Mathematicians have devised a set of axioms from which the ordinary principles of arithmetic can be derived. They regard them as though they were the source of all arithmetical truths. Although they may not admit it they treat them as though they were built into the structure of the universe, even to the extent that they were devised by God at the time of the supposed creation. In my blog "The Uncreated Universe" the concepts of creation and God are dismissed as meaningless. If that is the case then the principles of arithmetic and other mathematical principles were not built into the the universe.
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Arithmetic is a system that we have devised for describing certain aspects of the universe. The elements of the system are based on what we find empirically, that is, what we find from the observation of things. Thus the "six" face of a dice demonstrates clearly that three twos are the same as two threes. The axioms of arithmetic are not the ultimate source of arithmetical principles and are accepted because they are consistent with what is found empirically. Thus they are themselves empirical.
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The same is true of the principles of logic. In fact logic is a branch of applied mathematics. It can be expressed in terms of modular arithmetics, the arithmetics of a limited number of values, even though these may not provide the most convenient forms of expression. The two-value logic, that of truth and falsehood, can be expressed in terms of the two-value arithmetic. Thus the truth-value of the compound sentence "P or Q" where P and Q are component sentences can be expressed as
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p + q - pq
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where p and q represent the truth-values of P and Q, taking the values 1 when these are true and 0 otherwise. The expression takes the value 1, expressing truth, if either of or both P and Q are true and 0, for falsehood, otherwise.
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Likewise the logic of the three truth-values truth, falsehood and meaninglessness can be expressed in terms of the three-value arithmetic. In fact the three value logic can be expressed in terms of the two-value arithmetic if account is taken of the fact that it is true that one of the three values must apply to any sentence and false otherwise, and likewise that only one of them can apply. Other logics can be compressed into the two-value arithmetic when due account is taken of their distinguishing features.
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Mathematicians draw conclusions from the axioms of arithmetic and logic that extend beyond the realm of counting. They behave like theologians who attempt to determine the full implications of God's supposed works. Thus they study the mathematics of the various forms of infinity as though infinity were a counting number, which it is not. Being outside the realm of counting it is probably meaningless. ("Infinite" meaning "without limit" is not meaningless. Thus the ordered sequence of whole numbers is infinite.) Extending arithmetic beyond the useful realm of counting leads to inconsistencies which provide mathematicians with problems. And if mathematics had been devised by God these would be very big problems.
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Based on the axioms of arithmetic and logic Kurt Godel devised a sentence G written in an arithmetical code. The code is unique in the sense that the original axioms can be inferred unequivocally from the sentence. In effect G states "G cannot be proved true." The conclusion that this represented an inconsistency was rejected on the basis that anything derived from the axioms of arithmetic cannot be inconsistent, and so it was assumed that arithmetic is incomplete in that it cannot prove all of its own truths.
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The sentence G is its own subject. It is therefore what is known as a self-referential sentence and in consequence it is meaningless. Therefore Godel's conclusion is wrong.
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The predicates of ordinary simple sentences must distinguish between the subjects of the sentences to which they apply and, by negation, those to which they do not apply. If a sentence is false then negation provides a true sentence, and likewise for the removal of negation from a negated sentence. These changes do not change the subjects of sentences but they reverse the descriptions of them.
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Consider the self-referential sentence "This sentence contains the last letter of the alphabet." It is seemingly false, so let us negate it to make it true: "This sentence does not contain the last letter of the alphabet." "This sentence" does not now represent the original sentence but instead the negated sentence. Therefore attempted negation has replaced the original subject whereas true negation would have left it unchanged. Therefore self-referential sentences cannot be negated and are meaningless.
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That the attempted negation of a self-referential sentence changes the subject represents a form of inconsistency having regard the the requirement that negation must not do this. Godel has therefore shown that the axioms of arithmetic and logic can yield an inconsistency when systematically applied.
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The sentence G may be compared with a sentence S: "This sentence cannot be proved true." Each is self-referential and each attempts to state that it cannot be proved true. G is based on accepted arithmetical and logical principles that are used for counting and reasoning while S is based on accepted semantic and grammatical principles that are used in verbal communication. Just as G is supposed by many mathematicians to have meaning because of its firm basis, so S might likewise be supposed to have meaning. But by being self-referential neither has meaning.
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Consider "S is meaningless" a premise. As S is meaningless it cannot be proved true. That, apparently, is exactly what it states so that it is seemingly true and not meaningless. Thus the premise leads to the contradictory conclusion that S is not meaningless. If this were a valid reductio ad absurdum then it would have been demonstrated that S is not meaningless, so that the premise would be false, contrary to what is known about self-referential sentences in general. The fallacy that leads to this conclusion lies in the step that assumes thar S makes a statement despite being meaningless. The position of G is similar.
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Classes are very convenient concepts which we have devised to enable us to speak collectively of similar things. Thus in respect of the class of mammals we can say "All mammals have backbones" so that we do not have to make such a statement in respect of each member of the class, that is, of each species of mammal. But Lord Russell discovered the inconsistent class of all classes that are not members of themselves. If it is a member of itself then it is not a member of itself, and vice versa. This shows how the concept class breaks down when its is taken beyond the limits of its usefulness. It is not a perfect feature of the universe so it is not God-given element of the universe.
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Various axioms have been devised to obviate the inconsistency but they appear to be no more than instructions for avoiding it.
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Things that are associated in one way or another are said to be related. But the concept of relationship breaks down when taken outside its useful range. "A and B are not related" defines a relationship between A and B, that of not being related.
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It may seem that measurement takes part in natural physical behaviour, so that it is not a man-made device. Thus the gravitational force between two objects is effectively inversely proportional to the square of the distance between them, the distance being a measurement.
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If measurement is man-made then natural behaviour may be due to the cumulative effect of very large and varying numbers of fundamental units. Measurement would then be a simplified count of the very large numbers of the very small fundamental units.
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The fundamental unit of time for example could be that which is correlated in quantum terms with the total energy of the universe.
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