Unity+ Posted March 2, 2014 Author Share Posted March 2, 2014 Adding on to the earlier post: Because of the way theorems work, being branched off the Fundamental Theorem, or set of axioms, the equation must be a Dynamic Collatz-Matrix equation. This means that using a particular logic constant the parameters change per the change in distance of the initial point of the matrix solution. The logic constant is not a numerical value, but a set of logical statements that influence the change in the parameters per movement in the matrix solution. For example, the initial parameter could be that set A is equal to set B. Then, after such a statement is applied to the initial statement in the matrix solution, the statement could become set B's elements are equal to set C's elements. The following states that after two movements of the initial element in a matrix solution, the parameter of addition would become an operation of multiplication. [math]\Delta \Lambda _{a}\Rightarrow O_{a}\star O_{b},\left \{ \star : +\to\times \right \}[/math] Link to comment Share on other sites More sharing options...

Unity+ Posted March 4, 2014 Author Share Posted March 4, 2014 (edited) Here is a paradox I found while working on Dynamic Collatz-Matrix equations. If someone solves the paradox I will give reputation points.The Contradicting-Axiom ParadoxHere is the mathematical statement:[math]\delta_{F}\rightarrow\left\{A_a\star A_{b}\right \}\left( A_{a}: A_{b}\star_{!}A_{N}\right)[/math]This paradox is a statement about the nature of axioms. The paradox is that there could be a Fundamental Theorem, consisting of a set of axioms, that has the main axiom state "All other axioms and/or theorems must contradict each other." The paradox here is if all axioms must contradict each other, even the main axiom, then these axioms have complied with such an axiom and therefore do not contradict all axioms, even if the axioms or theorems afterword contradict each other. Therefore, either the axioms and theorems comply with the main axiom or do not comply with it. Edited March 4, 2014 by Unity+ Link to comment Share on other sites More sharing options...

John Posted March 4, 2014 Share Posted March 4, 2014 (edited) I'm not sure I understand your notation, but if I'm understanding you correctly, then you're considering a formal system which has a contradiction as a theorem. Such a system is inconsistent.It's a fact of logic that an inconsistent formal system (at least in the usual sense) can be used to prove any statement. Therefore it's trivially complete but its structure is fairly uninteresting.From the last few lines of your description, it sounds like you're thinking about something similar to Russell's paradox in naïve set theory (allowing the "set of all sets that aren't members of themselves") and the formal statement Gödel constructed as part of his proof of the incompleteness theorems. Edited March 4, 2014 by John 1 Link to comment Share on other sites More sharing options...

Unity+ Posted March 5, 2014 Author Share Posted March 5, 2014 (edited) I'm not sure I understand your notation, but if I'm understanding you correctly, then you're considering a formal system which has a contradiction as a theorem. Such a system is inconsistent. It's a fact of logic that an inconsistent formal system (at least in the usual sense) can be used to prove any statement. Therefore it's trivially complete but its structure is fairly uninteresting. From the last few lines of your description, it sounds like you're thinking about something similar to Russell's paradox in naïve set theory (allowing the "set of all sets that aren't members of themselves") and the formal statement Gödel constructed as part of his proof of the incompleteness theorems. The theorem isn't an actual theorem I made, persay, but it is something that I looked into when developing an example Fundamental Theorem that contains such an axiom to be used for Dynamic Collatz-Matrix equations that deal with logic functions I described in the earlier post. The idea is being able to take an axiom of a set of axioms to be able to develop an a branch of other theorems or axioms. The notation is simply saying that there is the Fundamental Theorem which contains axiom a which is now having to comply with axiom b, which axiom b must not comply with axiom N. It is also implying that it must not comply with axiom a. And actually, I should have done this earlier, but the notation should have been this to show that these axioms must not comply with axiom a. [math]\delta_{F}\rightarrow\left\{A_a\star A_{b}\right \}\left( A_{a}: A_{b}\star_{!}A_{N}\right)[/math] [math]A_{N}\ni A_{a}[/math] NOTE: Because of this, axiom a, which is an element of axiom N, or the axioms of the system, axiom a must also contradict axiom a. This doesn't make sense because the rule of the axiom is all axioms contradict each other. Therefore, axiom a contradicting itself because axiom is compliant with axiom a since they are both the same and have the same rules. I see now where you get the relation with Russell's paradox. To avoid an inconsistent contradiction with the statements, [math]A_{b}=\varnothing[/math] EDIT: Realized this was correct. Wouldn't it be that if the axioms have to all contradict that no new theorem could be introduced because even if you were trying to find one that contradicted axiom a, the axiom of contradiction, then it would still contradict and therefore the theorem could not be proved true? I can see, though a theorem that could develop that would say "axiom a contradicts axiom N, an axiom that represents axioms that contradict axiom a because they contradict each other for they all comply with axiom a." The theorem wouldn't hold because then it doesn't contradict axiom N, the axioms that exist within the Fundamental Theorem of such a system. Therefore, not theorem could be proven with such a system. The whole idea isn't meant to be formal. The idea is developing systems using axioms that are not perceived in the natural sense of Mathematics. Dynamic Collatz-Matrix equations use a completely new set of Fundamental axioms or a new Fundamental Theorem consisting of a set of axioms. The idea would be that such an inconsistent system could exist within the spectrum of axioms that produce systems. It goes off the concept of taking forms of reasoning to look at new ways of thinking of systems of mathematics. Instead of the formal axioms that exist to produce more theorems, this takes a whole new set of axioms that are forms of reason for a completely different system, which produce a completely new system. However, this means now systems now must be determined to be consistent with each other now. Here is a question: If you have two completely different systems that have a different set of axioms, is it possible that each could produce similar theorems? I would think not because if the logic of both systems are not similar then they would produce completely different results. EDIT: I took a further look at the statements presented in the Principle of Explosion, and here is the statement I found: "If one claims something is both true () and not true (), one can logically derive any conclusion ()." The paradox doesn't, or at least I don't see it does, address statements that are both true and false. It is merely stating that axiom A states that all what comes after must contradict itself, meaning nothing most comply with its logic. Therefore, if an axiom were to comply with its rules of logic then it already has defined itself as complying with its logic therefore it hasn't complied with its logic, making it not liable as an axiom. Edited March 5, 2014 by Unity+ Link to comment Share on other sites More sharing options...

