[FOM] iterative conception/cumulative hierarchy
d_obrien
d_obrien at telus.net
Wed Feb 29 21:04:43 EST 2012
> Michael Kremer wrote:
>
> > the van Aken idea of presuppositions is intuitive and explains why
> > foundation should hold ... the point is it explains the basic idea of
> > a cumulative hierarchy without any metaphor of sets being "formed"
>
> and Chris Menzel wrote:
>
> > the metaphor of set formation is cashed in terms of the idea of a set
> > *dependending on*, or *presupposing*, its members, a relation that is
> > reflected in the (static) structure of the hierarchy
>
> referring to Richard Heck's comment about "the idea that sets are
> *metaphysically dependent* upon their members".
>
A notion that a mathematical object can be "metaphysically dependent" on
something else is totally obscure, in the same way as "a green sound" is
totally obscure outside of some poetic setting. What discussion of
"metaphysics" is being referenced here?
A notion that a mathematical object can depend essentially on something
else, in the sense of it being by definition dependent, is clearer -- at
least in the sense that one might be expected to provide the relevant
definitions, with their explanations, in a more concrete way than one can
ever be expected to provide a "metaphysics".
> So there are two distinct questions here. The first is how we are to
repair our
> naive ideas about sets (viz., sets are extensions of concepts and every
> concept has an extension) in the face of the paradoxes. The second is to
> what extent our modified understanding will support the ZFC axioms.
>
No part of mathematics depends on anything described as "our naïve ideas
about x", whatever we substitute for 'x'.
A so called "naïve" idea that "sets are extensions of concepts and every
concept has an extension" is entirely bogus, and in truth I don't think it
would even occur to someone unless first implanted.
For example, sets are sometimes described as being single things,
totalities, (when we enumerate sets, we count each as one thing), whereas
extensions are only described as being single in the case of the extension
of an individual term or description of one (as S. of a sentence). No
presupposition that every plurality can be unified in a totality is required
to understand this.
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