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3 Mathematical Objects

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In the philosophical framework of this paper there are three types of mathematical `objects':
  1. finite sets,
  2. properties of finite sets and
  3. properties of properties.
Finite sets are abstract idealizations of what can exist physically.
Objective properties of finite sets, like being an even integer, are logically determined human creations. Whether a particular finite set has the property follows logically from the definition of the property. However, the property can involve questions that are logically determined but not determinable. One example is the set of Gödel numbers of Turing Machines (TM)1 that do not halt. What the TM does at each time step is logically determined and thus so is the question of whether it will halt, but, if it does not halt, there is no general way to determine this. The non-halting property is objectively determined but not in general determinable.

3.1  Properties of properties

The property of being a subset of the integers has led to the idea that uncountable collecitons exist. No finite or countable formal system can capture what mathematicians want to mean by the set of all subsets of the integers. One can interpret this as the human mind transcending the finite or that mathematics is a human creation that can always be expanded. The inherent incompleteness of any sufficiently strong formal system similarly suggests that mathematics is creative.

3.2  Gödel and mathematical creativity

Gödel proved that any formalization of mathematics strong enough to embed the primitive recursive functions (or alternatively a Universal Turing Machine) must be either incomplete or inconsistent[1]. In particular such a system, will be able to model itself, and will not be able to decide if this model of itself is consistent unless it is inconsistent.
This is often seen as putting a limit on mathematical knowledge. It limits what we can be certain about in mathematics, but not what we can explore. A divergent creative process can, in theory, pursue every possible finite formalization of mathematics as long as it does not have to choose which approach or approaches are correct. Of course it can rule out those that are discovered to be inconsistent or to have other provable flaws. This may seem to be only of theoretical interest. However the mathematically capable human mind is the product of just such a divergent creative process known as biological evolution.

3.3  Cantor's incompleteness proof

One can think of Cantor's proof (when combined with the Löwenheim-Skolem theorem) as the first great incompleteness proof. He proved the properties defining real numbers can always be expanded. The standard claim is that Cantor proved that there are "more" reals than integers. That claim depends on each real existing as a completed infinite totality. From an objective standpoint, Cantor's proof shows that any formal system that meets certain prerequisites can be expanded by diagonalizing the real numbers provably definable within the system. Of course this can only be done from outside the system.
Just as one can always define more real numbers one can always create more objective mathematics. One wants to include as objective all statements logically determined by a recursively enumerable sequence of events, but that can only be precisely defined relative to a particular formal system and will always be incomplete and expandable just as the reals are.
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Solmon Feferman, John W. Dawson Jr., Stephen C. Kleene, Gregory H. Moore, Robert M. Solovay, and Jean van Heijenoort, editors. Publications 1929-1936, volume 1 of Kurt Gödel Collected Works. Oxford University Press, New York, 1986.


1There is overwhelming evidence that one can create a Universal Turing Machine that can simulate every possible computer. Assuming this is true, one can assign a unique integer to every possible program for this universal computer. This is called a Gödel number because Gödel invented this idea in a different, but related, context a few years before the concept of a Universal Turing Machine was proposed.

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