Talk:Skolem's paradox/Archive 1

Error

The article states:

In mathematics, specifically model theory, Skolem's paradox is a direct result of the Löwenheim-Skolem Theorem, which states that every infinite model has an elementarily equivalent countably infinite submodel.

which is not quite right: the theorem only states this if the model is of a language with no more than countably many constants and functions. I'm not going to correct this all that soon. --- Charles Stewart 21:27, 14 Jun 2005 (UTC)

Paradox? section--poss copyvio, pov problems

The "Is it a paradox?" section copies significant amounts of text from http://uk.geocities.com/frege@btinternet.com/cantor/skolem_moore.htm . Is this asserted to be fair use?

Another problem with this section is the following passage:

But this in turn is not possible unless we endorse the error that there is a set containing all the sets we mean to talk about.

It needs to be clarified whether this is a description of Moore's (or Skolem's) position, or whether the article itself is asserting that this is an error. In the last case it would be POV and would have to be removed (or recast as someone's position). --Trovatore 20:35, 3 October 2005 (UTC)

I agree that this sentence by Moore should be clarified. The page (referenced at the end of the article) on extracts from Moore's article does not clarify it.MikalZiane 15:31, 11 April 2007 (UTC)

I removed the part that seemed to be copied. It needs to be rewritten from scratch to be included; as it stood it was much too close to the original. It would be worth referencing Putnam's paper "Models and reality" here. CMummert · talk 16:13, 11 April 2007 (UTC)

Could P(w) be countable?

It says:

If Skolem's explanation is true, ideas such as countability and uncountability are inherently relative. Our belief that the power set of the natural numbers, P(w), as uncountable, is correct, but must be understood relative to our own current "viewpoint". From another viewpoint this set may in fact be countable.

I'm not sure if this is true. Doesn't the uncountability of P(w) follow from the axioms of ZF? -- Baarslag 21:58, 15 November 2005 (UTC)

Yes, it does. What is probably intended here is that what we think of as P(omega), though uncountable in the model we're considering, may be countable in some larger model. To be sure, the point isn't worded very clearly, and IMHO that's because it's kind of a muddled idea to start with. But even though I don't agree with it, there's no question that some otherwise sensible people subscribe to the idea, so it should probably be represented somehow. --Trovatore 22:00, 15 November 2005 (UTC)

Sorry, still not with you; if the uncountability of P(w) follows from the axioms of ZF, then this would hold in any model of ZF, so how could it be countable in some larger model then? -- Baarslag 21:57, 22 November 2005 (UTC)

The larger model would think the smaller model's version of P(omega) is countable. It would not believe its own version of P(omega) to be countable. --Trovatore 22:03, 22 November 2005 (UTC)

suspicious peter suber edits

I am a mathematician but not an expert in the subfield of math logic. However I find the edits promoting the work of one "Peter Suber" extremely suspicious. He appears to be an expert in legal and philosophical matters, but as far as I can see has no credentials in mathematical logic. I am reverting his edits until he presents proof that his course notes are authoritative.--98.224.223.201 (talk) 19:06, 19 May 2008 (UTC)

i have restored the original entry as your comments are just blatant censoring of views which orthodxy find hard to handle.Suber is a philosopher at a respected university running a philosophy course if you want to dismiss his article its up to you to prove it is not authoritative. - by pointing out what is wrong mathematically with it And its up to to you to prove he has no credentials in mathematical logic before you start criticism even if he has not these credentials its up to you to prove his article is wrong rather than just making unsubstantiated claims to hush his view

Secondly you have deleted even the references to the quotes to the uncited sources in the original entry - unconnected with Suber-so that with your censoring people cant know what the sources are for the quotes. You have deleted even the references to others who say the skolem pardox is a contradiction ie Bunch.

