premjan Posted November 18, 2004 Share Posted November 18, 2004 you have some rather nice graphics of different logical systems there, Doron. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 19, 2004 Share Posted November 19, 2004 Thank you dear premjan. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 19, 2004 Share Posted November 19, 2004 Thank you dear premjan. Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 The Axiom of the unreachable weak limit:No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. The logical basis of both of them can be found in pages 4' date='5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf Also please read http://www.geocities.com/complementarytheory/SegPoint.pdf Thank you.[/quote']Hi Doron. For the last century, the foundations of mathematics has used axiomatic systems where the principle concepts are uninterpreted. For instance, in ZF set theory, you ultimately don't have to worry about what a "set" is. In fact, you can substitute "set" and "belongs to" with "blob" and "bamboozles", make your inferences, substitute back and still have theorems of ZF set theory. Allow me to demonstrate (since I'm using rhetoric rather than symbolism, you'll have to put aside your issue with treating "any" and "all" as synonyms): Axiom of Extensionality: Suppose we have two blobs (or the same blob with two different names), blobA and blobB. If, given any blob, blobC, blobC bamboozles both blobA and blobB, or blobC bamboozles neither, then blobA and blobB are the same blob. Axiom of SmartBlob: There exists a blob, SmartBlob, who is bamboozled by no other blob. Theorem 1: SmartBlob is the only blob bamboozled by no other blob. Proof Suppose there is another blob, CleverBlob, who is bamboozled by no other blob. Then, given any blob, blobC, blobC bamboozles neither SmartBlob nor CleverBlob, and thus, by the Axiom of Extensionality, SmartBlob and CleverBlob are the same blob. QED. Now, we subsitute back, and we have a theorem of ZF: The Empty Set is unique. So the question is, can you substitute the words "Emptiness", "Fullness", "input", "{}", "{___}" and "stands for", for "blob", "slob", "flob", "snob", "bob" and "bamboozles", or some other nonsense, make inferences, substitute back and end up with the same theorems* you had before? If you can't, then you're not using a formal approach, and if you're not using a formal approach, I, at least, want a good reason why, since I consider formality and rigour to be a great achievement of modern mathematics and the very things which attract me to the subject. EDIT: *You do have some theorems, right? Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 The Axiom of the unreachable weak limit:No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. The logical basis of both of them can be found in pages 4' date='5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf Also please read http://www.geocities.com/complementarytheory/SegPoint.pdf Thank you.[/quote']Hi Doron. For the last century, the foundations of mathematics has used axiomatic systems where the principle concepts are uninterpreted. For instance, in ZF set theory, you ultimately don't have to worry about what a "set" is. In fact, you can substitute "set" and "belongs to" with "blob" and "bamboozles", make your inferences, substitute back and still have theorems of ZF set theory. Allow me to demonstrate (since I'm using rhetoric rather than symbolism, you'll have to put aside your issue with treating "any" and "all" as synonyms): Axiom of Extensionality: Suppose we have two blobs (or the same blob with two different names), blobA and blobB. If, given any blob, blobC, blobC bamboozles both blobA and blobB, or blobC bamboozles neither, then blobA and blobB are the same blob. Axiom of SmartBlob: There exists a blob, SmartBlob, who is bamboozled by no other blob. Theorem 1: SmartBlob is the only blob bamboozled by no other blob. Proof Suppose there is another blob, CleverBlob, who is bamboozled by no other blob. Then, given any blob, blobC, blobC bamboozles neither SmartBlob nor CleverBlob, and thus, by the Axiom of Extensionality, SmartBlob and CleverBlob are the same blob. QED. Now, we subsitute back, and we have a theorem of ZF: The Empty Set is unique. So the question is, can you substitute the words "Emptiness", "Fullness", "input", "{}", "{___}" and "stands for", for "blob", "slob", "flob", "snob", "bob" and "bamboozles", or some other nonsense, make inferences, substitute back and end up with the same theorems* you had before? If you can't, then you're not using a formal approach, and if you're not using a formal approach, I, at least, want a good reason why, since I consider formality and rigour to be a great achievement of modern mathematics and the very things which attract me to the subject. EDIT: *You do have some theorems, right? Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Dear VazScep, Please think simple. The set concept is nothing but a technical environment, where we can explore the associations between the concepts, which we are using in our formal framework. But before we are doing this, we have to determine the lowest and the highest concepts that clearly define the limitations of our framework, and in the case of a framework that is based of information, no information can be found and explored if we cannot use it as an input that can be manipulated by the rules and tools of our framework. I have found, by using a symmetrical approach, that Emptiness is the lowest concept that cannot be used as an input by any formal or informal language, where Fullness is the highest concept that cannot be used as an input by any formal or informal language. This elementary understanding comes before any use of logical reasoning, formal proposition or axiomatic system that determines our framework, exactly as the set concept itself is not determinate by the system, and we ultimately don't have to worry about what a "Fullness" or "Emptiness" are. All we need is two axioms, that put these concepts within our formal framework, and this is exactly what I did by these to axioms, which are: The Axiom of the unreachable weak limit: No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. The logical reasoning, which is related to these two concepts can be found in pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf Because the formal approach, sometimes replaces simple understanding by, so called, rigorus logical/germial rules, it totally failed to help Cantor to understand the Infinity concept. The Cantorian aleph0 cannot be a Natural number if aleph0+n = aleph0, and this is exactly some result of a Cantorian transfinite cardinals arithmetic. So the Cantorian transfinite system is based on self contradiction, if aleph0 is a Natural number, therefore aleph0 must be beyond any Natural number, and in this case, it does not belong anymore to any model, which is based on infinitely many elements. The only Cantorian alternative is that aleph0 is Fullness (an infinitely long pointless solid element) and then it cannot be manipulated by the language of Mathematics. Fullness can be clearly shown by this model: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Dear VazScep, Please think simple. The set concept is nothing but a technical environment, where we can explore the associations between the concepts, which we are using in our formal framework. But before we are doing this, we have to determine the lowest and the highest concepts that clearly define the limitations of our framework, and in the case of a framework that is based of information, no information can be found and explored if we cannot use it as an input that can be manipulated by the rules and tools of our framework. I have found, by using a symmetrical approach, that Emptiness is the lowest concept that cannot be used as an input by any formal or informal language, where Fullness is the highest concept that cannot be used as an input by any formal or informal language. This elementary understanding comes before any use of logical reasoning, formal proposition or axiomatic system that determines our framework, exactly as the set concept itself is not determinate by the system, and we ultimately don't have to worry about what a "Fullness" or "Emptiness" are. All we need is two axioms, that put these concepts within our formal framework, and this is exactly what I did by these to axioms, which are: The Axiom of the unreachable weak limit: No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. The logical reasoning, which is related to these two concepts can be found in pages 4,5,6 of http://www.geocities.com/complementarytheory/CompLogic.pdf Because the formal approach, sometimes replaces simple understanding by, so called, rigorus logical/germial rules, it totally failed to help Cantor to understand the Infinity concept. The Cantorian aleph0 cannot be a Natural number if aleph0+n = aleph0, and this is exactly some result of a Cantorian transfinite cardinals arithmetic. So the Cantorian transfinite system is based on self contradiction, if aleph0 is a Natural number, therefore aleph0 must be beyond any Natural number, and in this case, it does not belong anymore to any model, which is based on infinitely many elements. The only Cantorian alternative is that aleph0 is Fullness (an infinitely long pointless solid element) and then it cannot be manipulated by the language of Mathematics. Fullness can be clearly shown by this model: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 The Axiom of the unreachable weak limit:No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. Please prove one theorem from these two axioms, or provide additional axioms which allow you to prove a theorem. I cannot deduce anything of interest from them as they stand. Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 The Axiom of the unreachable weak limit:No input can be found by {} which stands for Emptiness. The Axiom of the unreachable strong limit: No input can be found by {__} which stands for Fullness. Please prove one theorem from these two axioms, or provide additional axioms which allow you to prove a theorem. I cannot deduce anything of interest from them as they stand. Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Dear VazScep, My axiomatic system, which is based on Included-middle reasoning (http://www.geocities.com/complementarytheory/CompLogic.pdf), can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Dear VazScep, My axiomatic system, which is based on Included-middle reasoning (http://www.geocities.com/complementarytheory/CompLogic.pdf), can be found in: http://www.geocities.com/complementarytheory/My-first-axioms.pdf Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 Can you state a theorem and provide its proof, appealing to nothing but your axioms, to such a sufficient level of rigour that I can substitute the principle terms with random words, follow your inferences, substitute back and still end up with your conclusions? Link to comment Share on other sites More sharing options...
