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.

They are not. Arithmetical truth is already not arithmetical.
Arithmetic seen from inside is *vastly* bigger than arithmetic. This needs 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.

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!




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