Your deletion is just a blatant attempt to hide from the public a view which threatens the foundation of set theory

In case you come back and statr vandalsing I will put the orginal entry here so that it can at least be seen by others

[quote]The "paradox" is viewed by most logicians as something intriguing, but not a paradox in the sense of being a logical contradiction (i.e., a paradox in the same sense as the Banach–Tarski paradox rather than the sense in Russell's paradox). Timothy Bays has argued in detail that there is nothing in the Löwenheim-Skolem theorem, or even "in the vicinity" of the theorem, that is self-contradictory. Peter Suber on the contrary argues there are a number of contradictions that result from the skolem paradox and that mathematicians claim skolems paradox is not a contradiction but they dont know how to prove it is not a contradiction

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it. [The Löwenheim-Skolem Theorem, http://www.earlham.edu/~peters/courses/logsys/low-skol.htm#amb3]

In agreement with Suber B.Bunch in "Mathematical fallacies and paradoxes” Dover 1982" notes p.167

"no one has any idea of how to re-construct axiomatic set theory so that this paradox does not occur”

However, some philosophers, notably Hilary Putnam and the Oxford philosopher A.W. Moore, have argued that it is in some sense a paradox.

Peter Suber argues that the skolem paradox is a paradox in the ancient sense [ibid]

Insofar as this is a paradox it is called Skolem's paradox. It is at least a paradox in the ancient sense: an astonishing and implausible result. Is it a paradox in the modern sense, making contradiction apparently unavoidable?

Now Suber shows that a reading of LST gives us a serious contradiction

ne reading of LST holds that it proves that the cardinality of the real numbers is the same as the cardinality of the rationals, namely, countable. (The two kinds of number could still differ in other ways, just as the naturals and rationals do despite their equal cardinality.) On this reading, the Skolem paradox would create a serious contradiction, for we have Cantor's proof, whose premises and reasoning are at least as strong as those for LST, that the set of reals has a greater cardinality than the set of rationals. [ibid]

The difficulty lies in the notion of "relativism" that underlies the theorem. Skolem says:

In the axiomatization, "set" does not mean an arbitrarily defined collection; the sets are nothing but objects that are connected with one another through certain relations expressed by the axioms. Hence there is no contradiction at all if a set M of the domain B is nondenumerable in the sense of the axiomatization; for this means merely that within B there occurs no one-to-one mapping of M onto Z0 (Zermelo's number sequence). Nevertheless there exists the possibility of numbering all objects in B, and therefore also the elements of M, by means of the positive integers; of course, such an enumeration too is a collection of certain pairs, but this collection is not a "set" (that is, it does not occur in the domain B).

But Skolem admitted that his relativism destroyed the notion that set theory was a foundation of mathematics

"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

Even John von Neumann noted that Skolems relativism was one more reason to upset set theory and destroy it

"At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.

Abraham Fraenkel noted that Skolems relativism did not satisfactoraly disprove the antinomy and that there was no agreement as to his relativist solution

"Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached." – ([[Abraham Fraenkel] in "Einleitung in die Mengenlehre" 3rd ed p. 333, 1928, quoted in "The Bulletin of symbolic logic"" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps

Moore (1985) has argued that if such relativism is to be intelligible at all, it has to be understood within a framework that casts it as a straightforward error. This, he argues, is Skolem's Paradox.

Peter Suber points out the problem with Skolems relativism

This means that there simply are no sets whose cardinality is absolutely uncountable. For many, this view guts set theory, arithmetic, and analysis. It is also clearly incompatible with mathematical Platonism which holds that the real numbers exist, and are really uncountable, independently of what can be proved about them. [ibid]

Suber goes on to pount out contradictions due to Skolems paradox in non-relativistic accounts

If we want to insist on the non-relativity of our set theoretic notions, and if we hold that our formal systems to date fail fully to capture the secret of the real numbers, then we must choose between the unattractive options (1) that the theory of real numbers is inconsistent, hence has no model, and (2) that the secret of the real numbers cannot be captured by any first-order formal system, i.e. that every attempt will fail either by having no model or by "incurring" a merely countable model. LST puts us to the choice between inconsistency and non-categoricity. If we discard the first of these, then we are left with a view that implies that our notions of uncountable infinities, including the continuum, cannot be fully formalized. As John Myhill put it, in LST we have proved an insurmountable limitation of formalization itself. [ibid]

