On 9/27/2012 4:37 AM, Bruno Marchal wrote:

On 26 Sep 2012, at 19:37, Craig Weinberg wrote:in which case, how are they really arithmetic.## Advertising

They are not. Arithmetical truth is already not arithmetical.Arithmetic seen from inside is *vastly* bigger than arithmetic. Thisneeds a bit of "model theory" to be explained formally.

Hi Bruno,

`Is this not just the direct implication of the Löwenheim--Skolem`

`theorems <http://mathworld.wolfram.com/Loewenheim-SkolemTheorem.html>?`

`What is missing? The discussion here is wonderful!`

`http://www.earlham.edu/~peters/courses/logsys/low-skol.htm#review`

`<http://www.earlham.edu/%7Epeters/courses/logsys/low-skol.htm#review> It`

`seems to run parallel to what I have been trying to discuss with you`

`regarding the possibility that non-standard models allow for a "true"`

`plurality of 1p in extensions of modal logic.`

`http://www.earlham.edu/~peters/courses/logsys/low-skol.htm#skolem`

`<http://www.earlham.edu/%7Epeters/courses/logsys/low-skol.htm#skolem>`

*Skolem's Paradox*

`" LST has bite because we believe that there are/un/countably many real`

`numbers (more than_0 ). Indeed, let's insist that we/know/it; Cantor`

`proved it in 1873, and we don't want to open the question again. What is`

`remarkable about LST is the assertion that even if the intended`

`interpretation of S is a system of arithmetic about the real numbers,`

`and even if the system is consistent and has a model that makes its`

`theorems true, its theorems (under a different interpretation) will be`

`true for a domain too small to contain all the real numbers. Systems`

`about uncountable infinities can be given a model whose domain is only`

`countable. Systems about the reals can be interpreted as if they were`

`about some set of objects no more numerous than the natural numbers. It`

`is as if a syntactical version of "One-Thousand and One Arabian Nights"`

`could be interpreted as "One Night in Centerville".`

`This strange situation is not hypothetical. There are systems of set`

`theory (or number theory or predicate logic) that contain a theorem`

`which asserts in the intended interpretation that the cardinality of the`

`real numbers exceeds the cardinality of the naturals. That's good,`

`because it's true. Such systems therefore say that the cardinality of`

`the reals is uncountable. So the cardinality of the reals must`

`really/be/uncountable in all the/models/of the system, for a model is an`

`interpretation in which the theorems come out/true/(for that`

`interpretation). Now one would think that if theorems about uncountable`

`cardinalities are true for a model, then the domain of the model must`

`have uncountably many members. But LST says this is not so. Even these`

`systems, if they have models at all, have at least one countable model.`

`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? We know from history all too many cases of`

