Philosophical Monologues: Non-well-founded infinity

Wednesday, March 13, 2024

Non-well-founded infinity

Suppose we wish to define an enigmatic property $\Pi(x)$ of a set $x$. We have the condition

\[\Pi(x) \leftrightarrow \Pi_1(x) \vee \exists y \in x. \Pi(y) \]\[ \neg \exists y. y \in x \, \rightarrow \, \Pi(x)\] That is, if $a$ is an atom then $\Pi(a)$. Our question: is it true that $ZFC \vdash \forall x. \Pi(x)$ ? In what sense in ZFC is the whole always 'more' in some special sense than the part ?

1 comment:

Philosophical InquirerMarch 25, 2024 at 11:29 AM
This post is an attempt to formalize some aspects of the first proposition of Proclus' Elements of Theology.
ReplyDelete
Replies

Add comment

Logic, therefore, is not a tool; it is an open-ended, evolving science having an ontology of its own. Mathematics is but one of its parts, and its full scope is yet to be discovered. Given the definition of validity, every necessary thought whose analysis contains only purely logical objects is valid. So every time we discover a purely logical analysis of a concept previously thought to belong to a discipline outside logic, the acknowledged scope of logic must be expanded accordingly. The purely logical analysis of number is just one case in point, and I submit that there are many more (...) If there is anything in the analyses I will offer, then the conception of logic that emerges is very far indeed from that of Aristotle, for whom logic is primarily a tool; instead, it is more like that of Plato, for whom logic is akin to reason itself. George Bealer

Wednesday, March 13, 2024

Non-well-founded infinity

1 comment:

Prolegomena to a future logico-mathematical metaphysics