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**1 - 3**of**3**### Incompleteness versus a Platonic multiverse

"... The Platonic multiverse view is that there are multiple and incompatible concepts of a set with corresponding Platonic universes. For example the continuum hypothesis may be true in some of these universes and false in others. This philosophical view leads to an approach for exploring mathematics th ..."

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The Platonic multiverse view is that there are multiple and incompatible concepts of a set with corresponding Platonic universes. For example the continuum hypothesis may be true in some of these universes and false in others. This philosophical view leads to an approach for exploring mathematics that is similar to an approach that stems from the more conservative view that infinity is a potential that can never br fully realized. My version of this view sees infinite collections as human conceptual creations that can have a definite meaning even if they cannot exist physically. The integers and recursively enumerable sets are examples. In this view infinite sets are definite things only if they are logically determined by events that could happen in an always finite but potentially infinite universe with recursive laws of physics. This includes much of generalized recursion theory, but can never include absolutely uncountable sets. “Logically determined ” is a philosophical principal that can be partially defined rigorously, but will always be expandable. In this view uncountable sets can be definite

### An overview of the ordinal calculator

"... An ordinal calculator has been developed as an aid for understanding the countable ordinal hierarchy and as a research tool that may eventually help to expand it. A GPL licensed version is available in C++. It is an interactive command line calculator and can be used as a library. It includes notati ..."

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An ordinal calculator has been developed as an aid for understanding the countable ordinal hierarchy and as a research tool that may eventually help to expand it. A GPL licensed version is available in C++. It is an interactive command line calculator and can be used as a library. It includes notations for the ordinals uniquely expressible in Cantor normal form, the Veblen hierarchies and a form of ordinal projection or collapsing using notations for countable admissible ordinals and their limits. The calculator does addition, multiplication and exponentiation on ordinal notations. For a recursive limit ordinal notation, α, it can list an initial segment of an infinite sequence of notations such that the union of the ordinals represented by the sequence is the ordinal represented by α. It can give the relative size of any two notations and it determines a unique notation for every ordinal represented. Input is in plain text. Output can be plain text and/or L ATEX math mode format. This approach is motivated by a philosophy of mathematical truth that sees objectively true mathematics as connected to properties of recursive processes. It suggests that computers are an essential adjunct to human intuition for extending the combinatorially complex parts of objective mathematics.

### Mathematical incompleteness and objectively guided intrinsic values

"... As science and technology grow at an accelerating rate they create a similar growth in our power to manipulate the physical world. This growth is only possible through the objective guidance of mathematics and experiments. This has led to many great advances. However the use of this power is largely ..."

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As science and technology grow at an accelerating rate they create a similar growth in our power to manipulate the physical world. This growth is only possible through the objective guidance of mathematics and experiments. This has led to many great advances. However the use of this power is largely determined by human values and ultimately what is believed to have intrinsic value. Here no objective guide exists. This puts us in the position of the sorcerer’s apprentice with far more power than we know how to control. The lack of an objective guide to values has helped to create the existential threats of human caused global warming and nuclear weapons. It contributes to many other problems with immense and needless human suffering. This article argues that intrinsic value exists only in conscious experience. It links consciousness with physical structure by making the simplest possible assumptions consistent with what we know. This is the starting point for an objective understanding of intrinsic value. Mathematical incompleteness implies that any consistent mathematical system, can