VazScep Posted November 20, 2004 Share Posted November 20, 2004 Can you state a theorem and provide its proof, appealing to nothing but your axioms, to such a sufficient level of rigour that I can substitute the principle terms with random words, follow your inferences, substitute back and still end up with your conclusions? Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Some examples of my work: ------------------------------------------------------------------------------ A proof that cannot be accomplished by using standard N members: Theorem: 1*5 not= 1+1+1+1+1 Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1 To understand this proof, please read at least page 13 of http://www.geocities.com/complementarytheory/ONN2.pdf ------------------------------------------------------------------------------ A test that shows the advantage of - and + operations in an included-middle logical reasoning framework, can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf ------------------------------------------------------------------------------- Complementary relations between Multiplication and Addition binary operations can be found in pages 7-8 of http://www.geocities.com/complementarytheory/ONN1.pdf ------------------------------------------------------------------------------- A fundamental new approach about the Natural numbers can be found in: http://www.geocities.com/complementarytheory/ONN1.pdf http://www.geocities.com/complementarytheory/ONN2.pdf http://www.geocities.com/complementarytheory/ONN3.pdf ------------------------------------------------------------------------------- A new approach about 0.9999... = 1 can be found here: http://www.geocities.com/complementarytheory/9999.pdf ------------------------------------------------------------------------------- A new approach about the Limit concept can be found here: http://www.geocities.com/complementarytheory/Anyx.pdf ------------------------------------------------------------------------------- A new approach about Russell's first paradox, can be found here: http://www.geocities.com/complementarytheory/Russell1.pdf ------------------------------------------------------------------------------- A new approach about Cantor's diagonal methods can be found here: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf http://www.geocities.com/complementarytheory/TRANSFINITES.pdf ------------------------------------------------------------------------------- A new approach about Collatz' problem can be found here: http://www.geocities.com/complementarytheory/3n1proof.pdf ------------------------------------------------------------------------------- A new approach about the Real numbers can be found here: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf ------------------------------------------------------------------------------- A new approach about the Infinity concept can be found here: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf ------------------------------------------------------------------------------- A new approach about the Function concept can be found here: http://www.geocities.com/complementarytheory/Function.pdf ------------------------------------------------------------------------------- A new approach about the Logic concept can be found here: http://www.geocities.com/complementarytheory/CompLogic.pdf ------------------------------------------------------------------------------- Link to comment Share on other sites More sharing options...
Guest Doron Shadmi Posted November 20, 2004 Share Posted November 20, 2004 Some examples of my work: ------------------------------------------------------------------------------ A proof that cannot be accomplished by using standard N members: Theorem: 1*5 not= 1+1+1+1+1 Proof: 1*5 = {1,1,1,1,1} not= {{{{1},1},1},1},1} = 1+1+1+1+1 To understand this proof, please read at least page 13 of http://www.geocities.com/complementarytheory/ONN2.pdf ------------------------------------------------------------------------------ A test that shows the advantage of - and + operations in an included-middle logical reasoning framework, can be found in pages 22-29 of http://www.geocities.com/complementarytheory/My-first-axioms.pdf ------------------------------------------------------------------------------- Complementary relations between Multiplication and Addition binary operations can be found in pages 7-8 of http://www.geocities.com/complementarytheory/ONN1.pdf ------------------------------------------------------------------------------- A fundamental new approach about the Natural numbers can be found in: http://www.geocities.com/complementarytheory/ONN1.pdf http://www.geocities.com/complementarytheory/ONN2.pdf http://www.geocities.com/complementarytheory/ONN3.pdf ------------------------------------------------------------------------------- A new approach about 0.9999... = 1 can be found here: http://www.geocities.com/complementarytheory/9999.pdf ------------------------------------------------------------------------------- A new approach about the Limit concept can be found here: http://www.geocities.com/complementarytheory/Anyx.pdf ------------------------------------------------------------------------------- A new approach about Russell's first paradox, can be found here: http://www.geocities.com/complementarytheory/Russell1.pdf ------------------------------------------------------------------------------- A new approach about Cantor's diagonal methods can be found here: http://www.geocities.com/complementarytheory/NewDiagonalView.pdf http://www.geocities.com/complementarytheory/TRANSFINITES.pdf ------------------------------------------------------------------------------- A new approach about Collatz' problem can be found here: http://www.geocities.com/complementarytheory/3n1proof.pdf ------------------------------------------------------------------------------- A new approach about the Real numbers can be found here: http://www.geocities.com/complementarytheory/No-Naive-Math.pdf ------------------------------------------------------------------------------- A new approach about the Infinity concept can be found here: http://www.geocities.com/complementarytheory/RiemannsLimits.pdf ------------------------------------------------------------------------------- A new approach about the Function concept can be found here: http://www.geocities.com/complementarytheory/Function.pdf ------------------------------------------------------------------------------- A new approach about the Logic concept can be found here: http://www.geocities.com/complementarytheory/CompLogic.pdf ------------------------------------------------------------------------------- Link to comment Share on other sites More sharing options...
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