PROOF INFINITY DOES NOT EXIST! Not even ONE type
in [Math]
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From: FredJeffries on 9 Jul 2010 22:14 On Jul 9, 10:17 am, Dan Christensen <Dan_Christen...(a)sympatico.ca> wrote: > > I can't imagine that you would be able to do very much using > "finitist" methods. How do they handle such basic concepts as the > square root of 2? > See for instance Jan Mycielski "Analysis Without Actual Infinity" in The Journal of Symbolic Logic, Vol. 46, No. 3. (Sep., 1981), pp. 625-633.
From: George Greene on 10 Jul 2010 00:00 On Jul 7, 6:37 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote: > Take the Heine-Borel theorem. It is not in any apparent sense "about > words and their definitions" nor can it be with any accuracy described > as "study of formal language and what can be said using formal > language". Nor is it "math" in any general sense. It is A SPECIFIC piece of math, and to the extent that it is in formal language, it is a confirming example and not a counter-example for the thesis. Besides, the kinds of things that the Heine-Borel theorem CAN be said to be "about" ARE THEMselves *formal*! > The study of formal languages and their expressiveness is a > very marginal part of mathematics. You are using "expressiveness" in a very technical sense which YOU MUST surely have known HAS LITTLE TO NOTHING to do with the sense in which he intended it.
From: Nam Nguyen on 10 Jul 2010 00:43 K_h wrote: > "Curt Welch" <curt(a)kcwc.com> wrote in message > news:20100708093928.442$LY(a)newsreader.com... >> "|-|ercules" <radgray123(a)yahoo.com> wrote: >>> "Curt Welch" <curt(a)kcwc.com> wrote... >>> I realize the difficulty in confirming a rock exists. But all you have >>> to do is confirm *something* exists. Even if you're in error the >>> conclusion is still true. E x >>> >>> Herc >> That's an interesting point. I don't see any argument against the idea >> that something exists is an absolute truth. I think therefore I am. That >> might be the one and only absolute truth. > > Mathematical truth exists. Sure. In your mind for example! > To my recollection, I have never seen anybody claim > that 2x7=14 is false or fails to be true after somebody dies. How did you mean by that? Did you mean the dead couldn't claim 2x7=14 is false? (If so, there are many things they couldn't possibly claim, naturally!) > The equation > 10+20=30 is an absolute truth and that truth does exist. Again, in your mind perhaps. Others working in modulo arithmetic may state 10+20=0 is absolutely true, just as you stated "10+20=30 is an absolute truth". What's the difference anyway? > It is obvious that > there are an infinite number of such truths so infinity, as a platonic truth, > must exist. "Must exist" in your mind obviously, for example. > >> But sadly, our ability to express the idea with language still runs into >> the problem of potential failure to correctly communicate with some small >> probability of error, and likewise, our ability to even think the idea >> comes with the same error. So even if the idea is itself, when expressed >> correctly, an absolute truth, it's not an absolute truth that our brain can >> ever correctly express or understand the idea. > > Humans do have limitations but those limitations are not limitations on > existence. What does "limitations on existence" mean? > > So you have existential doubts about the truth of 4+5=9? People have no doubt that 4+5=9 is false in some modulo arithmetic. -- ---------------------------------------------------- There is no remainder in the mathematics of infinity. NYOGEN SENZAKI ----------------------------------------------------
From: The Raven on 10 Jul 2010 01:22 STOP THE IRRELEVANT CROSSPOSTING!
