Papers by Nicholas K Swenson
This paper directly refutes the motivating points of §8: Application of the diagonal process from... more This paper directly refutes the motivating points of §8: Application of the diagonal process from Alan Turing’s paper On Computable Numbers. After briefly touching upon the uncontested fact that computational machines are necessarily fully enumerable, we will discuss an alternative to Turing’s algorithm for computing direct diagonal across the computable numbers. This alternative not only avoids an infinite recursion, but also any sort of decision paradox. Then, by using techniques described in §3 of how to resolve a halting paradox to correct the interface of decision machine 𝓓, we will mitigate the decision paradox that occurs in Turing’s attempt at computing a direct diagonal, and show that it can actually compute a direct diagonal. Finally, we will analogously fix the decision paradox found in trying to compute an inverse diagonal, but in this case we will demonstrate that the resulting computation is not sufficient to produce a complete inverse diagonal. Opposed to Turing’s several objections, there is no way to utilize a paradox-resistant correction of 𝓓 to compute an inconsistency that would disprove its existence. This undermines the foundation which Turing builds his uncomputability arguments on, and leaves us with an open question on the true nature of computability.
In 1936 Alan Turing published the groundwork math paradigms we still use today as our foundations... more In 1936 Alan Turing published the groundwork math paradigms we still use today as our foundations for computing. He spent the first half of this paper describing the model we now call Turing machines, but the second half was dedicated to proofs attempting to establish inherent incompleteness in computing as a theory: including the halting problem. Since then the halting problem has stood as a relatively unquestioned fundamental limit to computing. The paradoxes encountered when hypothetically applying halting deciders in self-referential analysis are interpreted to be some kind of ultimate algorithmic limit to reality. This paper proposes alternatives to the accepted consensus on the matter, and attempts to demonstrate two methods in which we might circumvent those paradoxes through refining the interfaces we use in halting computation, in order to make the programmatic forms of those paradoxes decidable.
Both methods hinge on utilizing multiple decision machines, or deciders, in distinct ways, in order to mitigate attempts at creating self-defeating logic. This paper is focused on just resolving the paradoxes involved in halting analysis under self-reference, and to be clear: it is not then presenting a general halting algorithm. This paper does not attempt to present at depth arguments or reasons for why we should accept either of these proposals vs a more conventional perspective, it is mostly an objective description of the conceptions for further musing upon. Lastly, we will stick to solely the basic halting paradoxes found within computing. We will not try to address or apply these techniques to other problems of logical undecidability, either within computing, or greater math such as Gödel’s Incompleteness.
In Alan Turing's seminal paper 1 , he attempted to prove through several means that computable nu... more In Alan Turing's seminal paper 1 , he attempted to prove through several means that computable numbers were not enumerable, and invented a formal definition for modern computing in the process. A computable number is a real number that can be computed to the nth-digit, by a finite-state, computational machine that takes input n. Unfortunately, even though he spent half the paper proving that said machines can be fully described by a natural number (Turing calls these D.N. for designated numbers), he missed the full gravity of this connection that debunks much of his paper's followup discussion. Here is the resulting counterproof: Computable numbers are bijectable on a set of finite-state machines which uniquely compute them, which in turn are bijectable on a set of natural numbers which uniquely and fully describe them, so therefore computable numbers must be enumerable, as all sets of natural numbers are enumerable Q.E.D.
Drafts by Nicholas K Swenson
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Papers by Nicholas K Swenson
Both methods hinge on utilizing multiple decision machines, or deciders, in distinct ways, in order to mitigate attempts at creating self-defeating logic. This paper is focused on just resolving the paradoxes involved in halting analysis under self-reference, and to be clear: it is not then presenting a general halting algorithm. This paper does not attempt to present at depth arguments or reasons for why we should accept either of these proposals vs a more conventional perspective, it is mostly an objective description of the conceptions for further musing upon. Lastly, we will stick to solely the basic halting paradoxes found within computing. We will not try to address or apply these techniques to other problems of logical undecidability, either within computing, or greater math such as Gödel’s Incompleteness.
Drafts by Nicholas K Swenson