Cantor's Diagonal?
in [Math]
|
From: Transfer Principle on 8 Feb 2010 01:55 On Feb 7, 11:20 am, zuhair <zaljo...(a)gmail.com> wrote: > > You are being either disingenious, or willfully obtuse. > same to be said of you also. This is the week that I plan on returning to work on the so-called "crank" ellipsis problem. But as I return to regular posting here at sci.math, I can't help but notice this current zuhair thread. Before my extended absence, I often looked to zuhair as being an ex-"crank," an example of a poster who started out as a "crank" who attacked standard set theory, but is now regarded as someone who merely wants to create his own theories without attacking ZF. But this thread is making many poster reconsider whether zuhair has really reformed after all or returning to his old "crank"-ish habits. Now is as good a time as any to summarize what I believe now to be three chief differences between the "cranks" and the standard theorists. Keep in mind that these are mainly generalizations and may not necessary describe every individual poster. I. Standard theorists think of mathematics in terms of the objects that they want to _exist_ in their own conception of the mathematical universe. For example, the complete ordered field is certainly a desirable object (since a field allows addition, subtraction, multiplication, and nonzero division to be always defined, order allows one to compare objects in terms of less than and greater than, and completeness allows for certain sequences to converge and establish continuity). Thus, standard theorists work in theories which prove the existence of such an object, the complete ordered field. "Cranks," on the other hand, think of mathematics in terms of the objects that they _don't_ want to exist. Thus, a "crank" might declare that they don't want any uncountable sets to exist. Since the complete ordered field is provably uncountable, the "crank" would rather work in a theory in which no complete ordered field can exist than one in which any uncountable set exists. In other words, avoiding all uncountable sets is more important to "cranks" than having a complete ordered field. II. Standard theorists believe that physics favors their side of the debate. Physical processes can be modeled by differential equations, which require the existence of a complete ordered field to solve. Thus, they believe that standard analysis is necessary for the sciences, and so science is impossible in the "crank" theories. "Cranks," on the other hand, believe that physics favors their own side of the debate. It's possible that space and time are quantized, and there may be only finitely many particles in the physical universe. Thus, "cranks" believe that all of physics can be modeled using only finite mathematics, and the standard theorists have theories which prove the existence of sets such as P(P(P(R))), which to them have no physical justification. III. Standard theorists prefer the use of symbolic object language to natural metalanguage. To standard theorists, natural languages such as English lack the mathematical precision of symbolic language, and so all axioms must be stated using symbolic language only. An axiom written in metalanguage isn't truly an axiom to standard theorists. "Cranks," on the other hand, prefer the use of metalanguage to object language. To "cranks," purely symbolic language lacks the direct applicability to real-world phenomena that can be described in metalanguage, and that working with symbolic language is merely playing around with symbols. An axiom written in object language isn't truly an axiom to the "cranks." To me, both sides have a point. For example, in III, it's known that human metalanguages aren't precise enough for computer programming, which is why computer languages had to be invented. Indeed, computers can now verify simple proofs written in symbolic language (Metamath). This is a point in favor of the standard theorists. But when I write axioms, I sometimes prefer the use of metalanguage to object language in certain situations. For to me, it's easier to understand the axiom: R is an equivalence relation than it is to understand: Axyz (xRx & (xRy <-> yRx) & ((xRy & yRz) <-> xRz)). And some axioms, such as: R is a wellorder would be especially cumbersome to write, and even if the axiom were written out completely, it would take several minutes to decipher what all the symbols mean, when one can read "R is a wellorder" and understand what the axiom is saying in mere seconds. These concepts of "equivalence relation," "wellorder," etc., are expected to be well known to most readers of sci.math, and so it's easier for me to write "R is a wellorder" and have it be understood than writing the same in pure object language. But on the other hand, if I am writing about a brand new concept rather than an established concept, then I'd rather write in object language in order to introduce the new concept. I've decided that I would expanded my definition of "crank" and finally admit that for some individuals, the label "crank" is actually justified. I like to defend nonstandard theories, but I'm doing no one, not even the "_cranks_," any favors by defending almost any proposed theory. Indeed, a "crank" may get a "boost" by seeing my defense of his theory, until he sees me defend a theory even crazier than his own, at which point the "crank" is