiom of infinity. I do not even have a axiom for the empty set. I haven't found it to be necessary. If I want to do number theory, I start with an initial premise giving the equivalent of Peano's Axioms. They are entirely separate from my set theory. (I made these innovations primarily for pedagogical reasons. I think they really simplify matters for students first learning how to write formal proofs.) He will keep trying to > say that > the fact that this proof (allegedly) implies the existence of higher > infinities > proves that there must be something OBVIOUSLY wrong with it. I invite him to examine each step of my proof and point out where I might have gone wrong. It might help me improve my system. > THAT is what you are dealing with (or not dealing with, since you keep > trying to explain the proof, which is completely off-topic here, since > the burden > of proof is actually ON HERC to identify SOME ERROR IN the proof, and > he > thinks he can refuse to shoulder it because he can cry "self- > reference!" and "superinfinity!" I always welcome a challenge. ;^) Dan
From: Dan Christensen on 20 Jun 2010 00:53
Did you have chance to look at my proof, Hercules? Dan On Jun 18, 1:57 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> wrote: > On Jun 17, 5:26 pm, George Greene <gree...(a)email.unc.edu> wrote: > > > > > > > On Jun 17, 10:42 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> > > wrote: > > > > On Jun 17, 7:00 am, "|-|ercules" <radgray...(a)yahoo.com> wrote: > > > > > CANTOR'S POWER PROOF! > > > > Superinfinity is based on the circular reasoning > > > > "no box contains the box numbers that don't contain their own box number". > > > > I don't know about "boxes" and "box numbers," but it is relatively > > > easy to formally prove the power set of any set s is larger than s: > > > > Suppose p is the power set of s. Suppose further that f is a > > > surjective function mapping s onto p. Select a subset k containing > > > those and only those elements of s that are not elements of their > > > images under f. (Do you see a problem with that?) > > > Of course he does. First of all, you said "those and only those", so > > this > > subset is UNIQUE -- it is THE subset, NOT *a* subset, of those > > elements. > > True. > > > Second of all, you are talking about elements of s under THEIR OWN > > images, > > so he's going to claim THAT'S SELF-referential! It ISN'T, and even if > > it WERE > > it would NOT be a PROBLEM, BUT THAT'S THE PROBLEM HE CLAIMS TO SEE! > > Examples: > > Suppose f(1)={1, 2, 3} and f(2)={4, 5, 6} > > Then 1 e f(1) and ~ 2 e f(2) and ~ 1 e k and 2 e k. > > No self-reference. Not that I see anything inherently wrong with self- > reference. In my opinion, banning it just makes for a more complicated > system of axioms. There are simpler ways of dealing with so-called > paradoxes of self-reference. > > > Seriously, you have been around here long enough to know better than > > this. > > This has been going on for close to a decade now and you HAVE been > > here > > for most of it! > > Thanks for noticing! > > > > k is an element of p, so there must be a pre-image of k under f, say k'. Applying the > > > definition of k, you can then obtain the contradiction k' e k and ~k e k. > > > > I have generated a formal proof using my DC Proof software (download > > > it free atwww.dcproof.com): > > > This will be a proof FROM SOME AXIOMS. > > The only set theoretic axiom used in this proof is the subset axiom. > The rest is just FOL. > > > Herc WILL NOT UNDERSTAND your axioms, let alone think they mean > > something real about sets. He will also not understand the point that > > THE AXIOM OF INFINITY IS NOT USED in this proof and that therefore, > > infinity simply has nothing to do with this. > > He may be interested to know that, in my set theory, there is no axiom > of infinity. I do not even have a axiom for the empty set. I haven't > found it to be necessary. If I want to do number theory, I start with > an initial premise giving the equivalent of Peano's Axioms. They are > entirely separate from my set theory. (I made these innovations > primarily for pedagogical reasons. I think they really simplify > matters for students first learning how to write formal proofs.) > > He will keep trying to > > > say that > > the fact that this proof (allegedly) implies the existence of higher > > infinities > > proves that there must be something OBVIOUSLY wrong with it. > > I invite him to examine each step of my proof and point out where I > might have gone wrong. It might help me improve my system. > > > THAT is what you are dealing with (or not dealing with, since you keep > > trying to explain the proof, which is completely off-topic here, since > > the burden > > of proof is actually ON HERC to identify SOME ERROR IN the proof, and > > he > > thinks he can refuse to shoulder it because he can cry "self- > > reference!" and "superinfinity!" > > I always welcome a challenge. ;^) > > Dan |