Re: The seven step-Mathematical preliminaries
2009/6/17 Torgny Tholerus tor...@dsv.su.se: Bruno Marchal skrev: Torgny, I agree with Quentin. You are just showing that the naive notion of set is inconsistent. Cantor already knew that, and this is exactly what forced people to develop axiomatic theories. So depending on which theory of set you will use, you can or cannot have an universal set (a set of all sets). In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the collection of all sets is not a set. It is not the naive notion of set that is inconsistent. It is the naive *handling* of sets that is inconsistent. This problem has two possible solutions. One possible solution is to deny that it is possible to create the set of all sets. This solution is chosen by ZF and VBG. The second possible solution is to be very careful of the domain of the All quantificator. You are not allowed to substitute an object that is not included in the domain of the quantificator. It is this second solution that I have chosen. What is illegal in the two deductions below, is the substitutions. Because the sets A and B do not belong to the domain of the All quantificator. You can define existence by saying that only that which is incuded in the domain of the All quantificator exists. In that case it is correct to say that the sets A and B do not exist, because they are not included in the domain. But I think this is a too restrictive definition of existence. It is fully possible to talk about the set of all sets. But you must then be *very* careful with what you do with that set. That set is a set, but it does not belong to the set of all sets, it does not belong to itself. It is also a matter of definition; if you define set as the same as belonging to the set of all sets, then the set of all sets is not a set. This is a matter of taste. You can choose whatever you like, but you must be aware of your choice. But if you restrict yourself too much, then your life will be poorer... In NF, some have developed structure with universal sets, and thus universe containing themselves. Abram is interested in such universal sets. And, you can interpret the UD, or the Mandelbrot set as (simple) model for such type of structure. Your argument did not show at all that the set of natural numbers leads to any trouble. Indeed, finitism can be seen as a move toward that set, viewed as an everything, potentially infinite frame (for math, or beyond math, like it happens with comp). The problem of naming (or given a mathematical status) to all sets is akin to the problem of giving a name to God. As Cantor was completely aware of. We are confused on this since we exist. But the natural numbers, have never leads to any confusion, despite we cannot define them. The proof that there is no biggest natural number is illegal, because you are there doing an illegal deduction, you are there doing an illegal substitution, just the same as in the deductions below with the sets A and B. You are there substituting an object that is not part of the domain of the All quatificator. No the proof is based on PA and in PA you do not have an axiom restricting the successor function and as such it is defined in the axiom that you don't have an upper bound limit. The proof is *valid* against the axioms. *You* are doing an illegal deduction by not taking into accound the rules with wich you work. Regards, Quentin -- Torgny Tholerus You argument against the infinity of natural numbers is not valid. You cannot throw out this little infinite by pointing on the problem that some terribly big infinite, like the set of all sets, leads to trouble. That would be like saying that we have to abandon all drugs because the heroin is very dangerous. It is just non valid. Normally, later I will show a series of argument very close to Russell paradoxes, and which will yield, in the comp frame, interesting constraints on what computations are and are not. Bruno On 13 Jun 2009, at 13:26, Torgny Tholerus wrote: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you
Re: The seven step-Mathematical preliminaries
Bruno Marchal skrev: Torgny, I agree with Quentin. You are just showing that the naive notion of set is inconsistent. Cantor already knew that, and this is exactly what forced people to develop axiomatic theories. So depending on which theory of set you will use, you can or cannot have an universal set (a set of all sets). In typical theories, like ZF and VBG (von Neuman Bernay Gödel) the collection of all sets is not a set. It is not the naive notion of set that is inconsistent. It is the naive *handling* of sets that is inconsistent. This problem has two possible solutions. One possible solution is to deny that it is possible to create the set of all sets. This solution is chosen by ZF and VBG. The second possible solution is to be very careful of the domain of the All quantificator. You are not allowed to substitute an object that is not included in the domain of the quantificator. It is this second solution that I have chosen. What is illegal in the two deductions below, is the substitutions. Because the sets A and B do not belong to the domain of the All quantificator. You can define existence by saying that only that which is incuded in the domain of the All quantificator exists. In that case it is correct to say that the sets A and B do not exist, because they are not included in the domain. But I think this is a too restrictive definition of existence. It is fully possible to talk about the set of all sets. But you must then be *very* careful with what you do with that set. That set is a set, but it does not belong to the set of all sets, it does not belong to itself. It is also a matter of definition; if you define set as the same as belonging to the set of all sets, then the set of all sets is not a set. This is a matter of taste. You can choose whatever you like, but you must be aware of your choice. But if you restrict yourself too much, then your life will be poorer... In NF, some have developed structure with universal sets, and thus universe containing themselves. Abram is interested in such universal sets. And, you can interpret the UD, or the Mandelbrot set as (simple) model for such type of structure. Your argument did not show at all that the set of natural numbers leads to any trouble. Indeed, finitism can be seen as a move toward that set, viewed as an everything, potentially infinite frame (for math, or