John Posted March 5, 2014 Share Posted March 5, 2014 (edited) If your system is inconsistent, then it isn't a good foundation on which to build a mathematical theory. There are some attempts to develop logics that tolerate inconsistencies, but they involve abandoning certain intuitive properties of logical reasoning (see the Wiki page for details).It is not correct to say an inconsistent formal system doesn't admit theorems. On the contrary, as noted above, an inconsistent formal system has all statements as theorems. Keep in mind I am speaking formally here. If we take a system containing both p and ¬p as axioms, then the statement p ∧ ¬p is a theorem of the system, despite the fact that such a statement doesn't really "make sense." The fact that p ∧ ¬p is a theorem of the system then allows us to deduce any statement (this is explained in some detail on the Wiki page linked above). However, you say you aren't being formal. I'm not sure precisely what you mean by that, but in the usual sense, a lack of formal rigor may hinder the development and acceptance of any mathematical theory. I'm also not sure what you mean when you talk about the "natural sense of Mathematics" or axioms not perceived in that way.I'm not sure what the requirements are for two systems to be "completely different." The axioms of Peano arithmetic (the first-order theory of the natural numbers with addition and multiplication) are mostly different from the axioms of Zermelo-Fraenkel set theory (though the former's axiom schema of induction is similar to the latter's axiom of infinity, and maybe some other similarities can be found), but there are theorems deducible from both. In fact, I think the axioms of PA are deducible from ZF. Edited March 5, 2014 by John Link to comment Share on other sites More sharing options...

Unity+ Posted March 5, 2014 Author Share Posted March 5, 2014 If your system is inconsistent, then it isn't a good foundation on which to build a mathematical theory. There are some attempts to develop logics that tolerate inconsistencies, but they involve abandoning certain intuitive properties of logical reasoning (see the Wiki page for details). It is not correct to say an inconsistent formal system doesn't admit theorems. On the contrary, as noted above, an inconsistent formal system has all statements as theorems. Keep in mind I am speaking formally here. If we take a system containing both p and ¬p as axioms, then the statement p ∧ ¬p is a theorem of the system, despite the fact that such a statement doesn't really "make sense." The fact that p ∧ ¬p is a theorem of the system then allows us to deduce any statement (this is explained in some detail on the Wiki page linked above). However, you say you aren't being formal. I'm not sure precisely what you mean by that, but in the usual sense, a lack of formal rigor may hinder the development and acceptance of any mathematical theory. I'm also not sure what you mean when you talk about the "natural sense of Mathematics" or axioms not perceived in that way. I'm not sure what the requirements are for two systems to be "completely different." The axioms of Peano arithmetic (the first-order theory of the natural numbers with addition and multiplication) are mostly different from the axioms of Zermelo-Fraenkel set theory (though the former's axiom schema of induction is similar to the latter's axiom of infinity, and maybe some other similarities can be found), but there are theorems deducible from both. In fact, I think the axioms of PA are deducible from ZF. Just to remind you, the system I am talking about is different than Dynamic Collatz-Matrix equations that is logic functions. It is simply something that I found that I thought would be interesting. Also, the system doesn't contain any axioms besides the first axiom of contradiction. The problem is encountered when you do add more axioms. So, the only axiom that would exist within the system would be that particular axiom. What I was saying when referring to formality is that the axioms that are used in Dynamic Collatz-Matrix equations are not the common system of axioms used in mathematics. They are simply systems from the logic equations. That is what I mean, if you understand what I am saying. The problem with that example, as you stated, is there are similar axioms within both systems, which means they, of course, will have similar axioms that arrive from both systems. I apologize if I am being confusing. Link to comment Share on other sites More sharing options...

