The most radical variant of the possible (impossible?) world

What does the term " Infinity " mean? There are mathematical, physical and metaphysical definitions of the concept of limitlessness. This study will focus on the scription of the three philosophical foundations of mathematics – formalism, intuitionism and logicism – in set theory. Examples will also be provided of the concept of infinity for these three schools of thought. However, none of them cannot prove whether there is an infinite set or the existence of infinity. It forms the foundational crisis of mathematics. Further elaboration on these schools of philosophy leads to the ideas of actual, potential and absolute boundlessness. These correspond to three basic aforementioned definitions of infinity. Indeed for example, by using Basic Metaphor Infinity, cognitive mechanisms such as conceptual metaphors and aspects, one can appreciate the transfinite cardinals' beauty fully (Nũńez, 2005). This implies the portraiture for endless is anthropomorphic. In other words, because there is a connection between art and mathematics through infinity, one can enjoy the elegance of boundlessness (Maor, 1986). Actually, in essence this is what mathematics is: the science of researching the limitless.

The Southern Journal of Philosophy, 1989

Axiomathes, 2018

This is an article whose intended scope is to deal with the question of infinity in formal mathematics, mainly in the context of the theory of large cardinals as it has developed over time since Cantor’s introduction of the theory of transfinite numbers in the late nineteenth century. A special focus has been given to this theory’s interrelation with the forcing theory, introduced by P. Cohen in his lectures of 1963 and further extended and deepened since then, which leads to a development and further refinement of the theory of large cardinals ultimately touching, especially in view of the discussion in the last section, on the metatheoretical nature of infinity.

2013

Forewords.- Introduction.- A. Conceptual Foundations.- 1. What is a model of ultimate reality?.- 2. Meta-theoretical questions about models of ultimate reality.- B. Specific Models of Ultimate Reality.- Overview of specific models Ted Peters, James E. Taylor.- 1. Classical and neo-classical theism.- 2. Pantheism.- 3. Process theology.- 4. Open theism.- 5. Panentheism.- 6. Deism.- 7. Ground of being theology.- 8. Religious naturalism/naturalistic theism.- 9. Dualism.- 10. Polytheism.- 11. Communotheism.- 12. Via Negativa/apophatic tradition (against all models).- 13. Skeptical or non-theistic views.- C. Diversity of Models of Ultimate Reality.- D. Practical Impacts of Models of Ultimate Reality.- References.- Index.

2011

We show that the theory ZFC-, consisting of the usual axioms of ZFC but with the power set axiom removed-specifically axiomatized by extensionality, foundation, pairing, union, infinity, separation, replacement and the assertion that every set can be well-ordered-is weaker than commonly supposed and is inadequate to establish several basic facts often desired in its context. For example, there are models of ZFC-in which ω 1 is singular, in which every set of reals is countable, yet ω 1 exists, in which there are sets of reals of every size ℵn, but none of size ℵω, and therefore, in which the collection axiom fails; there are models of ZFC-for which the Loś theorem fails, even when the ultrapower is well-founded and the measure exists inside the model; there are models of ZFC-for which the Gaifman theorem fails, in that there is an embedding j : M → N of ZFC-models that is Σ 1 -elementary and cofinal, but not elementary; there are elementary embeddings j : M → N of ZFC-models whose cofinal restriction j : M → j " M is not elementary. Moreover, the collection of formulas that are provably equivalent in ZFC-to a Σ 1 -formula or a Π 1 -formula is not closed under bounded quantification. Nevertheless, these deficits of ZFC-are completely repaired by strengthening it to the theory ZFC − , obtained by using collection rather than replacement in the axiomatization above. These results extend prior work of Zarach [Zar96].

In the elucidation of his Set Theory, Cantor had made tacit assertions about the nature of the de finiteness of the set theoretical universe through the various principles he had introduced to establish the mathematical rigour of the (then) highly controversial subject. However, such assertions appear to be, themselves, controversial, for they seem to propound two mutually incompatible views; while the Domain Principle seems to assert the "completed" nature of the universe, the Generating Principles, instead, appear to advocate its "incompletability". This article examines the reasons behind why one might be inclined towards arguing for either the potentiality or the actuality of the set-theoretical universe. This paper also attempts to question whether these two conflicting views are indeed as mutually incompatible as they appear to be, and proposes a way in which one could hold on to both these views, and yet retain a consistent understanding of the set-theoretical universe.

Cartesian Idea of God as the Infinite

This paper discusses presuppositions of the so-called trademark argument for the existence of God presented by René Descartes

Universality in Set Theories