From: David Libert on 10 Jul 2010 06:36
Transfer Principle (lwalke3(a)lausd.net) writes: > On Jul 6, 12:23=A0pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> INFERENCE there is no oo > > Herc has joined the growing number of sci.math finitists. > > We already know that Han de Bruijn and R. Srinivasan are > finitists as well. A few posters, including Archimedes > Plutonium and WM, defy classification. AP believes that > all numbers larger than 10^500 are infinite -- so > technically speaking, he still believes in infinity, but > his infinity is much less than standard infinity. (Note > that both Herc and AP use Chapernowne's constant in their > arguments regarding infinity.) WM, on the other hand, > believes that some standard natural numbers don't really > exist, although he has no upper bound on the largest > natural number. It had been said that for that reason, > one can't call WM an _ultra_finitist, although it's > possible that WM is merely a finitist. > > In the theory ZF-Infinity, one can't prove that an > infinite set exists (assuming consistency), and if we > add an axiom such as ~Infinity, we can actually prove > that no infinite set exists. (There's no need to point > out yet again that ~Infinity asserts that there is no > _inductive_ set as opposed to no _infinite_ set. In > several other threads, a proof in ZF-Infinity+~Infinity > that all sets are finite has been presented, and the > proof uses Replacement Schema. So let's not travel down > that road for the umpteenth time.) > > Srinivasan has also proposed another axiom, D=3D0, which > implies that all sets are finite, as well as a logic, > NAFL, which proves the same. > > I recommend that Herc read some of the other sci.math > finitists to see what they have to say. (Of course, he > already knows about WM.) You are writing above about the 2 axioms ~Infinity and D=0. These are closely related, and came up in discussions in [1] "How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)" sci.logic 551 articles also sci.math and others http://groups.google.com/group/sci.logic/browse_thread/thread/4c784919ee892256/ I am in basic agreement with what was written about these axioms in [1], but I have always wanted to note some technical points about it. I didn't get around to it originally in [1], and since you mentioned it here i will do so now. R. Srinivasan introduced that D=0 in [2] R. Srinivasan "How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)" sci.logic,sci.math and others June 27, 2010 http://groups.google.com/group/sci.logic/msg/045183a90426396f with: >There is a simple way to sidestep this controversy. Suitably extend >the language of ZF-Inf to admit the set D, where > >D = {x: An (x not in P_n(0))} > >Here 0 is the null set, n ranges over the non-negative integers and >P_n(0) is the power set operation iterated n times on 0 with P_0(0) = >P(0). > >Note that by definition, D does not include any hereditarily finite >set, but it will contain every other set. > >Consider the theory F = ZF-Inf+{D=0} It is clear that F will only >admit models with hereditarily finite sets. Use the theory F instead >of ZF-Inf+~Inf in my post. There were various points of discussion related to this in [1]. One question was defining the P_n(0) hierarchy. Some other discussion gave a name to the compliment of this D, which in other posts I recall had also been locally named D (different posts using the same name for compliementaey pairs) and the point was raised that there may be in these ZF-Inf models no such set as the second D. You pointed out this last point could be handled instead as a definable proper class, like L in ZF. I think all that is ok. Handle the other D that way, and P_n(0) definitions are not a major problem. And in fact this D=0 axiom is itself closely related to ~Infinity axionm anyway. For me the D=0 or ~Infinity formulations are not a major point, things can be translated back and forth. And the other issues raised above are not big problems. And the claims made above about these, everything hereditarily finite, or nom infinite sets, are ultimately ok. But there is an additional technical point on the way to all that that needs to be cleared up. Once it is the above becomes ok. The definition above of D : >D = {x: An (x not in P_n(0))} > >Here 0 is the null set, n ranges over the non-negative integers and >P_n(0) is the power set operation iterated n times on 0 with P_0(0) = >P(0). depended on having n range over all natural numbers. So it needs to have the collection of all natural numbers defined as at least a class. ~Infinity is just negating thge usual statement of the axiom of infinity, so is no problem to state. But discussion of D=0 and ~Infinity continued to discuss whether there are any infinite sets, or to discuss hereditarily finite. So all the discussion rests on the definitons of natural number, finite and heritarily finite. Nothing was specified about these definitions, so the default would be presumably the usual defintions. I claim the usual definitions must be reworked in the context of ZF-Inf. With that the above becomes ok, but a complete account should mention this point. Namely the usual definition I am used to of finite is having cardinality a member of omega. (And no: we don't define omega as the set of all finite ordinals :) ). If we take the usual Russell definite description expansion of definitions, this definition amounts to saying x is finite <-> there exists y such that y satisifies the defintion of omega and x member y. So on existential exansion of omega in ZF - Infinity + ~Infinity, nothing satisifies the definition of omega so the definition always vacuously fails and everything is actually infinite by this definition. Or idf you take instead univeral reading of the definite description, everything is vacuously finite in ZF - Infinity + ~Infinity, seems closer to what we wanted but this definition being vacious doesn't readily prove the familar