View source for The Correspondence Between the Axioms of Quantum Mechanics and Imaginary and Transfinite Number Forms

{{Infobox paper
| title = The Correspondence Between the Axioms of Quantum Mechanics and Imaginary and Transfinite Number Forms
| author = [[William M Honig]]
| keywords = [[algebraic logic]], [[transfinite ordinals]], [[forcing]], [[non-Boolean fields]], [[undecidability]], [[imaginary numbers and exponentials]]
| published = 1988
| journal = [[Physics Essays]]
| volume = [[1]]
| number = [[4]]
}}

==Abstract==

<em>A<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>presentation is made showing how imaginary numbers, exponentials, and transfinite<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>ordinals can be given logical meanings that are applicable to<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>the definitions for the axioms of quantum mechanics (QM). This<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>is based on a proposed logical definition for axioms that<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>includes an axiom statement and its negation as parts of<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>an undecidable statement that is forced to the tautological truth<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>value ?true.? The logical algebraic expression for this is shown<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>to be isomorphic to the algebraic expression defining the imaginary<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>numbers</em> ?i(<img border="0" alt="sqrt( - 1)" align="middle" src="http://physicsessays.aip.org/servlet/GetImg?key=PHESEM000001000004000247000001%3A0%3A0%3A28&amp;t=a&amp;d=a" />). ''This supports a progressive and Hegelian view of<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>theory development This means that thesis and antithesis axioms in<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>the QM theory structure, which should be carried along at<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>present, could later on be replaced by a synthesis to<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>a deeper theory prompted by subsequently discovered new experimental facts<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>and concepts. This process could repeat at a later time,<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>since the synthesis theory axioms would then be considered as<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>a new set of thesis statements from which their paired<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>antithesis axiom statements would be derived. The present epistemological methods<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>of QM, therefore, are considered to be a good way<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>of temporarily leapfrogging defects in our conceptual and experimental knowledge<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>until a deeper determinate theory is found. These considerations bring<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>logical meaning to exponential forms such as the psi and<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>wave functions. This is derived from the set theoretic meaning<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>for simple forms such as 2''''<sup><span style="FONT-SIZE: x-small">A</span></sup>, which is known to<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>be the set of all subsets of the (discrete) set''<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>A. ''The equal symbol in equations that are axioms, and<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>all its other symbols, can be mapped to a transfinite<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>ordinal Imaginary exponential forms (such as'' e<sup><span style="FONT-SIZE: x-small">i<img border="0" alt="theta" align="bottom" src="http://physicsessays.aip.org/stockgif3/thgr-script.gif" /></span></sup>'') can be shown<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>to stand for the (continuous) set of all subsets or<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>the set of all experimental situations (which thus includes arbitrary<sup><span style="FONT-SIZE: x-small">&nbsp;</span></sup>sets of experimental situations) which are based on the axiom<span style="FONT-SIZE: x-small"><sup>&nbsp;</sup><img border="0" alt="theta" align="bottom" src="http://physicsessays.aip.org/stockgif3/thgr.gif" /></span>, a transfinite ordinal''.

[[Category:Scientific Paper|correspondence axioms quantum mechanics imaginary transfinite number forms]]