There are a number of contradictions that result from the skolem paradox as pointed out by suber

If all the models of a system are isomorphic with one another, we call the system categorical. LST proves that systems with uncountable models also have countable models; this means that the domains of the two models have different cardinalities, which is enough to prevent isomorphism. Hence, consistent first-order systems, including systems of arithmetic, are non-categorical. We might have thought that, even if a vast system of uninterpreted marks on paper were susceptible of two or more coherent interpretations, or even two or more models, at least they would all be "equivalent" or "isomorphic" to each other, in effect using different terms for the same things. But non-categoricity upsets this expectation. Consistent systems will always have non-isomorphic or qualitatively different models. LST proves in a very particular way that no first-order formal system of any size can specify the reals uniquely. It proves that no description of the real numbers (in a first-order theory) is categorical. Very Very Serious Incurable Ambiguity: Upward and Downward LST If the intended model of a first-order theory has a cardinality of 1, then we have to put up with its "shadow" model with a cardinality of 0. But it could be worse. These are only two cardinalities. The range of the ambiguity from this point of view is narrow. Let us say that degree of non-categoricity is 2, since there are only 2 different cardinalities involved. But it is worse. A variation of LST called the "downward" LST proves that if a first-order theory has a model of any transfinite cardinality, x, then it also has a model of every transfinite cardinal y, when y > x. Since there are infinitely many infinite cardinalities, this means there are first-order theories with arbitrarily many LST shadow models. The degree of non-categoricity can be any countable number. There is one more blow. A variation of LST called the "upward" LST proves that if a first-order theory has a model of any infinite cardinality, then it has models of any arbitrary infinite cardinality, hence every infinite cardinality. The degree of non-categoricity can be any infinite number. A variation of upward LST has been proved for first-order theories with identity: if such a theory has a "normal" model of any infinite cardinality, then it has normal models of any, hence every, infinite cardinality. [ibid]

Suber notes that mathematician claim skolems paradox is not a contradiction but they dont know how to prove it is not a contradiction

Most mathematicians agree that the Skolem paradox creates no contradiction. But that does not mean they agree on how to resolve it. [ibid]

[edit] Quotations

Zermelo at first declared the Skolem paradox a hoax. In 1937 he wrote a small note entitled "Relativism in Set Theory and the So-Called Theorem of Skolem" in which he gives (what he considered to be) a refutation of "Skolem's paradox", i.e. the fact that Zermelo-Fraenkel set theory—guaranteeing the existence of uncountably many sets—has a countable model. His response relied, however, on his understanding of the foundations of set theory as essentially second-order (in particular, on interpreting his axiom of separation as guaranteeing not merely the existence of first-order definable subsets, but also arbitrary unions of such). Skolem's result applies only to the first-order interpretation of Zermelo-Fraenkel set theory, but Zermelo considered this first-order interpretation to be flawed and fraught with "finitary prejudice". Other authorities on set theory were more sympathetic to the first-order interpretation, but still found Skolem's result astounding:

At present we can do no more than note that we have one more reason here to entertain reservations about set theory and that for the time being no way of rehabilitating this theory is known." – ([[John von Neumann]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 148 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps. ])[citation needed]

"Skolem's work implies 'no categorical axiomatisation of set theory (hence geometry, arithmetic [and any other theory with a set-theoretic model]...) seems to exist at all'." – (John von Neumann)[citation needed]

"Neither have the books yet been closed on the antinomy, nor has agreement on its significance and possible solution yet been reached." – ([[Abraham Fraenkel] in "Einleitung in die Mengenlehre" 3rd ed p. 333, 1928, quoted in "The Bulletin of symbolic logic"" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps ])[citation needed]

"I believed that it was so clear that axiomatization in terms of sets was not a satisfactory ultimate foundation of mathematics that mathematicians would, for the most part, not be very much concerned with it. But in recent times I have seen to my surprise that so many mathematicians think that these axioms of set theory provide the ideal foundation for mathematics; therefore it seemed to me that the time had come for a critique." – ([[Skolem]"The Bulletin of symbolic logic" Vol.6, no 2. June 2000, pp. 147 http://www.math.ucla.edu/~asl/bsl/0602/0602-001.ps.])[citation needed] [/quote] —Preceding unsigned comment added by 211.27.78.23 (talk) 01:32, 14 June 2008 (UTC)