Unity+ Posted April 18, 2014 Author Share Posted April 18, 2014 (edited) Well, I developed a Collatz-Matrix equation that works with cellular automation. The equation looks like the following. Here is how it is done. First, find all the matrix solutions for the Collatz-Matrix equation. Then, for each matrix equation represent each element as a 2x2 square that has four elements in total. Using that, convert the number into binary. Then, take that binary sequence and represent each square as a binary node and spiral the sequence in the square, from left to right. If there are more nodes than spaces in the 2x2 square then wrap the binary sequence around the square(making a spiral). Then, connect the squares together as if they are 2x2 squares. If there are nodes that overlap, the rule to use is if a 0 and 0 node overlap, make it a 0 node. If the node is a 1 and 1 node then make it a 0 node. Of course, these rules are defined by the mew in the equation. You can define the rules to change it up. After all that, combine all the resulting matrix solutions and treat them as steps like on Conway's game of life. Put them in the order from which they were found(going downward in a matrix solution to going leftward in a matrix solution). Here is the result of the equation above. Where s(1,1) I call this the "pulsating fish" because of its pulse-like nature and how it pulses. EDIT: Adding onto this, this type of equation came from the idea that equations are the result of cellular automations while the behavior of the cellular automations are a result of the equations produced by the cellular automations. EDIT2:Here are more examples: s(2,2) s(1,2) Edited April 19, 2014 by Unity+ Link to comment Share on other sites More sharing options...

Unity+ Posted April 26, 2014 Author Share Posted April 26, 2014 (edited) A new rule that is being added is the ratio [math]\frac{d_{e}}{d_{i}}[/math] which is it cannot be an irrational number. The following is the reason why. Let us take some example, such as [math]\pi[/math]. Since it cannot be represented as a fraction, a summation can be used to use it within a Collatz-Matrix equation. [math]\frac{d_{e}}{d_{i}}=\sum_{k=1}^{n}\frac{4(-1)^{k+1}}{2k-1}=\frac{\sum_{k=1}^{n}4(-1)^{k+1}}{\sum_{k=1}^{n}2k-1}[/math] Since an element of a matrix solution must be a real while number, the ratio must be represented as above. The bottom part of the ratio would turn out to be infinity, therefore making every element infinity. Therefore, irrational numbers cannot be used for the ratio. Edited April 26, 2014 by Unity+ Link to comment Share on other sites More sharing options...

Unity+ Posted April 27, 2014 Author Share Posted April 27, 2014 (edited) Here is a way to deal with irrational ratios: The way to deal with irrational ratios would be to represent the ratio as a function of f(s). For example, of the ratio is pi then the following would occur. [math]C(x)_{k\times d}\begin{Bmatrix} a_{f} &b_{f} \\ u_{f} & v_{f} \end{Bmatrix}\left [ f(s)\rightarrow \frac{\sum_{k=1}^{s}4(-1)^{k+1}}{\sum_{k=1}^{s}2k-1} \right ],s(k_{p},d_{p})[/math] This means that after each step in a matrix solution the next step would occur in the sum, which is assumed in the functions in the Collatz-Matrix equation. EDIT: Here is an example for a Collatz-Matrix equation. [math]C(x)_{k\times d}\begin{Bmatrix} \frac{x}{2} &\frac{x-\sum_{k=1}^{n}4(-1)^{k+1}}{\sum_{k=1}^{n}2k-1} \\ x\sum_{k=1}^{n}2k-1+\sum_{k=1}^{n}4(-1)^{k+1} &2x \end{Bmatrix}\left [ f(s)\rightarrow \frac{\sum_{k=1}^{n}4(-1)^{k+1}}{\sum_{k=1}^{n}2k-1} \right ],s(k_{p},d_{p})[/math] Where n represents each step in the matrix solution(s). There is also a way to find the equation for the numbers specific to these set of functions. Edited April 27, 2014 by Unity+ Link to comment Share on other sites More sharing options...

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