properties of finite. So instead we must provide an alternative definition of finiteness. Similarly to define natural number has similar issues. I will discuss the pitfalls of doing this in weak fragments of ZF, and solutions. Above I spoke about the difficulties created by dropping Infinity. I will also discuss issues with AC and regularity R dropped. So ultimately below I will speak of solutions that work well even in ZF - Infinity - R. I will also discuss some points about dropping replacement, back to Z. One possibility would be Dedekind finiteness. See my next reference for details. But the proof that Dedekind finiteness supports proofs by ordinary mathematical induction uses AC (there are counter models of ZF if ZF is consistent). I seek definitions of finiteness that make it work reasoinably in weak theories. The definitions I will give below over the theories in question will then support the discussion above about D=0 and ~Infinity, namely allow the defintion of the P_n(0), and show the claims about no infinite sets, ie with these defintitions of infinite (not finite as finite to be defined). One defintion of finiteness that gets induction and function defintions by ordinary recursion was discussed in [3] David Libert "How Can ZFC/PA do much of Math - it Can't Even Prove PA is Consistent (EASY PROOF)" sci.logic,sci.math and others June 27, 2010 http://groups.google.com/group/sci.logic/msg/7011b0c9510d6764 [3] discusses some points about the definitions and references an older article that gives it. The definition will give induction and recusrion in Z - Infinity - R . So that theory would at least get the discussion started, to work from those definitions. However to get from ~Infinity to every set is finite, you need replacement. This is the result Transfer Principle mentioned above, which had been discussed several times in [1] as Transfer Pirncile noted. Other people had mentioned this point, that replacement is needed. My older article referenced in [3], also gave this proof as well as contructing a counterexample model of Z + ~Replacement, showing replacement really is needed as claimed. That is handling ~Infinity from the [3] definition of finiteness. What about D=0 ? That definition depended on the definition of natural number. So we can define natural number as a finite von Neumann ordinal, using that definition of finiteness. But in this weak theory Z - Infinity - R there are some complicatiions about defining von Neumann ordinals. They can be overcome, but some definitions need to be restated. The usual ZF definition of von Neumann ordinal is transitive and hereditarily transitive. The usual defition for property P that x is heriditarily P is every member of TC(x) has property P, TC being transitive closure. I posted in ZFC - Infinity transitive closures might not exist as a set: [4] David Libert "Axiom of infinity and the set of all hereditary finite sets" sci.logic Oct 3, 2007 http://groups.google.com/group/sci.logic/msg/7593d4adf17732b7 So if just expand the usual definition above as Russell is problems as for omega above. In [5] David Libert "Recursive cardinals" sci.logic, sci.math Jan 3, 2010 http://groups.google.com/group/sci.logic/msg/02248254025cb4c8 I posted how to defince TC as a class in ZF - Infinity. The same definition would work in Z - Infinity - R . I remarked in [5] we could also define TC as having a finite sequence of members below the top, except how to define finite. So an alternative would be to define it this way, using [3] 's definition of finiteness. With either of these, we can redefine heritarily transitive, and so redefine von Neumann ordinal. In the special case of von Neumann ordinals, since they are also defined to be transitive they do have a transitive closure, namely themselves. So you could also define them as being transtive and every member is transitive. Or you could just return to the usual definition of transitive closure, because the definition expands properly by Russell in this special case. I still think this is worth pointing out, when the general definition can fail. Anyway, in all these ways we can restate a formulation of von Neumann ordinal. But we are not done yet. The restatement so far would work in Z - Infinity. But when we drop regularity this defintion allows for ordinals to be not linearly ordered. So instead for Z - Inifinity - R, define von Neumann ordinal to be the above reworkings of transtive and hewreditarily transitive and also well-ordered by epsilon. With this definition we get a reasonable recasting of von Neumann ordinals in Z - Infinity - R. With this, we want to define a natural number to be a finite von Neumann ordinal. Use the [3] definition of finiteness. With these reworkings, we do get proofs in Z - Infinity - R that D=0 gives every set is finite and is hereditarily finite, using those definitions of finite and hereditarily finite. Zuhair, in the same thread as [5], posted other defintions of finiteness for von Neumann ordinals: [6] Zuhair "Recursive cardinals" sci.logic, sci.math Jan 8, 2010 http://groups.google.com/group/sci.logic/msg/65ef21f30e4578de Defining von Neumann ordinal as I just said, we can also use these definitions, to replace [3] 's definition of finiteness. Another definition of finite von Neumann ordinal that works reasonably in Z - Infinity - R is von Neumann ordinal as defined above which is not a limit ordinal and has no members which are limit ordinals. With any of these Z - Infinity - R and D=0 give the claims every set is finite and is hereditarily finite. We can use these alternative definitions of natural number for the D definition. To define finite use [3], or use had cardinality a natrual number the current definition of natural number. Abd hereditarily finite in the conclusion defined as above in the discussion of von Neumann ordinal. So with these roeworked defintions, I think the claims above are ok. -- David Libert ah170(a)FreeNet.Carleton.CA |