thinking, "Oh, the only person who'll defend my theory is that guy who'll defend almost any garbage theory." Thus, it will be helpful to both standard theorists and "cranks" alike for me to be more selective as to which theories I'll defend. Formerly, I've stated that I'll defend any theory that isn't contradicted by ultrafinitism. Now, I'll tighten up and defend only theories not contradicted by _finitism_. The problem with ultrafinitism is that whenever a ultrafinitist tries to claim an upper bound M on the set of admissible natural numbers, a standard theorist can merely counter that the choice of upper bound M is arbitrary, even if the ultrafinitist attempts to give a justification from physics for the bound M. Therefore, _ultra_finitists such as WM and Yessenin-Volpin will now be considered cranks (and that's _without_ the scare quotes, since the label is deserved). On the other hand, mere finitism only holds that every set is finite, but there is no upper bound on how large a set is allowed to be. As for now, finitists who are labeled by standard theorists as "cranks" will still retain the scare quotes. (On the other hand, if Ed Nelson successfully completes his proof of the inconsistency of PA and ZFC based on the existence of large numbers, then ultrafinitism would at once be _justified_, and if Nelson shows that it's the existence of numbers larger than a certain number M that leads to the inconsistence, then that M wouldn't be an _arbitrary_ upper bound, but rather an upper bound that's necessary to avoid a contradiction, in order to invent a new theory that seeks to avoid the inconsistency he finds in PA/ZFC.) We now return to zuhair, and use the above criteria to determine whether he now deserves the label "crank" or not: I. Right now, zuhair appears to be denying the existence of uncountable sets, and is trying to come up with theories which preclude their existence. This is a point in favor of the standard theorists that the label "crank" is justified. II. Now zuhair rarely refers to physics, so this criterion is irrelevant to the discussion. III. We see that zuhair regularly uses both object language and metalanguage to write his axioms. Thus zuhair writes: > However this bijective correspondence might fail in other set > theories, and I gave an example of such a theory T that do not have > an empty set, and that have Quine atoms, and lets say it has two > primitive constants O and H so now the set of all maps f:Q -> {O,H} > would be {<Q,O>, <Q,H>} which is bigger than the power set of Q > which is Q itself, since we have Q={Q} since Q is a Quine atom. which contains metalanguage, but then he writes a modified Axiom of Separation: > Exist y (P(y)) -> For all A exist x for all y ( y e x <-> ( y e A & > P(y) ) ) which is pure object language. We notice that zuhair here is describing what I once called "zuhair's Original Theory (ZOT)," because it is one of the first theories that he posted here at sci.math. The idea behind ZOT was that Russell's Paradox prohibits naive comprehension: ExAy (yex <-> P(y)) So zuhair replaced this with a similar-looking schema: ExAy (yex <-> (y=x v P(y)) This was supposed to avoid Russell's paradox because if one let P(y) be the formula "~yey," then we obtain: ExAy (yex <-> (y=x v ~yey)) Combining this with another axiom, which, among other things, precludes the existence of the empty set: Ax (xex) we derive the theorem: ExAy (yex <-> y=x) The set whose existence is guaranteed by this theorem is the Quine atom Q mentioned in zuhair's post above. Notice that denying the existence of the _empty_ set may seem even more "crank"-ish than merely denying the existence of uncountable sets. But if zuhair can really come up with an elegant comprehension schema that's as simple as naive comprehension but without the contradiction, and the cost of that schema is that the empty set no longer exists, then I would consider it a small price to pay for the new schema, and so zuhair wouldn't deserve to be called "crank." But unfortunately, the set theorist Randall Holmes (the inventor of Pocket Set Theory) proved that (depending on how exactly it is written) ZOT is either inconsistent or admits only models of cardinality 1 (i.e., they prove that every set is equal to the Quine atom Q, so that Q is the only set). So zuhair has no longer considered ZOT. This is why it seems interesting that zuhair has suddenly returned to a theory which, like ZOT, precludes the existence of the empty set and proves that the simplest set is Q, the Quine atom. Then again, this theory isn't equivalent to ZOT in that his new Separation Schema only prevents the empty set from existing without requiring every set to contain itself as an element. So the final verdict on whether I'll consider calling zuhair a "crank" to be justified? If by dropping the empty set and admitting the Quine atom, zuhair comes up with an interesting theory with some nontrivial sets that are non-Cantorian (i.e., fail to be strictly smaller than their own powersets), which may be finite or possibly infinite, then I'll consider his theory in more detail, and declare that the "crank" label is definitely unjustified. If, on the other hand, the only set that turns out to be non-Cantorian is the Quine atom itself, then I'll declare the theory to be pointless and lean toward the standard theorists in dropping the scare quotes when labelling zuhair.