`shocking results initially misperceived as contradictions. Think about`

`the existence of pairs of numbers with no common divisor, no matter how`

`small, or the property of every infinite set that it can be put into`

`one-to-one correspondence with some of its proper subsets."`

`What broke the Skolem prison for me was a remark by Louis Kauffman`

`that self-referencing systems allows for finite models of infinities as`

`they can capture the mereology of infinite sets ( the property that`

`there is a one-to-one correspondence between whole and proper subsets).`

`We get stuck on what is "proper"....`

`From: http://www.earlham.edu/~peters/courses/logsys/low-skol.htm#amb1`

`<http://www.earlham.edu/%7Epeters/courses/logsys/low-skol.htm#amb1>`

`"To talk about incurable ambiguity suggests calamity, and this is how it`

`seems to some logicians. LST shows that the real numbers cannot be`

`specified uniquely by any first order theory. If that is so, then one of`

`the most important domains in mathematics cannot be reached with the`

`precision and finality we thought formal systems permitted us to attain.`

`But this is not a calamity from another point of view. If formal`

`languages were not ambiguous or capable of many interpretations, they`

`would not be formal. From this standpoint, LST is less about deficiences`

`in our ability to express meanings univocally, or about deficiencies in`

`our ability to understand the real numbers, than it is about the gap`

`between form and content (syntax and semantics).`

`Other metatheorems prove this ambiguity for statements and systems`

`in general. LST proves this ambiguity for any system attempting to`

`describe uncountable infinities: even if they succeed on one`

`interpretation, there will always be other interpretations of the same`

`underlying syntax by which they describe only countably many objects.`

`LST has this similarity to Gödel's first incompleteness theorem.`

`While Gödel's theorem only applies to "sufficiently strong" systems of`

`arithmetic, LST only applies to first-order theories of a certain`

`adequacy, namely, those with models, hence those that are consistent.`

`Gödel's theorem finds a surprising weakness in strength; sufficiently`

`powerful systems of arithmetic are incomplete. LST also finds a`

`surprising weakness in strength; first-order theories with models are`

`importantly ambiguous in a way that especially hurts set theory,`

`arithmetic, and other theories concerned to capture truths about`

`uncountable cardinalities.*`

Serious Incurable Ambiguity: Plural Models*

`But LST proves a kind of ambiguity much more important than the`

`permanent plurality of interpretations. There can be plural/models/,`

`that is, plural interpetations in which the theorems come out/true/.`

`As we become familiar with formalism and its susceptibility to various`

`interpretations, we might think that plural interpretations are not that`

`surprising; perhaps they are inevitable. Plural/models/are more`

`surprising. We might think that we must go out of our way to get them.`

`But on the contrary, LST says they are inevitable for systems of a`

`certain kind.`

*Very Serious Incurable Ambiguity: Non-Categoricity*

`But the ambiguity is stronger still. The plural permissible models are`

`not always isomorphic with one another. The isomorphism of models is a`

`technical concept that we don't have to explain fully here. Essentially`

`two models are isomorphic if their domains map one another; their`

`elements have the same relations under the functions and predicates`

`defined in the interpretations containing those domains.`

`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_."`

`This non-isomorphism is the point I have been trying to make, we`

`cannot extract a true plurality of minds from a single Sigma_1 COMP`

`model unless we allow for an inconsistency in the ontologically`

`primitive level. Continuing the quote:`

`" We don't even approach univocal reference "at the limit" or`

`asymptotically, by increasing the number of axioms or theorems`

`describing the real numbers until they are infinite in number. We might`

`have thought that, even if a certain vast system of bits sustained`

`non-isomorphic models, we could approach unambiguity (even if we could`

`not reach it) by increasing the size of the system. After all, "10"`

`could symbolize everything from day and night to male and female, and`

`from two to ten; but a string of 1's and 0's a light-year in length must`

`at least narrow down the range of possible referents. But this is not`

`so, for LST applies even to infinitely large systems. 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."`

`This reasoning is parallel to my own argument that there must be a means`

`to "book keep" the differences and that this cannot be done "in the`

`arithmetic" itself. This is the fundamental argument that I am making`

`for he necessity of physical worlds, which we can represent faithfully`

`as topological spaces and we get this if we accept the Stone duality as`

`a ontological principle. But that is an argument against your thesis of`

`immaterial monism. :_(`

Continuing the quote: "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."`

`Why not allow for arbitrary extensions via forcing? Why not the`

`unnameable towers of cardinalities of Cantor, so long as it is possible`

`to have pair-wise consistent constructions from the infinities?`

`"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."`

`Implying the existence of sets of countable numbers within each`

`model, subject to some constraint?`

`"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."`

Thus an argument for the Tower!

`"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.`

Coping

`Most mathematicians agree that the Skolem paradox creates no`

`contradiction. But that does not mean they agree on how to resolve it.`

`First we should note that the ambiguity proved by LST is curable in the`

`sense that LST holds only in first-order theories. Higher-order logics`

`are not afflicted with it, although they are afflicted with many`

`weaknesses absent in first-order logic. The ambiguity is also curable as`

`soon as we add ordering to our collection of domain objects supposed to`

`be real numbers. Once ordering is added, systems intended to capture the`

`reals can become categorical.`

`But the ambiguity remains baffling and frustrating for first-order`

`theories prior to the introduction of ordering.`

`Can such a system really assert the uncountability of the reals if the`

`assertion is "just as much" about some merely countable infinite? Or can`

`it really assert that the cardinality of the continuum is 1 (assuming`

`the continuum hypothesis) if the assertion is "just as much" about every`

`other infinite cardinality? LST may not force us to retract our belief`

`that the reals are uncountable; but on one terrifying reading it does,`

`and to avoid that reading we may well have to alter the modality of our`

`belief that the reals are uncountable.`

`What of models that do not assume CH? We get a plenum of`

`continua... (At least between Aleph_0 and Aleph_1) No? Do we necessarily`

`lose countability so long as ordering can be imposed by some rule?`

`"In metalogic the term "model" is used in (at least) two senses. We have`

`used the term in the more technical sense, as an interpretation of a`

`system in which its theorems come out true for that interpretation. But`

`the term "model" may also be used more casually to refer to the domain`

`of things on which we want to focus, such as the real numbers,`

`especially if we assume that such things have an existence independent`

`of formal systems and human logicians. In this less strict second sense`

`of "model", Platonists in mathematics who believe that the real numbers`

`exist independently of human minds and formal systems can say that there`

`is an uncountable model of the real numbers: namely, the real numbers`

`themselves. However, they must (by LST) admit that first-order formal`

`systems that seem to capture the real numbers can always be satisfied by`

`a merely countable domain. For this reasons, Platonists will remain`

`Platonic and will not pin their hopes on formalization."`

Bad news for Bruno! :_(

`"One 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.`

`The good news is that this strongly paradoxical reading is optional. The`

`bad news is that the obvious alternatives are very ugly. The most common`

`way to avoid the strongly paradoxical reading is to insist that the real`

`numbers have some elusive, essential property not captured by system S.`

`This view is usually associated with a Platonism that permits its`

`proponents to say that the real numbers have certain properties`

`independently of what we are able to say or prove about them.`

`The problem with this view is that LST proves that if some new and`

`improved S' had a model, then it too would have a countable model.`

`Hence, no matter what improvements we introduce, either S' has no model`

`or it does not escape the air of paradox created by LST. (S' would at`

`least have its own typographical expression as a model, which is`

`countable.) As Morris Kline put it, while Gödel's first incompleteness`

`theorem showed that certain strong formal systems always prove less than`

`we'd like, LST shows that they also prove more than we'd like."`

Please note the discussion of Platonism! -- Onward! Stephen http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html -- You received this message because you are subscribed to the Google Groups "Everything List" group. To post to this group, send email to everything-list@googlegroups.com. To unsubscribe from this group, send email to everything-list+unsubscr...@googlegroups.com. For more options, visit this group at http://groups.google.com/group/everything-list?hl=en.

`<<inline: aleph.gif>>`