more on this theme | more from this thinker
Full Idea
The argument that the relation of dependence is well-founded ...is a version of the classical arguments for substance. ..Any conceptual scheme which genuinely represents a world cannot contain infinite backward chains of meaning.
Gist of Idea
If dependence is well-founded, with no infinite backward chains, this implies substances
Source
Michael Potter (Set Theory and Its Philosophy [2004], 03.3)
Book Ref
Potter,Michael: 'Set Theory and Its Philosophy' [OUP 2004], p.40
A Reaction
Thus the iterative conception of set may imply a notion of substance, and Barwise's radical attempt to ditch the Axiom of Foundation (Idea 13039) was a radical attempt to get rid of 'substances'. Potter cites Wittgenstein as a fan of substances here.
Related Idea
Idea 13039 Foundation:∀x(∃y(y∈x) → ∃y(y∈x ∧ ¬∃z(z∈x ∧ z∈y))) [Kunen]
| 10702 | Set theory's three roles: taming the infinite, subject-matter of mathematics, and modes of reasoning [Potter] |
| 10703 | Supposing axioms (rather than accepting them) give truths, but they are conditional [Potter] |
| 10704 | We can formalize second-order formation rules, but not inference rules [Potter] |
| 13041 | Collections have fixed members, but fusions can be carved in innumerable ways [Potter] |
| 10707 | Mereology elides the distinction between the cards in a pack and the suits [Potter] |
| 10708 | Nowadays we derive our conception of collections from the dependence between them [Potter] |
| 13042 | If dependence is well-founded, with no infinite backward chains, this implies substances [Potter] |
| 10709 | Priority is a modality, arising from collections and members [Potter] |
| 10712 | If set theory didn't found mathematics, it is still needed to count infinite sets [Potter] |
| 10713 | Usually the only reason given for accepting the empty set is convenience [Potter] |
| 13043 | A relation is a set consisting entirely of ordered pairs [Potter] |
| 13044 | Infinity: There is at least one limit level [Potter] |
| 17882 | It is remarkable that all natural number arithmetic derives from just the Peano Axioms [Potter] |
| 13546 | The 'limitation of size' principles say whether properties collectivise depends on the number of objects [Potter] |