The topic is worthy of a serious edit

I tried to read this article, but (not knowing enough theory in advance of reading the article) I could make no sense of it at all. Unfortunately, the clumsy "style" is quite off-putting and hinders understanding -- the grammar and spelling are abysmal, and the attributions/references in very poor shape (not conforming to any wiki-standards that I know of). But I'd like to know more -- antinomies make for interesting topics; the article is worthy of, and due for, some serious editing. Bill Wvbailey (talk) 18:33, 15 June 2008 (UTC)

I started working on it this afternoon. At some point I plan to add more content on Putnam's work and its reception - the key contemporary significance of the paradox is philosophical, as mathematically there is not much to it.

I looked for publications of Peter Suber about the topic (a previous version quoted at length from [1]) but I think he has not actually published anything about the theorem. Coincidentally, Suber is a member of the Wikimedia Foundation board [2]. — Carl (CBM · talk) 19:54, 15 July 2008 (UTC)

This is a vast improvement over what was there before. A question: doesn't the Lowenheim-Skolem theorem imply the use the axiom of choice? Suppose there is something "wrong" with the axiom of choice (in particular related its non-constructive nature, in turn implying a non-constructive use of the LoEM). Then perhaps the Skolem "anomaly" can be traced back to the LoEM being extended over the infinite? Bill Wvbailey (talk) 17:18, 16 July 2008 (UTC)

The Lowenheim-Skolem theorem doesn't require the axiom of choice per se, at least not to obtain Skolem's result that if ZFC is consistent then it has a countable model. The fact that any syntactically consistent sequence of first-order formulas in a countable language has a finite or countable model is provable in a very weak subsystem of second-order arithmetic.

As it is understood by mathematicians today, the source of the "anomaly" is simply that countability is not absolute.

There is another wrinkle that is often overlooked in non-technical presentations. There are two issues with the claim that "there is a model of set theory in which P(ω) is countable.". The first issue is that "countable" is not an absolute property. The second is that "X = P(ω)" is not an absolute property of a set X. In a countable model M, the set that M believes is P(ω) is not the set we recognize as the "real" P(ω), it's the set of elements of the "real" P(ω) that happen to appear in the model M. The fact that M is countable means that most of the elements of the "real" P(ω) have been omitted. Skolem's result does not show that there is another model in which the "real" P(ω) is countable. — Carl (CBM · talk) 20:53, 16 July 2008 (UTC)

Countable first-order axiomatisation

Forgive the nitpick, but in reference to this: what other kind is there? I mean infinitary formulas are generally not included in the definition first-order logic right? Or do we have subcountability in mind? --Unzerlegbarkeit (talk) 17:12, 18 July 2008 (UTC)

The issue may be that you are reading "axiomatisation" in an effective sense. If it is read simply as a set of formulas, there is the possibility of an uncountable language, in which case a countable model can't be expected. — Carl (CBM · talk) 13:50, 19 July 2008 (UTC)

Sorry what do you mean, uncountable language? I mean I get that we can have non-effective axiomoatisations like the subset of all true arithmetic sentences, but it would still be countable in the classical sense. Now the alleged bijection between the integers and that subset would not exist constructively, in that sense the axioms are not countable, but I am assuming this is not what we are talking about here right? --Unzerlegbarkeit (talk) 18:48, 21 July 2008 (UTC)

An uncountable language means you have uncountably many non-logical symbols. For example you might have a collection of constant symbols indexed by the real numbers, one for each real. --Trovatore (talk) 18:57, 21 July 2008 (UTC)

These arise most often in the context of model theory; see Theory_(mathematical_logic)#Theories_associated_with_a_structure. — Carl (CBM · talk) 19:27, 21 July 2008 (UTC)

Hah, I see. Thanks! --Unzerlegbarkeit (talk) 21:05, 21 July 2008 (UTC)