details re Z-R |- PA consistent
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From: Transfer Principle on 14 Jul 2010 17:42 On Jul 14, 8:51 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 12, 6:04 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > In this case, we can write Herc's Axiom of No Infinity as: > > {} _is_ a (Frege) natural number. > > This forces the universe to be finite > Why? Suppose (for some bizarre reason) there are no sets of > cardinality 43. Why, despite this, could there not be sets of all > cardinalities greater than 43? The answer depends on the context. In the context of Herc, when I asked him a similar question, he gave the following response: Herc: "I could redefine pseudo infinity to be any finite length = any computable length with a program that terminates, if you can compute n then you can (theoretically) compute n-1, so pseudo infinity might be the initial sequence of N up to the largest physically computable number." Thus, according to Herc, if a set of cardinality n (i.e., a sequence of length n) exists, then one of size n-1 exists. And so if 44 exists then so does 43. What about the context of NFU itself? My knowledge of NFU is limited to what Randall Holmes writes about it: http://math.boisestate.edu/~holmes/holmes/nf.html And Holmes states the Axiom of Infinity as: {} is not a natural number without discussing its negation. Presumably, for natural numbers m < n, one can show that if card(n) = x, then there exists a stratified formula phi such that: card({yex | phi(y)} = m but I'm not quite sure how to accomplish this. Since Holmes includes the Axiom of Choice, perhaps we can wellorder V, then use it to form the set of the first m elements of x according to this wellorder. Other than that, I can't see how Holmes would preclude having an empty Frege natural number with a nonempty successor. > Or could it not be that there only exist sets of even cardinality and > none of odd cardinality? Or that there are no sets with cardinality a > prime number? Notice that unlike Herc and Holmes, there is a sci.math poster whose ideas are as described here -- namely WM. According to WM, the number googolplex exists, but not all the classical naturals less than googolplex exist. Thus, there really are WM-sets of large cardinality such that no WM-set of certain smaller cardinalities exist. But this number wouldn't be 43, but some (most) numbers between googol and googolplex.
From: Transfer Principle on 14 Jul 2010 18:01 On Jul 14, 9:01 am, FredJeffries <fredjeffr...(a)gmail.com> wrote: > On Jul 12, 4:57 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > So this represents a bias towards infinite sets. > Elsewhere you state there is nothing infinite in the real world. So > doesn't everyone have a natural bias towards finitism? Therefore by > your principle shouldn't anyone who is able to overcome this naive > finitism and intelligently discuss infinite sets be supported and > commended rather than those who have hitherto been unable to overcome > their finitistic bias? Touche. So Jeffries cleverly finds a way to make my argument sound as if the _finitists_ are closed-minded, while the _infinitists_ are the ones who really are open-minded. Thus, how can I tell who really are the open-minded posters? On the surface, one might argue that infinitists accept finite sets (since, after all, their infinite sets have finite subsets), but finitists don't accept infinite sets. But then by this logic, we should call anyone who doesn't accept the largest of the large cardinals (including those whose existence contradicts AC) closed-minded. Instead, I want to consider those who shut out their opponents ideas to be closed-minded. Presumably, there exist finitists who at least are open-minded to infinite sets, even if the finitists themselves can't formalize infinite ideas. So a finitist who works with, say, the finite naturals of PA can be open-minded to ZFC, even though PA (assuming consistent) can't prove that ZFC is consistent (while ZFC -- even the subset Z-R according to the OP of this thread -- can prove that PA is consistent). And even if we did declare all finitists to be closed-minded, what about those posters like Tony Orlow who accept infinite sets that don't work the same way that they work in ZFC? I can make the argument that there is more symmetry in TO vs. ZFC as opposed to finitism vs. ZFC. Thus, for TO to be open-minded would be for him to be open to ZFC with its standard cardinality, while for those opposing TO to be open-minded would be for them to be open to idea that proper subsets can have different "sizes" (Bigulosities)? Therefore, there is still a way to go before I am convinced that the ZFC infinitist majority is really open-minded.
From: Transfer Principle on 14 Jul 2010 18:39 On Jul 14, 12:44 am, Tim Little <t...(a)little-possums.net> wrote: > On 2010-07-14, Marshall <marshall.spi...(a)gmail.com> wrote: > > None of the cases you've ever fussed about could even > > remotely be described as "absent any supporting information." > Not entirely true. They could be considered "absent any supporting > information" to someone who is too dense to comprehend any of the > supporting information. Of course in all those cases a sufficient > amount of information has been comprehensible to a 10-year-old, but in > the case of [TP] that may be too much to expect. But then this can go on forever in a vicious circle -- I can then go on to claim that it should be obvious to any 10-year-old that _I_ am right, that the majority of sci.math posters are closed-minded. And so on and so on. So instead of this, what if I were to assume for the sake of argument that Little is right, that MoeBlee is right, that MoeBlee's opponents are dogmatic, and that MoeBlee isn't himself dogmatic. So now what? My stated reason for posting is to discuss theories other than ZFC. If, as it turns out, the only posters discussing alternatives to ZFC are dogmatists, and that the non-dogmatists are not posting alternatives to ZFC, then I'm still going to stick with those who are dogmatic. I'd much rather have a discussion that I enjoy with a dogmatist than one that I don't with a non-dogmatist. And so I'm going to continue to have discussions about alternatives to ZFC, no matter how dogmatic the posters discussing it become.
From: Transfer Principle on 14 Jul 2010 18:49 On Jul 12, 7:29 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote: > Transfer Principle <lwal...(a)lausd.net> writes: > > But I'm not sure whether I want to resort to NFU to make Herc's > > theory work. There might also be a trick using NBG and Srinivasan > > to declare some object similar to D -- such as the class of all sets > > of cardinality greater than n -- to equal the empty set 0. > Herc doesn't have a theory. Herc is, as we all know, a radically > deluded individual. > Of all the persons that you "defend", Herc is clearly the worst choice. > I'm not sure why you think that he is a mathematician at all. He is > honestly incapable of rational argument. I don't say this to be mean or > to defend my own biases regarding the existence of infinite sets (about > which I really have no coherent philosophical views) but rather because > we can all see that Herc is a disturbed individual who believes that he > is Adam, that an unfortunate lady is his Eve and that satellites are > tormenting him with sonar. These assertions are not wholly irrelevant > to understanding his "mathematical" claims. And here we go again. I'd like to be able to discuss Herc's mathematics independently of his unorthodox religious beliefs about Adam and Eve, but according to Jesse Hughes, his mathematical and non-mathematical claims are related. This also comes up in the AP threads, where I'd like to discuss AP's mathematics without becoming an Atom Totalitarian. So what should I do? I want to discuss alternatives to ZFC without being associated with the Adam-Herc or Atom Totality theories, but posters continue to bring these non-mathematical ideas over and over. This is something that I have to think about for a while. There are still enough posters who oppose ZFC without posting unorthodox non-mathematical ideas that I can still have the type of discussion that I want to have. So I will decide soon whether to consider Herc indefensible, even wrt mathematics, due to his non-mathematical posts.
From: MoeBlee on 14 Jul 2010 18:51
On Jul 14, 5:39 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > and that MoeBlee isn't himself dogmatic. My dogmas: (1) In unmarked intersections, pedestrians have the right of way. (2) The Axis powers were the bad guys in WWII. (3) In public laundromats, one should clean the lint trap after using a dryer. Other than that, I don't know on what points I'm supposed to have succumbed to dogma. MoeBlee |