From: Peter Webb on 8 Feb 2010 04:33 I don't think zuhair is a "crank" in the more general Usenet sense. He does appear to eventually agree he was wrong, which cranks don't - they either change the topic, make an ad-hominem attack, or do both. And of course your views on crankdom are very specific to logic. One thing I disagreed on: II. Standard theorists believe that physics favors their side of the debate. Physical processes can be modeled by differential equations, which require the existence of a complete ordered field to solve. Thus, they believe that standard analysis is necessary for the sciences, and so science is impossible in the "crank" theories. ______________________________________ All physical science can be done using only the countable (but uncomputable) set of computable Reals. Calculus also works fine over this subset of R. There is no hard evidence to me that uncountable sets need exist in nature at all. I grant that it would be somewhat arbitrary for magnetic field strengths to only take only computable values, but for all we know they may do so. You might call this UltraCountabilism in comparison to UltraFinitism ! This tie back to physical processes is also (IMHO) of no value at all to validating set theory. At the risk of sounding somewhat crankish myself, the classical analogy between AxC and Euclid's parallel postulate demonstrates the point. In this Universe, parallel lines do in fact grow closer and quite possibly intersect 20 billion light years away. That doesn't invalidate Euclidean geometry, but the parallel postulate is simply not true in the real world. Its not inconceivable (just very unlikely) that physical processes could exist which demonstrate that AxC is, or isn't, true in this Universe. Quantum mechanics gives us a possible starting place; a photon or particle has to choose in a sense what polarisation/position/momentum etc to show when an observation is made. Picking one value over another purely on the basis of probability is a form of Choice. Indeed, if you accept that you can make precise measurements of anything at all in nature, and you believe in QM, then in this Universe I can well order the Reals. I do so as follows. I place an electron inside an energy well such that its location between 0 and 1 is randomly determined by QM. The location at which I precisely measure the electron is the lowest (first) Real. Then I continue doing so for the second and subsequent measurements, making sure each time I haven't listed that Real already. In this manner, I can select a random Real from any subset of Reals, which in ZFC requires a limitted form of choice. The random outcomes in QM provides a mechanism not directly available inside ZF for choosing elements, and may provide a choice function in this Universe. The point of all this being it is very dangerous to try and use the physical world to validate set theory.
From: Tim Little on 8 Feb 2010 19:41 On 2010-02-04, zuhair <zaljohar(a)gmail.com> wrote: > On the other hand the Diagonal argument seems to require well > ordering The *mathematical* content of the diagonal argument is simply this: for any f:X->A^X, show that there exists g in A^X having the property that for all x, g(x) != f(x)(x). Therefore no such f is surjective. No well-ordering is used, in fact no ordering at all. The argument gets its name from the f(x)(x) terms which lie on the "diagonal" of a table of the function. The illustrative table that gives the argument its name is not required for the proof itself, though, which works for sets that cannot possibly be illustrated. - Tim
From: Tim Little on 8 Feb 2010 19:49 On 2010-02-04, zuhair <zaljohar(a)gmail.com> wrote: > I was speaking of the diagonal argument that uses infinite binary > sequences, and not of the other kinds of arguments (although Arturo > say they the same). An infinite binary sequence *is defined as* a function from N to {0,1}, and the set of all such functions is 2^N. The theorem is that no f:N->2^N is surjective. The diagonal proof is that there exists g in 2^N such that for all n, g(n) != f(n)(n). The sequence h in 2^N defined by h(n) = f(n)(n) is called "the diagonal". Now do you see why they are the same? >> The argument using the map that Russell's Paradox presented, and which >> gives the proof for the above almost at once, remains the same... > > what Russell's paradox has to do with this? It is an instance of a very similar type of construction. However, you are confused enough about the basic diagonal argument that I doubt it would be fruitful to discuss it any further. - Tim
From: Nam Nguyen on 8 Feb 2010 23:05
Transfer Principle wrote: > > I. Standard theorists think of mathematics in terms of the > objects that they want to _exist_ in their own conception > of the mathematical universe. > > "Cranks," on the other hand, think of mathematics in terms > of the objects that they _don't_ want to exist. What about those posters who subscribe to none of those "thinking"? Are they "crank", or "standard"? |