beyond math, like it happens with comp). The problem of naming (or given a mathematical status) to all sets is akin to the problem of giving a name to God. As Cantor was completely aware of. We are confused on this since we exist. But the natural numbers, have never leads to any confusion, despite we cannot define them. The proof that there is no biggest natural number is illegal, because you are there doing an illegal deduction, you are there doing an illegal substitution, just the same as in the deductions below with the sets A and B. You are there substituting an object that is not part of the domain of the All quatificator. -- Torgny Tholerus You argument against the infinity of natural numbers is not valid. You cannot throw out this little infinite by pointing on the problem that some terribly big infinite, like the set of all sets, leads to trouble. That would be like saying that we have to abandon all drugs because the heroin is very dangerous. It is just non valid. Normally, later I will show a series of argument very close to Russell paradoxes, and which will yield, in the comp frame, interesting constraints on what computations are and are not. Bruno On 13 Jun 2009, at 13:26, Torgny Tholerus wrote: Quentin Anciaux skrev: 2009/6/13 Torgny Tholerus tor...@dsv.su.se: What do you think about the following deduction? Is it legal or illegal? --- Define the set A of all sets as: For all x holds that x belongs to A if and only if x is a set. This is an general rule saying that for some particular symbol- string x you can always tell if x belongs to A or not. Most humans who think about mathematics can understand this rule-based definition. This rule holds for all and every object, without exceptions. So this rule also holds for A itself. We can always substitute A for x. Then we will get: A belongs to A if and only if A is a set. And we know that A is a set. So from this we can deduce: A beongs to A. --- Quentin, what do you think? Is this deduction legal or illegal? It depends if you allow a set to be part of itselft or not. If you accept, that a set can be part of itself, it makes your deduction legal regarding the rules. OK, if we accept that a set can be part of itself, what do you think about the following deduction? Is it legal or illegal? --- Define the set B of all sets that do not belong to itself as:
Abstract: Cognitive Reflectivity
I tripped over my own boot-laces and stumbled – whacking my head. Bent over to rub my head and retie the damn laces, all a sudden I had the following quick rough thoughts for a paper: 'Cognitive Reflectivity' Marc Geddes Melbourne, Australia 18th June, 2009 Abstract A change in the goal-system of an agent is equivalent to a change in the way in which knowledge is represented by the agent. It follows that it is equivalent to a change in the complexity of the program representing the agent. Thus we require a method of comparing the complexity of strings in order to ensure that relevant program structure is preserved with state transitions over time. Standard probability theory cannot be used because; (1) Consistent probability calculations require implicit universal generalizations, but a universal measure of the complexity of finite strings is a logical impossibility (fromGodel, Lob theorems); and (2) Standard measures of complexity (e.g Kolmogorov complexity) from information theory deal only with one aspect of information (i.e. Shannon information), and fail to consider semantic content. The solution must resolve both these problems. Regarding (2) the solution is as follows:, information theory is generalized to deal with the actual meaning of information (i.e . the semantics of Shannon information) .The generalized definition of the complexity of a finite string is based on the conceptual clustering of semantic categories specifying the knowledge a string represents. The generation of hierarchical category structures representing the knowledge in a string is also associated with a generalization of Occam’s razor. The justification for Occam’s razor and the problem of priors in induction is resolved by defining ‘utility’ in terms of ‘aesthetic goodness’, which is the degree of integration of different concept hierarchies. This considers the process through which a theory is generated; it is a form of process-oriented evaluation. Regarding (1); The Godel limitation is bypassed by using relative complexity measures of pairs of strings . This requires generalizing standard Bayesian induction ; in fact induction is merely a special case of a new form of case-based reasoning (analogical reasoning) . Analogical reasoning can be formalized by utilizing concepts from category theory to implement prototype theory, where mathematical categories are regarded as semantic categories. Semantic concepts representing the knowledge encoded in strings can be considered to reside in multi-dimensional feature space, and this enables mappings between concepts; such mappings are defined by functors representing conceptual distance; this gives a formal definition of an analogy. The reason this overcomes the Godel limitation and is more general than induction is because it always enables relative comparisons of the complexity of pairs of strings. This is because case-based reasoning depends only on the specific details on the strings being compared, whereas induction makes implicit universal generalizations, and thus fails. To summarize: Induction is shown to be merely a special case of a new type of generalized case-based (analogical) reasoning. Concepts from category theory enable a formal definition of an analogy, which is based on the notion of conceptual distance between concepts. The notion of complexity is generalized to deal with semantics, where the information in a string is considered to be a concept hierarchy. This enables comparisons between pairs of strings; relations between strings are defined in terms of the mappings between concepts, and the mapping is evaluated in terms of its aesthetic goodness. Godelian limitations are overcome, since analogical reasoning always enables a comparison of the relative complexity between any two finite strings. Further the new metric of aesthetic goodness ensures that the relevant program structure is preserved between state transitions and thus maintains a stable goal system. Cheers --~--~-~--~~~---~--~~ 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 -~--~~~~--~~--~--~---
